A symmetric Bloch–Okounkov theorem

Abstract

The algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra \(\mathcal {T}\) consisting of symmetric polynomials in the part sizes and multiplicities.

Introduction

Partitions of integers are related in interesting ways to modular forms, starting with the observation that the generating series of partitions is closely related to the Dedekind \(\eta \)-function, i.e.,

$$\begin{aligned} \sum _{\lambda \in \mathscr {P}} q^{|\lambda |}\, =\, \prod _{n>0} (1-q^n)^{-1}\, =\, q^{1/24}\eta (\tau )^{-1} \quad \quad (q=e^{2\pi i\tau }), \end{aligned}$$

where \(\mathscr {P}\) denotes set of all partitions and \(|\lambda |\) denotes the integer \(\lambda \) is a partition of. Another example is the occurrence of modular forms in the proof of the partition congruences which go back to Ramanujan [1].

More recently, partitions were connected to (quasi)modular forms via the q-bracket. Given a function \(f:\mathscr {P}\rightarrow \mathbb {Q}\), the q-bracket of f is defined as the following power series

$$\begin{aligned} \langle f \rangle _q \, =\, \frac{\sum _{\lambda \in \mathscr {P}}f(\lambda ) q^{|\lambda |}}{\sum _{\lambda \in \mathscr {P}} q^{|\lambda |}} \in \mathbb {Q}[[q]]. \end{aligned}$$
(1)

Before continuing, note that it is not surprising at all that for a well-chosen function f the q-bracket \(\langle f \rangle _q\) is a quasimodular form, since it is easily seen that the map (1) from \(\mathbb {Q}^\mathscr {P}\) to \(\mathbb {Q}[[q]]\) is surjective. What is surprising is that one can find graded subalgebras A of \(\mathbb {Q}^\mathscr {P}\) which (i) are “interesting” in the sense that they have an interpretation in combinatorics, enumerative geometry or another field of mathematics and (ii) have the property that the q-bracket of a homogeneous function \(f\in A\) is quasimodular of the same weight as f. In this case we call A a quasimodular algebra. Note that the q-bracket is linear but not multiplicative, so in order to show that an algebra is quasimodular, it is not sufficient to show that the q-brackets of the generators of such an algebra are quasimodular. The aim of this paper is to introduce new quasimodular algebras.

The Bloch–Okounkov theorem [3, Theorem 0.5] provided the first quasimodular algebra \(\Lambda ^*\). Write a partition \(\lambda \) as a non-increasing sequence \((\lambda _1,\lambda _2,\ldots )\) of non-negative integers with \(|\lambda |=\sum _{i=1}^\infty \lambda _i\) finite. The \(\mathbb {Q}\)-algebra \(\Lambda ^*\) is freely generated by the so-called shifted symmetric power sums

$$\begin{aligned} Q_k(\lambda ) \, =\, c_k \,+\, \sum _{i=1}^\infty \bigl ((\lambda _i-i+\tfrac{1}{2})^{k-1}-(-i+\tfrac{1}{2})^{k-1}\bigr ) \quad \quad (k\ge 2), \end{aligned}$$
(2)

where the \(c_k\) are constants given by \(\frac{1}{x}+\sum _{k} c_k \frac{x^{k-1}}{(k-1)!} = \frac{1}{2\sinh (x/2)}\) . The function \(Q_3\) naturally occurs in the simplest case of the Gromov–Witten theory of an elliptic curve, as discovered by Dijkgraaf [7] and for which quasimodularity was proven rigorously in [14]. Quasimodularity of \(\Lambda ^*\) is used in many recent works in enumerative geometry [4,5,6, 12, 13]. There are many other functions in invariants of partitions which turn out to be elements of \(\Lambda ^*\), for example symmetric polynomials in de modified Frobenius coordinates [23, Eq. 19]; the hook-length moments [5, Theorem 13.5] (see Sect. 7.1); central characters of the symmetric group [15, Proposition  3] and symmetric polynomials in the content vector of a partition [15, Proof of Theorem 4].

Previously, the Bloch–Okounkov algebra \(\Lambda ^*\) and some generalizations to higher levels (see, e.g., [8, 9]), were the only known quasimodular algebras. However, there are many examples of functions on partitions admitting a quasimodular q-bracket (and in general not belonging to \(\Lambda ^*\)) [23, Sect. 9], for example the Möller transformation of functions with quasimodular q-bracket (defined by [23, Eq. 45] and recalled in Sect. 7), invariants \(\mathcal {A}_P\) for every even polynomial defined in terms of the arm- and leg-lengths of a partition and the moment functions

$$\begin{aligned} S_k(\lambda ) \, =\, -\frac{B_k}{2k}\,+\,\sum _{i=1}^\infty \lambda _i^{k-1} \quad \quad \quad (k \text { even}, B_k = k\text {th Bernoulli number}) \end{aligned}$$
(3)

that also occur in the study of so-called spin Hurwitz numbers in the algebra of supersymmetric polynomials [10] (in that reference, these functions are only evaluated at strict partitions—partitions without repeated parts—and quasimodularity is shown for a correspondingly adapted q-bracket).

In this paper, we prove the stronger result that the algebra \(\mathcal {S}\) generated by these moment functions \(S_k\) is quasimodular. Moreover, besides the pointwise product of functions on partitions, we define a second associative product \(\odot \), called the induced product as it is inherited from the product of power series. The vector space \(\mathrm {Sym}^\odot (\mathcal {S})\) generated by the elements in \(\mathcal {S}\) under the induced product is strictly bigger than \(\mathcal {S}\), is a quasimodular algebra for either of the two products, and has a particularly nice description in terms of functions \(T_{k,l}\) depending not only on the parts of a partition, but also on their multiplicities. Here, the multiplicity \(r_m(\lambda )\) of parts of size m in a partition \(\lambda \) is defined as the number of parts of \(\lambda \) of size m. More precisely, let \(\mathcal {F}_l\) be the Faulhaber polynomial of positive integer degree l, defined by \(\mathcal {F}_l(n)=\sum _{i=1}^n i^{l-1}\) for all \(n\in \mathbb {Z}_{>0}\). Then, \(T_{k,l}\) is given by

$$\begin{aligned} T_{k,l}(\lambda ) \, =\, C_{k,l}\,+\,\sum _{m=1}^\infty m^k \mathcal {F}_l(r_m(\lambda )) \quad \quad \quad (k\ge 0, l\ge 1, k+l \text { even}) \end{aligned}$$
(4)

with \(C_{k,l}\) a constant equal to \(-\frac{B_{k+l}}{2(k+l)}\) if \({k=0}\) or \({l=1}\) and 0 else. Let \(\mathcal {T}\) be the algebra generated by all these \(T_{k,l}\) under the pointwise product.

We show that \(\mathrm {Sym}^\odot (\mathcal {S})\) and \(\mathcal {T}\) are algebras for the pointwise product as well as for the induced product. In fact, the expression of elements of \(\mathrm {Sym}^\odot (\mathcal {S})\) in terms of the \(T_{k,l}\) implies that \(\mathrm {Sym}^\odot (\mathcal {S})\) is a strict subalgebra of \(\mathcal {T}\) (with respect to both products). Our main result is the following:

Theorem 1.1

The algebras \(\mathrm {Sym}^\odot (\mathcal {S})\) and \(\mathcal {T}\) are quasimodular algebras with respect to the induced product.

With respect to the pointwise product, these algebras are not quasimodular because of the following subtlety: The q-bracket of a homogeneous function f in \(\mathcal {T}\) (with respect to the pointwise product) often is of mixed weight (i.e., a linear combination of quasimodular forms of weights bounded by the weight of f). By making use of the induced product, one can explain these lower weight quasimodular forms, as we do in Sect. 6. For example,

$$\begin{aligned} \langle T_{0,2}^2 \rangle _q \, =\, G_2^2 \,+\, \frac{5}{6}G_4 \,+\, \frac{1}{6}G_2 \,+\, \frac{1}{288}, \end{aligned}$$

where \(G_2\) and \(G_4\) are the Eisenstein series defined by (6). The right-hand side is a quasimodular form of mixed weight, which is explained by the fact that

$$\begin{aligned} T_{0,2}^2 \, =\, T_{0,2}\odot T_{0,2} \,+\, \frac{5}{6}T_{0,4} \,+\, \frac{1}{6}T_{0,2} \,+\, \frac{1}{288}, \end{aligned}$$

is a linear combination of elements of \(\mathcal {T}\) of different weights with respect to the induced product.

A main theme throughout this paper is the principle to establish all identities in \(\mathbb {Q}^\mathscr {P}\) or \(\mathcal {T}\) before taking the q-bracket, instead of doing these computations in \(\mathbb {Q}[[q]]\) or the space of quasimodular forms \(\widetilde{M}\). By doing so, we discover the algebraic structure of \(\mathcal {T}\). Without having the induced product at one’s disposal, for example when studying the shifted symmetric algebra \(\Lambda ^*\), this seems impossible. See the following table for an overview of situations where the principle is appliedFootnote 1:

Previous definitions and results Definitions and results in this work Sections
Multiplication in \(\mathbb {Q}[[q]]\) Induced product \(\odot \) on \(\mathbb {Q}^\mathscr {P}\) 3.2
q-bracket\(\mathbb {Q}^\mathscr {P}\rightarrow \mathbb {Q}[[q]]\) \(\underline{u}\)-bracket\(\mathbb {Q}^\mathscr {P}\rightarrow \mathbb {Q}[[u_1,u_2,\ldots ]]\) 3.2
Connected q-bracket\(\mathrm {Sym}^\otimes (\mathbb {Q}^\mathscr {P})\rightarrow \mathbb {Q}[[q]]\) Connected product\(\mathrm {Sym}^\otimes (\mathbb {Q}^\mathscr {P})\rightarrow \mathbb {Q}^\mathscr {P}\) 3.2
Derivative \(q\frac{\mathrm {d}}{\mathrm {d}q}\) on \(\mathbb {Q}[[q]]\) Derivative on \(\mathbb {Q}^\mathscr {P}\) 5.1
\(\mathfrak {sl}_2\)-action on \(\widetilde{M}\) \(\mathfrak {sl}_2\)-action on \(\mathcal {T}\) 5.2
Rankin–Cohen brackets on \(\widetilde{M}\) Rankin–Cohen brackets on \(\mathcal {T}\) 5.3
Formula for \(\langle H_p f\rangle _q\) in [5, Eq. 152] Formula for \( T_{k,l}f\) 6.2

A further main result of the paper is the following:

Theorem 1.2

The q-bracket is an equivariant mapping \(\mathcal {T}\rightarrow \widetilde{M}\) with respect to \(\mathfrak {sl}_2\)-actions by derivations on both spaces.

Motivated by the fact that many functions in invariants of partitions are elements of \(\Lambda ^*\), in Sect. 7 we describe many functions on partitions which are elements of \(\mathcal {T}\) or are closely related. Among those are the border strip moments, generalizing the hook-length moments, which are defined in terms of the representation theory of the symmetric group. The corresponding space \(\mathcal {X}\) of border strip moments is the image of a space \(\mathcal {U}\) under the aforementioned Möller transform \(\mathcal {M}\), where \(\mathcal {U}\) is generated by the double moment functions \(T_{k,l}\in \mathcal {T}\) as well as the odd double moments functions (those for which \(k+l\) is odd). The q-brackets of these functions are contained in the space \(\mathcal {C}\) of so-called combinatorial Eisenstein series, having the space of quasimodular forms as a subspace. Moreover, the space of hook-length moments \(\mathcal {H}\) is contained in both \(\Lambda ^*\) and \(\mathcal {X}\)—this contrasts the situation for \(\mathcal {T}\), which by Remark 4.1.6 has a trivial intersection with \(\Lambda ^*\). See the commutative diagram below for an overview of the spaces related to \(\mathcal {T}\) with their corresponding mappings.                               

figurea

                              

We hope that this work—besides advocating the notion of a ‘quasimodular algebra’ by giving a new example of such an algebra and studying its algebraic structure—may serve as a tool for enumerative geometers trying to show that generating series are quasimodular forms.

The contents of the paper are as follows. In Sect. 2 we recall notions (known to the experts) related to quasimodular forms, partitions and special families of polynomials. Next, in Sect. 3 we motivate all new notions in this work and prove quasimodularity of the algebra \(\mathcal {S}\). A study of the symmetric algebra \(\mathcal {T}\), including a proof of our main theorem, can be found in Sect. 4. The \(\mathfrak {sl}_2\)-action by differential operators, the proof of  Theorem 1.2 and Rankin–Cohen brackets are the content of Sect. 5. In Sect. 6 further results that arise from comparing the two different products on \(\mathcal {T}\) are given, and finally, in Sect. 7 we provide many examples of functions in or closely related to \(\mathcal {T}\).

Preliminaries

Quasimodular forms

Let \(\mathrm {Hol}_0(\mathfrak {H})\) be the ring of holomorphic functions \(\varphi \) of moderate growth on the complex upper half plane \(\mathfrak {H}\), i.e., for all \(C>0\) one has \(\varphi (x+iy)=O(e^{Cy})\) as \(y\rightarrow \infty \) and \(\varphi (x+iy)=O(e^{C/y})\) as \(y\rightarrow 0\). A quasimodular form of weight k and depth at most p for \(\mathrm {SL}_2(\mathbb {Z})\) is a function \(\varphi \in \mathrm {Hol}_0(\mathfrak {H})\) such that there exist \(\varphi _0,\ldots , \varphi _p\in \mathrm {Hol}_0(\mathfrak {H})\) so that for all \(\tau \in \mathfrak {H}\) and all \(\gamma = \left( {\begin{matrix} a &{} b \\ c &{} d\end{matrix}}\right) \in \mathrm {SL}_2(\mathbb {Z})\), one has

$$\begin{aligned} (c\tau +d)^{-k}\varphi \Bigl (\frac{a\tau +b}{c\tau +d}\Bigr ) \, =\, \varphi _0(\tau )\,+\,\varphi _1(\tau )\frac{c}{c\tau +d}\,+\,\ldots \,+\,\varphi _p(\tau )\Bigl (\frac{c}{c\tau +d}\Bigr )^p. \end{aligned}$$
(5)

Equation (5) is called the quasimodular transformation property. Note that if \(\varphi \) is a quasimodular form, the functions \(\varphi _0,\ldots , \varphi _p\) are quasimodular forms uniquely determined by \(\varphi \) (the function \(\varphi _r\) has weight \(k-2r\) and depth \(\le p-r\)). For example, taking the identity \(I\in \Gamma \) yields \(\varphi _0=\varphi \). Quasimodular forms of depth 0 are called modular forms. Besides the constant functions, the simplest examples are the Eisenstein series

$$\begin{aligned} G_k(\tau ) \!=\! -\frac{B_k}{2k}\!+\!\sum _{r=1}^\infty \sum _{m=1}^\infty m^{k-1} q^{mr} \qquad (B_k \!=\! k\text {th Bernoulli number and } q=e^{2\pi i \tau })\nonumber \\ \end{aligned}$$
(6)

for positive even integers k. For \(k>2\) the Eisenstein series are modular forms of weight k. The Eisenstein series \(G_2\) is a quasimodular form of weight 2 and depth 1.

Denote by \(\widetilde{M}_k^{(\le p)}\) the vector space of quasimodular forms of weight k and depth at most p. Often we omit the depth and/or weight and simply write \(\widetilde{M}_k\) for the vector space of all quasimodular forms of weight k or \(\widetilde{M}\) for the graded algebra of all quasimodular forms. Let \(M\) denote the graded algebra of modular forms. The quasimodular form \(G_2\) generates the algebra of quasimodular forms as an algebra over the subalgebra of modular forms, that is, \(\widetilde{M}=M[G_2]\).

Often, when encountering an indexed collection of numbers or functions, we study its generating series. The generating series corresponding to the Eisenstein series is called the propagator or the Kronecker–Eisenstein series of weight 2 and given by

$$\begin{aligned} P(z,\tau )\, =\, P(z)\, :=\, \frac{1}{z^2} + 2\sum _{k=2}^\infty G_k\frac{z^{k-2}}{(k-2)!} . \end{aligned}$$
(7)

The propagator is closely related to the Weierstrass \(\wp \)-function and Jacobi theta series

$$\begin{aligned} \wp (z,\tau )\!=\!\wp (z) \!&:=\! \frac{1}{z^2}\,+\!\sum _{\begin{array}{c} \omega \in \mathbb {Z}\tau \!+\! \mathbb {Z}\\ \omega \ne 0 \end{array}} \!\Bigl (\frac{1}{(z+\omega )^2}-\frac{1}{\omega ^2}\Bigr ),\\ \theta (z)\,&:=\,\sum _{\nu \in \mathbb {Z}\!+\!{1\over 2}} (-1)^{\lfloor \nu \rfloor } e^{\nu z} q^{\nu ^2/2}\nonumber \end{aligned}$$
(8)

by

$$\begin{aligned} P(z)\, =\, \frac{1}{2\pi i}\wp (\tfrac{z}{2\pi i},\tau )+2G_2,\quad \quad P(z)\, =\, -\frac{\partial }{\partial z}\frac{\theta '(z)}{\theta (z)}. \end{aligned}$$

The action of \(\mathfrak {sl}_2\) on quasimodular forms by derivations

A way to produce examples of quasimodular forms is by taking derivatives of (quasi)modular forms under the differential operator \(D:\widetilde{M}_k^{(\le p)} \rightarrow \widetilde{M}_{k+2}^{(\le p+1)}\), given by

$$\begin{aligned} D\, =\, \frac{1}{2\pi i} \frac{\mathrm {d}}{\mathrm {d}\tau } \, =\, q\frac{\mathrm {d}}{\mathrm {d}q}. \end{aligned}$$

In fact, every quasimodular form can uniquely be written as a linear combination of derivatives of modular forms and derivatives of \(G_2\). For more details, see [22, p. 58–60]. It may happen that a polynomial in the derivatives of two modular forms \(f\in M_k\) and \(g\in M_l\) is actually modular. This is the case for the Rankin–Cohen brackets of f and g, defined by

$$\begin{aligned} {[}f,g]_n \, =\, \sum _{\begin{array}{c} r,s\ge 0\\ r+s=n \end{array}}(-1)^r\left( {\begin{array}{c}k+n-1\\ s\end{array}}\right) \left( {\begin{array}{c}l+n-1\\ r\end{array}}\right) D^r\!f\,D^s\!g \quad \quad (n\ge 0). \end{aligned}$$

That is, for all \(f\in M_k, g\in M_l\) and \(n\ge 0\), one has that \([f,g]_n\) is a modular form of weight \({k+l+2n}\).

Besides the differential operator D, an important differential operator on quasimodular forms is the operator \(\mathfrak {d}: \widetilde{M}_k^{(\le p)} \rightarrow \widetilde{M}_{k-2}^{(\le p-1)}\) defined by \(\varphi \mapsto 2\pi i\varphi _1\) (with \(\varphi _1\) defined in the quasimodular transformation property (5)). For example \({\mathfrak {d}G_2=-\frac{1}{2}}\) and in fact this property together with the fact that \(\mathfrak {d}\) annihilates modular forms defines \(\mathfrak {d}\) completely since \(\mathfrak {d}\) is a derivation and \({\widetilde{M}=M[G_2]}\).

Let W be the weight operator, which multiplies a quasimodular form by its weight. The triple \((D, \mathfrak {d}, W)\) forms an \(\mathfrak {sl}_2\)-triple with respect to the commutator bracket \({[A,B]=AB-BA}\):

Definition 2.2.1

A triple (XYH) of operators is called an \(\mathfrak {sl}_2\)-triple if

$$\begin{aligned} {[}H,X]\, =\, 2X, \quad [H,Y]\, =\, -2Y, \quad [Y,X]\, =\, H. \end{aligned}$$

Remark 2.2.2

By these commutation relations, for all \(n\ge 1\) one has

$$\begin{aligned} {[}\mathfrak {d},D^n] \, =\, n(W-n+1)D^{n-1}, \end{aligned}$$
(9)

which turns out to be useful later.

Following a suggestion of Zagier, we make the following definition:

Definition 2.2.3

Given a Lie algebra \(\mathfrak {g}\), a \(\mathfrak {g}\)-algebra is an algebra A together with a Lie homomorphism \(\mathfrak {g}\rightarrow \mathrm {Der}(A).\)

As \(D,\mathfrak {d}\) and W satisfy the Leibniz rule, the algebra \(\widetilde{M}\) becomes an \(\mathfrak {sl}_2\)-algebra.

Partitions as a partially ordered set

Given \(n\in \mathbb {Z}_{\ge 0}\), let \(\mathscr {P}(n)\) denote the set of all integer partitions of n and \(\Pi (n)\) the set of all partitions of the set \([n]:=\{1,2,\ldots , n\}\). Let \(\mathscr {P}=\bigcup _{n\in \mathbb {Z}_{\ge 0}} \mathscr {P}(n)\) and \(\Pi =\bigcup _{n\in \mathbb {Z}_{\ge 0}} \Pi (n)\) be the sets of all such partitions. Given \(\lambda \in \mathscr {P}(n)\) we write \(\lambda =(\lambda _1,\lambda _2,\ldots )\) with \(\lambda _1\ge \lambda _2\ge \ldots \) and \(|\lambda |:=\sum _{i=1}^\infty \lambda _i=n\). The largest index k such that \(\lambda _k>0\) is called the length of \(\lambda \), denoted by \(\ell (\lambda )\). Similarly, for \(\alpha \in \Pi (n)\) we write \(\ell (\alpha )\) for the cardinality of \(\alpha \). Moreover, for \(\lambda \in \mathscr {P}\) we let \(r_m(\lambda )\) denote the number of parts of \(\lambda \) equal to m, i.e., \(r_m(\lambda )=\#\{i\mid \lambda _i=m\}\), and denote by \(\lambda '\) the conjugate partition of \(\lambda \). We call a partition \(\lambda \) strict if there are no repeated parts, i.e., \(r_m(\lambda )\in \{0,1\}\) for all m. For two partitions \(\kappa ,\lambda \) we write \(\kappa \cup \lambda \) for the union of \(\kappa \) and \(\lambda \) as multisets, i.e., \({r_m(\kappa \cup \lambda )=r_m(\kappa )+r_m(\lambda )}\) for all \(m\in \mathbb {N}\).

Both \(\mathscr {P}\) and \(\Pi (n)\) form a locally finite partially ordered set, i.e., a partially ordered set P for which for all \(x,z\in P\) there exists finitely many \(y\in P\) such that \(x\le y\le z\). Namely, on \(\mathscr {P}\) we define a partial order by \(\kappa \le \lambda \) if \(r_m(\kappa )\le r_m(\lambda )\) for all \(m\ge 1\). The ordering on \(\Pi (n)\) is given by \(\alpha \le \beta \) if for all \(A\in \alpha \) there exists a \(B\in \beta \) such that \(A\subseteq B\). For instance, we have \(\alpha \le \mathbf {1}_n\) for all \(\alpha \in \Pi (n)\), where \(\mathbf {1}_n=\{[n]\}\).

Recall that on a locally finite partially ordered set P the Möbius function \(\mu :P^2\rightarrow \mathbb {Z}\) is defined recursively by (see for example [16]): \(\mu (x,z) \, =\, -\sum _{x\le y\le z} \mu (x,y)\) if  \(x<z\) with initial conditions \(\mu (x,x)=1\) and \(\mu (x,z)=0\) else. For the above partial order on \(\mathscr {P}\) the value of \(\mu (\kappa ,\lambda )\) depends on whether the difference of \(\kappa \) and \(\lambda \) considered as multisets, denoted by \(\lambda - \kappa \), is a strict partition. That is,

$$\begin{aligned} \mu (\kappa ,\lambda ) \, =\, {\left\{ \begin{array}{ll}(-1)^{\ell (\lambda )-\ell (\kappa )} &{} \lambda - \kappa \text { is a strict partition} \\ 0 &{} \text {else.}\end{array}\right. } \end{aligned}$$
(10)

The Möbius function \(\mu (\alpha ,\beta )\) of two elements \(\alpha ,\beta \in \Pi (n)\) is given by

$$\begin{aligned} \mu (\alpha ,\beta )\, =\, \prod _{B\in \beta }(-1)^{\ell (\alpha _B)-1}(\ell (\alpha _B)-1)!\, , \end{aligned}$$

where \(\alpha _B\) for \(B\subset [n]\) is the partition on B induced by \(\alpha \). A Möbius function satisfies the following two properties:

Theorem 2.3.1

Let fg be functions on a partially ordered set P. Then

(i):

\(\displaystyle \sum _{\alpha \le \gamma \le \beta } \mu (\alpha ,\gamma ) \, =\, \delta _{\alpha ,\beta } \, =\, \sum _{\alpha \le \gamma \le \beta } \mu (\gamma ,\beta )\)   for all \(\alpha ,\beta \in P\);

(ii):

\(\displaystyle f(\alpha )=\sum _{\gamma \le \alpha }g(\gamma ) \quad \forall \alpha \in P \quad \iff \quad g(\beta )=\sum _{\gamma \le \beta }\mu (\gamma ,\beta )\,f(\gamma ) \quad \forall \beta \in P.\)

The connected q-bracket

The q-bracket defined in the introduction (Eq. 1) is a map \(\mathbb {Q}^{\mathscr {P}}\rightarrow \mathbb {Q}[[q]]\). In this section we define the connected q-bracket following [5, p. 55–57], which naturally arises in enumerative geometric when counting connected coverings. In our setting, the connected q-bracket turns out to be easier to compute than the usual q-bracket.

For \(A\subset [n]\) we denote \(f_A = \prod _{a\in A} f_a\).

Definition 2.4.1

Given an integer \(n\ge 1\), the connected q-bracket is defined as the multilinear map

$$\begin{aligned} \langle \ \rangle _q:\underbrace{\mathbb {Q}^{\mathscr {P}}\otimes \cdots \otimes \mathbb {Q}^{\mathscr {P}}}_{n}\rightarrow \mathbb {Q}\end{aligned}$$

extending the q-bracket such that for all \(f,f_1,\ldots , f_n\in \mathbb {Q}^{\mathscr {P}}\) any of the following two equivalent conditions hold:

  1. (i)

    \(\displaystyle \langle f_1 \otimes \cdots \otimes f_n\rangle _q \, =\, \sum _{\alpha \in \Pi (n)} \mu (\alpha ,\mathbf {1}_n)\prod _{A\in \alpha }\langle f_A\rangle _q\,;\)

  2. (ii)

    \(\langle f_1\otimes \cdots \otimes f_n\rangle _q\) is the coefficient of \(x_1\cdots x_n\) in \(\log \langle \exp (\sum _{i=1}^n x_i f_i)\rangle _q\, .\)

By invoking the Möbius inversion formula (Theorem 2.3.1(ii)) condition (i) in Definition 2.4.1 implies that

$$\begin{aligned} \prod _{B\in \beta }\langle \otimes _{b\in B}f_b \rangle _q&\, =\, \sum _{\alpha \le \beta } \mu (\alpha ,\beta )\prod _{A\in \alpha }\langle f_A\rangle _q\, , \qquad \prod _{A\in \alpha }\langle f_A\rangle _q\, =\, \sum _{\beta \le \alpha } \prod _{B\in \beta }\langle \otimes _{b\in B}f_b \rangle _q \,. \end{aligned}$$

For example,

$$\begin{aligned} \langle f\otimes g\rangle _q&\, =\, \langle fg\rangle _q-\langle f\rangle _q\langle g\rangle _q\, , \\ \langle f\otimes g\otimes h\rangle _q&\, =\, \langle fgh\rangle _q-\langle f\rangle _q\langle gh\rangle _q-\langle g\rangle _q\langle fh\rangle _q-\langle h\rangle _q\langle fg\rangle _q+2\langle f\rangle _q\langle g\rangle _q\langle h\rangle _q\, , \end{aligned}$$

and

$$\begin{aligned} \langle fg\rangle _q&\, =\, \langle f\otimes g\rangle _q+\langle f\rangle _q\langle g\rangle _q \, , \\ \langle fgh \rangle _q&\, =\, \langle f\otimes g\otimes h\rangle _q+\langle f\rangle _q\langle g\otimes h\rangle _q+\langle g\rangle _q\langle f \otimes h\rangle _q+\langle h\rangle _q\langle f \otimes g\rangle _q+\langle f\rangle _q \langle g\rangle _q \langle h\rangle _q\, . \end{aligned}$$

We often make use of the fact that the connected q-bracket of functions \(f_1,\ldots , f_n\) vanishes if one of the \(f_i\) is constant.

Lemma 2.4.2

For all \(f_1,\ldots ,f_n\in \mathbb {Q}^{\mathscr {P}}\) one has

$$\begin{aligned} \langle 1 \otimes f_1\otimes \cdots \otimes f_n \rangle _q = 0. \end{aligned}$$

Proof

Write \(f_{n+1}=1\). Observe that \(\prod _{A\in \alpha }\langle f_A\rangle _q\) takes the same value for all \({\alpha \in \Pi (n+1)}\) which agree on [n] (but differ in the subset A of \(\alpha \) containing \({n+1}\)). Then, summing \(\mu (\alpha ,\mathbf {1}_n)\) over all such \(\alpha \) yields

$$\begin{aligned} a\cdot (-1)^{a-1}(a-1)! + (-1)^{a}a! = 0 \end{aligned}$$

as there are a choices for \(\alpha \) for which \(\{n+1\}\) is not a subset of \(\alpha \), where a is the length of such an \(\alpha \), and there is only one choice for \(\alpha \) for which \(\{n+1\}\) is a subset. By Definition 2.4.1(i) the result follows. \(\square \)

We will use the second condition in Definition 2.4.1 in our proof that \(\mathcal {S}\) is a quasimodular algebra.

The discrete convolution product and Faulhaber polynomials

Let \(\mathbb {N}\) denote the set of strictly positive integers. Given \(f,g:\mathbb {N}\rightarrow \mathbb {Q}\) we denote by \(f\cdot g\) or fg the pointwise product of f and g. We define the discrete convolution product of f and g by

$$\begin{aligned} (f*g)(n)\, =\, \sum _{i=1}^{n-1} f(i)\,g(n-i) \end{aligned}$$

and denote the convolution product of functions \(f_1,\ldots , f_n\) by

(11)

Let the discrete derivative \(\partial \) of \(f:\mathbb {N}\rightarrow \mathbb {Q}\) be defined by \(\partial f(n) = f(n)-f(n-1)\) for \(n\ge 2\) and \(\partial f(1)=f(1)\) and denote by \(\mathrm {id}\) the identity function \(\mathbb {N}\rightarrow \mathbb {N}\subset \mathbb {Q}\). Observe that

$$\begin{aligned} \partial (f*g)&= (\partial f)*g\, =\, f*(\partial g), \end{aligned}$$
(12)
$$\begin{aligned} \partial (fg)&= \partial (f)\,g+f\,\partial (g)-\partial (f)\,\partial (g), \end{aligned}$$
(13)
$$\begin{aligned} \mathrm {id}\cdot (f*g)&=(\mathrm {id}\cdot f)*g+f*(\mathrm {id}\cdot g), \end{aligned}$$
(14)
$$\begin{aligned} \partial ^2(f*\mathrm {id})&=f-\partial f. \end{aligned}$$
(15)

The Faulhaber polynomials \(\mathcal {F}_l\) for \(l\ge 1\) are defined as the unique polynomials with vanishing constant term satisfying \(\partial \mathcal {F}_l(n)=n^{l-1}\) for all \(n\in \mathbb {N}\), or equivalently by \(\mathcal {F}_l(n)=\sum _{i=1}^n i^{l-1}\). The first four are given by

$$\begin{aligned} \mathcal {F}_1(x)=x, \quad \mathcal {F}_2(x)=\frac{x(x+1)}{2}, \quad \mathcal {F}_3(x) = \frac{x(x+1)(2x+1)}{6}, \quad \mathcal {F}_4(x)=\frac{x^2(x+1)^2}{4}. \end{aligned}$$

Note that these polynomials are related to the Bernoulli polynomials \(B_n(x)\), the unique family of polynomials satisfying \(\int _{x}^{x+1}B_n(u)\,\mathrm {d}u = x^n\), by the formula \(l\mathcal {F}_l(x) = B_{l}(x+1)-B_{l}\). Hence, the Faulhaber polynomials admit the symmetry

$$\begin{aligned} \mathcal {F}_l(x) \, =\, (-1)^l\mathcal {F}_l(-x-1) \quad \quad (l \ge 2), \end{aligned}$$
(16)

which can also be deduced directly from the definition. The generating series \(\mathcal {F}\!(n)\) of the Faulhaber polynomials equals

$$\begin{aligned} \mathcal {F}\!(n)\, :=\, \sum _{l=1}^\infty \mathcal {F}_l(n)\frac{z^{l-1}}{(l-1)!} \, =\, e^z\frac{1-e^{nz}}{1-e^z}. \end{aligned}$$
(17)

The moment functions, their q-bracket and a second product

Three proofs of the quasimodularity of the moment functions

The q-bracket of the moment function \(S_k\) defined in (3) equals the Eisenstein series \(G_k\). To motivate the results in the rest of this work, we provide three different proofs—and three generalizations—of this statement using three different approaches. In the first approach, we motivate the definition of the \(T_{k,l}\) (see (4)), the second approach gives an interpretation for these functions, and the last approach gives an example of our main principle of establishing all identities before taking the q-bracket.

First approach The key observation in this first proof is that \(S_k\) can be rewritten as

$$\begin{aligned} S_k(\lambda )\, =\, -\frac{B_k}{2k}+\sum _{m=1}^\infty m^{k-1} r_m(\lambda ). \end{aligned}$$

More generally, for \(k> 0\) and \(f:\mathbb {N}\rightarrow \mathbb {Q}\) we set \(f(0)=0\) and we let

$$\begin{aligned} S_{k,f}(\lambda ) \, =\, -\frac{B_{k+1}}{2(k+1)}\delta _{f,\mathrm {id}} \,+\,\sum _{m=1}^\infty m^{k} f(r_m(\lambda )). \end{aligned}$$
(18)

In case when f is the identity, \(S_{k,f}=S_{k+1}\). Our first method of proof gives the following more general statement:

Proposition 3.1.1

Let f be a polynomial of degree l without constant term and k a positive integer satisfying \(k\equiv l \bmod 2\). Then,

(i):

if f equals a Faulhaber polynomial \(\mathcal {F}_l\), then \(\langle S_{k,f}\rangle _q\) equals

$$\begin{aligned} -\frac{B_{k+1}}{2(k+1)}\delta _{l,1}+\sum _{m,r\ge 1} m^{k}r^{l-1} q^{mr} \, =\, {\left\{ \begin{array}{ll} D^{l-1}G_{k-l+2} &{} k-l\ge 0, \\ D^k G_{l-k} &{} k-l\le 2;\end{array}\right. } \end{aligned}$$
(ii):

if \(\langle S_{k,f}\rangle _q\) is a quasimodular form, then f is a multiple of the Faulhaber polynomial \(\mathcal {F}_l\).

Proof

Let \(|x|\le 1\) and \(m\ge 1\). We compute

$$\begin{aligned} \langle x^{r_m}\rangle _q \, =\, \frac{\sum _{\lambda \in \mathscr {P}} x^{r_m(\lambda )}q^{|\lambda |}}{\sum _{\lambda \in \mathscr {P}} q^{|\lambda |}}. \end{aligned}$$
(19)

Observe that the multiplicities \(r_1(\lambda ), r_2(\lambda ), \ldots \) uniquely determine the partition \(\lambda \). Hence, for \(|q|<1\) we have that

$$\begin{aligned} \sum _{\lambda \in \mathscr {P}} x^{r_m(\lambda )}q^{|\lambda |}&\, =\, \sum _{r_1,r_2,\ldots \ge 0} x^{r_m} q^{r_1+2r_2+\ldots + m r_m + \ldots } \\&\, =\, \left( \sum _{r_m=0}^\infty x^{r_m}q^{m r_m} \right) \prod _{i\ne m} \left( \sum _{r_i=0}^\infty q^{i r_i}\right) \\&\, =\, \frac{1}{1-x q^m} \prod _{i\ne m} \frac{1}{1-q^i}\, . \end{aligned}$$

Substituting this result in the numerator of (19), we obtain

$$\begin{aligned} \langle x^{r_m}\rangle _q \, =\, \frac{1-q^m}{1-xq^m}\, . \end{aligned}$$

Hence,

$$\begin{aligned} \left\langle \tfrac{x}{1-x}(1-x^{r_m})\right\rangle _q \, =\, \frac{xq^m}{1-xq^m}\, . \end{aligned}$$
(20)

Observe that applying \(x\frac{\partial }{\partial x}\) to the right-hand side of (20) has the same effect as applying \(\frac{1}{m}D\), where D is defined in §2.1. After setting \(x=e^z\), we find that \(\frac{x}{1-x}(1-x^{r_m})\) equals \(\mathcal {F}\!(r_m)\) (see 17). Hence, by taking \({l-1}\) derivatives \(x\frac{\partial }{\partial x}=\frac{\partial }{\partial z}\) and setting \(z=0\), it follows that

$$\begin{aligned} \langle S_{k,\mathcal {F}_l} \rangle _q +\frac{B_{k+1}}{2(k+1)}\delta _{l,1}&\, =\, \sum _{m\ge 0} m^{k} \langle \mathcal {F}_l(r_m)\rangle _q \\&\, =\, \sum _{m\ge 0} m^k \left( x\frac{\partial }{\partial x}\right) ^{l-1} \frac{xq^m}{1-xq^m}\Big |_{x=1} \\&\, =\, \sum _{m\ge 0} m^{k} \left( \frac{1}{m}D\right) ^{l-1} \frac{q^m}{1-q^m} \\&\, =\, \sum _{m,r\ge 1} m^{k}r^{l-1} q^{mr}. \end{aligned}$$

Part (ii) of the statement follows by writing f as a linear combination of Faulhaber polynomials. \(\square \)

Second approach The double moment functions \(T_{k,l}\) (see (4)) are by definition equal to \(S_{k,\mathcal {F}_l}\) if \(k>0\). Given a partition \(\lambda \), let \(c_i(\lambda ) = \#\{j\le i \mid \lambda _i=\lambda _j\}\). Then, one has

$$\begin{aligned} T_{k,l}(\lambda ) \, =\, -\frac{B_{k+l}}{2(k+l)}(\delta _{l,1}+\delta _{k,0}) + \sum _{i=1}^\infty \lambda _i^k c_i(\lambda )^{l-1}. \end{aligned}$$

In this section we give a direct proof for the quasimodularity of the q-brackets of \(T_{k,l}\):

Proposition 3.1.2

For all \(k\ge 0,l\ge 1\) and \({k+l}\) even, one has

$$\begin{aligned} \langle T_{k,l}\rangle _q \, =\, {\left\{ \begin{array}{ll} D^{l-1}G_{k-l+2} &{} \text {if } k-l\ge 0, \\ D^k G_{l-k} &{} \text {if } k-l\le 2.\end{array}\right. } \end{aligned}$$

Proof

Denote by \(T_{k,l}^0(\lambda ) = \sum _{i=1}^\infty \lambda _i^k c_i(\lambda )^{l-1}\). The generating series of \(T_{k,l}^0\) is given by

$$\begin{aligned} W(X,Y)(\lambda ) \, =\, \sum _{i=1}^\infty X^{\lambda _i}Y^{c_i(\lambda )}, \end{aligned}$$

that is, \(T_{k,l}^0(\lambda )\) is the coefficient of \(\frac{x^ky^{l-1}}{k!(l-1)!}\) in \(W(e^x,e^y)(\lambda )\). Consider

$$\begin{aligned} \sum _{\lambda \in \mathscr {P}} W(X,Y)(\lambda )\, q^{|\lambda |}\, =\, \sum _{\lambda \in \mathscr {P}}\sum _{i=1}^\infty X^{\lambda _i}Y^{c_i(\lambda )}q^{|\lambda |}. \end{aligned}$$
(21)

Given \(a,b,n\in \mathbb {Z}_{\ge 0}\), denote by \(C_{a,b}(n)\), the coefficient in front of \(X^{a}Y^bq^{n}\) in (21), that is

$$\begin{aligned} \sum _{\lambda \in \mathscr {P}}\sum _{i=1}^\infty X^{\lambda _i}Y^{c_i(\lambda )}q^{|\lambda |} \;=:\; \sum _{a,b,n\ge 0} C_{a,b}(n) X^{a}Y^{b}q^{n}. \end{aligned}$$

Let p(n) denote the number of partitions of n. The coefficient \(C_{a,b}(n)\) equals the number of partitions of n with at least b parts of size a, i.e., \(C_{a,b}(n)=p(n-ab)\). Hence, writing \(m=n-ab\) we obtain

$$\begin{aligned} \sum _{\lambda \in \mathscr {P}}\sum _{i=1}^\infty X^{\lambda _i}Y^{c_i(\lambda )}q^{|\lambda |} \, =\, \left( \sum _{m=0}^\infty p(m)q^{m}\right) \left( \sum _{a,b\ge 0}X^{a}Y^{b} q^{ab}\right) . \end{aligned}$$

In other words,

$$\begin{aligned} \langle W(X,Y)\rangle _q \, =\, \sum _{a,b\ge 0} X^{a}Y^{b}q^{ab} \end{aligned}$$

so that expanding this equation for \(X=e^x\) and \(Y=e^y\) yields

$$\begin{aligned} \langle T_{k,l}^0 \rangle _q \, =\, \sum _{a,b\ge 0}a^kb^{l-1} q^{ab}. \end{aligned}$$

As \(T_{k,l}(\lambda ) = -\frac{B_{k+l}}{(k+l)}(\delta _{l,1}+\delta _{k,0}) + T_{k,l}^0(\lambda )\) we obtain the desired result. \(\square \)

Third approach In this last proof we start with the observation that one can rewrite the q-bracket as

$$\begin{aligned} \langle f\rangle _q \, =\, \frac{\sum _{\lambda \in \mathscr {P}} f(\lambda )\, u_{\lambda _1} u_{\lambda _2} \cdots }{\sum _{\lambda \in \mathscr {P}} u_{\lambda _1} u_{\lambda _2} \cdots }\Big |_{u_i=q^{i}}\, . \end{aligned}$$
(22)

In contrast to the previous two proofs, it is only in the last step of this proof that we take the q-bracket: First we rewrite (22) considering \(u_1,u_2,\ldots \) to be formal variables, and in the last step we let \(u_i=q^i\). We start with the denominator, where we encounter the Möbius function on partitions also defined in [17].

Proposition 3.1.3

There exists a function \(\mu :\mathscr {P}\rightarrow \{-1,0,1\}\) defined by any one of the following three equivalent definitions:

(i):

\(\mu (\lambda )\) is given by the Möbius function \(\mu (\emptyset , \lambda )\) on the partial order on the set of partitions in (10);

(ii):

\(\mu (\lambda ) \, =\, {\left\{ \begin{array}{ll} (-1)^{\ell (\lambda )} &{} \lambda \text { is a strict partition,} \\ 0 &{} \text {else;}\end{array}\right. }\)

(iii):

\(\displaystyle \frac{1}{\sum _{\lambda \in \mathscr {P}} u_{\lambda _1} u_{\lambda _2} \cdots } \, =\, \sum _{\lambda \in \mathscr {P}} \mu (\lambda )\, u_{\lambda _1} u_{\lambda _2} \cdots .\)

Proof

The first two definitions clearly coincide using (10). For the latter, it suffices to show that

$$\begin{aligned} \sum _{\alpha \cup \beta =\lambda } \mu (\alpha ) \, =\, \delta _{\lambda ,\emptyset }\, . \end{aligned}$$

Let \(f(\lambda )=1\) and \(g(\lambda )=\delta _{\lambda ,\emptyset }\) for \(\lambda \in \mathscr {P}\). Then, \(f(\alpha )=\sum _{\gamma \le \alpha }g(\gamma )\) for all \(\alpha \in \mathscr {P}\), so that by Möbius inversion and by using \(\mu (\gamma ,\beta ) = \mu (\emptyset ,\beta -\gamma )\) the last definition is equivalent. \(\square \)

The fact that \(\langle S_k\rangle _q=G_k\) follows directly from the following proposition:

Proposition 3.1.4

For all \(m\ge 1\) and \(f:\mathbb {N}\rightarrow \mathbb {Q}\) extended by \(f(0)=0\), one has

$$\begin{aligned} \frac{\sum _{\lambda \in \mathscr {P}} f(r_m(\lambda ))\, u_{\lambda _1} u_{\lambda _2} \cdots }{\sum _{\lambda \in \mathscr {P}} u_{\lambda _1}u_{\lambda _2}\cdots } \, =\, \sum _{r=1}^\infty \partial f(r)\, u_m^r. \end{aligned}$$

Proof

Fix \(m\ge 1\). By the previous proposition, we have

$$\begin{aligned} \frac{\sum _{\lambda \in \mathscr {P}} f(r_m(\lambda ))\, u_{\lambda _1} u_{\lambda _2} \cdots }{\sum _{\lambda \in \mathscr {P}} u_{\lambda _1}u_{\lambda _2}\cdots } \, =\, \left( \sum _{\lambda \in \mathscr {P}} f(r_m(\lambda ))\, u_{\lambda _1} u_{\lambda _2} \cdots \right) \left( \sum _{\lambda \in \mathscr {P}} \mu (\lambda )\, u_{\lambda _1} u_{\lambda _2} \cdots \right) . \end{aligned}$$

Denote by \(C(\lambda )\) the coefficient of \(u_{\lambda _1}u_{\lambda _2}\cdots \) after expanding the right-hand side of above equation. Observe that

$$\begin{aligned} C(\lambda ) \, =\, \sum _{\alpha \cup \beta =\lambda } (-1)^{\ell (\beta )} f(r_m(\alpha )), \end{aligned}$$

where \(\alpha \cup \beta \) denotes the union of \(\alpha \) and \(\beta \) considered as multisets and it is understood that \(\beta \) is a strict partition. Suppose \(\lambda \) admits a part equal to \(m'\ne m\). Then, define an involution \(\omega \) on all pairs \((\alpha ,\beta )\) satisfying that \(\alpha \cup \beta =\lambda \) and \(\beta \) is strict by

$$\begin{aligned} \omega (\alpha ,\beta ) \, =\, {\left\{ \begin{array}{ll} (\alpha \backslash \{m'\},\beta \cup \{m'\}) &{} \text {if } r_{m'}(\beta )=0, \\ (\alpha \cup \{m'\},\beta \backslash \{m'\}) &{} \text {if } r_{m'}(\beta )=1. \end{array}\right. } \end{aligned}$$

As \(\omega \) changes the sign of \((-1)^{\ell (\beta )}f(r_m(\alpha ))\), it follows that \(C(\lambda )=0\).

Observe that \(C(\emptyset )=0\) and that in case \(\lambda =(m,m,\ldots )\) consists of a strictly positive number of parts all equal to m one has

$$\begin{aligned} C(\lambda ) \, =\, f(r_m(\lambda ))-f(r_m(\lambda )-1)\, =\, \partial f(r_m(\lambda )). \end{aligned}$$

Therefore, the desired result follows. \(\square \)

The induced and connected product

Motivated by the last of the three approaches in the previous section, we define the \(\underline{u}\)-bracket of a function \(f\in \mathbb {Q}^\mathscr {P}\) by

$$\begin{aligned} \langle f\rangle _{\underline{u}} \, =\, \frac{\sum _{\lambda \in \mathscr {P}} f(\lambda )\, u_\lambda }{\sum _{\lambda \in \mathscr {P}} u_\lambda } \quad \quad \quad (u_\lambda =u_{\lambda _1} u_{\lambda _2}\cdots ). \end{aligned}$$

Then, for all \(f\in \mathbb {Q}^{\mathscr {P}}\) one has \(\langle f \rangle _q=\langle f \rangle _{(q, q^2, q^3,\ldots )}\). Observe that the \(\underline{u}\)-bracket defines an isomorphism of vector spaces

$$\begin{aligned} \mathbb {Q}^{\mathscr {P}}\xrightarrow {\sim } \mathbb {Q}[[u_1,u_2,u_3,\ldots ]], \quad \quad f \mapsto \langle f \rangle _{\underline{u}}\, . \end{aligned}$$

We now use the algebra structure of \(\mathbb {Q}[[u_1,u_2,u_3,\ldots ]]\) to define a product on \(\mathbb {Q}^{\mathscr {P}}\).

Definition 3.2.1

Given \(f,g\in \mathbb {Q}^{\mathscr {P}}\) we define their induced product \({f\odot g}\) by

$$\begin{aligned} \langle f\odot g \rangle _{\underline{u}}\, =\, \langle f\rangle _{\underline{u}}\langle g\rangle _{\underline{u}}\, , \end{aligned}$$

where the product of \(\langle f\rangle _{\underline{u}}\) and \(\langle g\rangle _{\underline{u}}\) is the usual product of power series.

Remark 3.2.2

Observe that \(\mathbb {Q}^{\mathscr {P}}\) is a commutative algebra with the constant function 1 as the identity for both the pointwise and the induced product. This observation should be compared with the q-bracket arithmetic in [17].

The following proposition gives an alternative definition for the induced product.

Proposition 3.2.3

For all \(\lambda \in \mathscr {P}\), one has

$$\begin{aligned} (f\odot g)(\lambda ) \;= \sum _{\alpha \cup \beta \cup \gamma =\lambda } f(\alpha )\,g(\beta )\,\mu (\gamma ). \end{aligned}$$

Proof

By definition

$$\begin{aligned} \sum _{\lambda \in \mathscr {P}}(f\odot g)(\lambda )\,u_\lambda \, =\, \frac{\left( \sum _{\lambda \in \mathscr {P}} f(\lambda )\, u_\lambda \right) \left( \sum _{\lambda \in \mathscr {P}} g(\lambda )\, u_\lambda \right) }{\sum _{\lambda \in \mathscr {P}} u_\lambda }. \end{aligned}$$

By Proposition 3.1.3 this equals

$$\begin{aligned} \left( \sum _{\lambda \in \mathscr {P}} f(\lambda )\, u_\lambda \right) \left( \sum _{\lambda \in \mathscr {P}} g(\lambda )\, u_\lambda \right) \left( \sum _{\lambda \in \mathscr {P}} \mu (\lambda )\, u_\lambda \right) . \end{aligned}$$

The result follows by expanding the products. \(\square \)

Analogous to the connected q-bracket, we define the connected product. For a set S and functions \(f_s\in \mathbb {Q}^{\mathscr {P}}\) for all \(s\in S\), we denote \(f_S = \prod _{s\in S}f_s\, .\)

Definition 3.2.4

For \(f_1,\ldots , f_n\in \mathbb {Q}^{\mathscr {P}}\), define the connected product \(f_1\,|\,\ldots | f_n\) to be the following function \(\mathscr {P}\rightarrow \mathbb {Q}\):

$$\begin{aligned} f_1\,|\,\ldots \,|\,f_n\, :=\, \sum _{\alpha \in \Pi (n)} \mu (\alpha ,\mathbf {1})\bigodot _{A\in \alpha } f_A\, . \end{aligned}$$
(23)

For example, for \(f,g,h\in \mathbb {Q}^{\mathscr {P}}\) one has

$$\begin{aligned} f\,|\,g&\, =\, fg- f\odot g, \\ f\,|\,g\,|\,h&\, =\, fgh- f \odot gh- g \odot fh- h \odot fg+2 f\odot g \odot h. \end{aligned}$$

The induced and connected product allow us to establish many identities before taking the q-bracket, as follows from the following result.

Proposition 3.2.5

For all \(f_1,\ldots , f_n\in \mathbb {Q}^{\mathscr {P}}\) one has

  • \(\displaystyle \langle f_1\odot f_2 \odot \cdots \odot f_n\rangle _q \, =\, \langle f_1\rangle _q \langle f_2\rangle _q \cdots \langle f_n\rangle _q\, \);

  • \(\displaystyle \langle f_1\,|\,\ldots \,|\,f_n\rangle _q \, =\, \langle f_1 \otimes \cdots \otimes f_n\rangle _q\,\).

Proof

Both statements follow directly from the definitions. For the first, note that for all \({f,g\in \mathbb {Q}^{\mathscr {P}}}\) one has

$$\begin{aligned} \langle f\odot g\rangle _q \, =\, \langle f\rangle _{\underline{u}}\langle g\rangle _{\underline{u}}|_{u_i=q^i} \, =\, \langle f\rangle _{\underline{u}}|_{u_i=q^i}\langle g\rangle _{\underline{u}}|_{u_i=q^i} \, =\, \langle f\rangle _q\langle g\rangle _q, \end{aligned}$$

so that the statement follows inductively. The second follows from the first, as

$$\begin{aligned} \displaystyle \langle f_1\,|\,\ldots \,|\,f_n\rangle _q&\, =\, \sum _{\alpha \in \Pi (n)} \mu (\alpha ,\mathbf {1})\prod _{A\in \alpha } \langle f_A\rangle _q \, =\, \langle f_1 \otimes \cdots \otimes f_n\rangle _q\,. \square \end{aligned}$$

Remark 3.2.6

Let \(\mathcal {R}\) be the space of functions having a quasimodular form as q-bracket, i.e., \({\mathcal {R}=\langle \, \cdot \,\rangle _q^{-1}(\widetilde{M})}\). Then, \(\mathcal {R}\) is a graded algebra with multiplication given by the induced product. Namely, if \(f\in \mathcal {R}\) and \(\langle f \rangle _q\in \widetilde{M}_k\), we define the weight of f to be equal to k. Note that if \(f,g\in \mathcal {R}\) and \(\langle f \rangle _q\) and \(\langle g \rangle _q\) are quasimodular forms of weight k and l, respectively, then \(\langle f \odot g\rangle _q = \langle f\rangle _q\langle g\rangle _q\) is a quasimodular form of weight \(k+l\).

When establishing identities on the level of functions on partitions (before taking the q-bracket), it turns out to be very useful to express the connected product of pointwise products of elements of \(\mathbb {Q}^{\mathscr {P}}\) in terms of connected and induced products. This can be done recursively using the following result.

Proposition 3.2.7

For all \(f_1,\ldots f_n\in \mathbb {Q}^{\mathscr {P}}\) one has

$$\begin{aligned} f_1f_2\,|\,f_3\,|\,f_4\,|\,\ldots \,|\,f_n&\, =\, f_1\,|\,f_2\,|\,\ldots \,|\,f_n \; \nonumber \\&\quad \,+\, \sum _{A\sqcup B=\{3,\ldots , n\}} (f_1\,|\,f_{A_1}\,|\,f_{A_2}\,|\,\ldots )\odot (f_2\,|\,f_{B_1}\,|\,f_{B_2}\,|\,\ldots ), \end{aligned}$$
(24)

where \(A_1,A_2,\ldots \) enumerate the elements of A (and similarly for B).

Proof

Observe that both sides of the equation in the statement are a linear combination of terms of the form \(\bigodot _{C\in \gamma }f_C\) over \(\gamma \in \Pi (n)\). We determine the coefficient of such a term on both sides of the equation.

First of all, assume \(\gamma \) is such that \(\{1,2\}\subset C\) for some \(C\in \gamma \). Then, on the right-hand side such a term only occurs in \(f_1\,|\,\ldots \,|\,f_n\) with coefficient \(\mu (\gamma ,\mathbf {1})\). Moreover, let \(\tilde{\gamma }\in \Pi (n-1)\) be given by \(\gamma \cap {\{2,\ldots ,n\}}\) subject to replacing i by \(i-1\) for all \(i=2,\ldots , n\). Note that the coefficient on the left-hand side equals \(\mu (\tilde{\gamma },\mathbf {1})\). As \(\ell (\tilde{\gamma })=\ell (\gamma )\), the coefficients on both sides agree.

Next, assume \(C_1,C_2\in \gamma \) with \(1\in C_1\) and \(2\in C_2\,\). Then, the coefficient of \(\bigodot _{C\in \gamma }f_C\) on right-hand side of (24) equals

$$\begin{aligned} \mu (\gamma ,\mathbf {1})+\sum \mu (\gamma |_{A},\mathbf {1})\mu (\gamma |_B,\mathbf {1}), \end{aligned}$$
(25)

where the sum is over all \(I\subset \{2,3,\ldots ,\ell (\gamma )\}\) and A and B are given by \(A=C_1\cup \bigcup _{i\in I}C_i\) and \(B=C_2\cup \bigcup _{i\in I^c}C_i\). Letting i be the number of elements of I, we find that (25) equals

$$\begin{aligned}&\mu (\gamma ,\mathbf {1})+\sum _{i=0}^{\ell (\gamma )-2} \left( {\begin{array}{c}\ell (\gamma )-2\\ i\end{array}}\right) \cdot (-1)^{i}i!\cdot (-1)^{\ell (\gamma )-i-2}(\ell (\gamma )-i-2)!\\&\quad =\mu (\gamma ,\mathbf {1})+\sum _{i=0}^{\ell (\gamma )-2} (\ell (\gamma )-2)!(-1)^{\ell (\gamma )-2}\\&\quad =\mu (\gamma ,\mathbf {1})-\mu (\gamma ,\mathbf {1})=0. \end{aligned}$$

Correspondingly, the coefficient of \(\bigodot _{C\in \gamma }\underline{f_C}\) on the left-hand side of (24) vanishes if there are \(C_1,C_2\in \gamma \) with \(1\in C_1\) and \(2\in C_2\,\). \(\square \)

Quasimodularity of pointwise products of moment functions

Not only do the moment functions \(S_k\) admit quasimodular q-brackets, but also the homogeneous polynomials in the moment functions admit quasimodular q-brackets; here, each moment function \(S_k\) has weight k in accordance with the fact that \(\langle S_k\rangle _q\) has weight k. Given a tuple \(\underline{k}=(k_1,...,k_n)\) of even integers, we write \(S_{\underline{k}} = S_{k_1}\cdots S_{k_n}.\) Note that, as a vector space, \(\mathcal {S}\) is spanned by these functions \(S_{\underline{k}}\). We provide two approaches to proving the quasimodularity of the q-brackets of the \(S_{\underline{k}}\). First, we give a direct proof of the statement in Theorem 3.3.1, after which, in accordance with our main principle of establishing all identities before taking the q-bracket, we prove a more general result which will be used frequently in the next section.

Theorem 3.3.1

The algebra \(\mathcal {S}\) is a quasimodular algebra. More precisely, for \(\underline{k}\in (2\mathbb {N})^n\) one has

$$\begin{aligned} \langle S_{\underline{k}}\rangle _q&\, =\, \sum _{\alpha \in \Pi (n)}\prod _{A\in \alpha } D^{\ell (A)-1}G_{|\underline{k}_A|-2\ell (A)+2}\, . \end{aligned}$$
(26)

Proof

Observe that it suffices to show that

$$\begin{aligned} \left\langle \bigotimes \nolimits _{k\in \underline{k}} S_{k}\right\rangle _q \, =\, D^{n-1} G_{|\underline{k}|-2n+2}\, \end{aligned}$$
(27)

as (26) follows from (27) by Möbius inversion. Recall that \(\langle f_1\otimes \cdots \otimes f_n\rangle _q\) is the coefficient of \({x_1\cdots x_n}\) in \(\log \langle \exp (\sum _{i=1}^n x_i f_i)\rangle _q\) (see Definition 2.4.1(ii)). Consider \(S_k^0(\lambda )=\sum _{i=1}^\infty \lambda _i^{k-1}\) for all positive even k. Euler’s formula for the generating series of partitions

$$\begin{aligned} \sum _{\lambda \in \mathscr {P}} q^{|\lambda |} \, =\, \prod _{m=1}^\infty (1-q^m)^{-1} \end{aligned}$$

follows from writing \(|\lambda |=\sum _{m\ge 1} m r_m(\lambda )\) and summing over all possible values of \(r_1(\lambda ), r_2(\lambda )\), etc. By the same idea, we find

$$\begin{aligned} \sum _{\lambda \in \mathscr {P}} \exp \left( \sum \nolimits _{k} S^0_{k}(\lambda )\, x_{k}\right) q^{|\lambda |} \, =\, \prod _{m=1}^\infty \left( 1-\exp \left( \sum \nolimits _{k} m^{k-1}x_{k}\right) q^m\right) ^{-1}. \end{aligned}$$
(28)

The logarithm of this expression equals

$$\begin{aligned} \sum _{m,r=1}^\infty \exp \left( r\sum \nolimits _{k} m^{k-1}x_{k}\right) \frac{q^{mr}}{r}. \end{aligned}$$
(29)

Now, assume all parts of \(\underline{k}\) are distinct. In the expansion of (29) the coefficient of \(x_{k_1}\cdots x_{k_n}\) equals

$$\begin{aligned} \sum _{m,r=1}^\infty m^{|\underline{k}|-n} r^{n-1} q^{mr} \, =\, D^{n-1}G_{|\underline{k}|-2n+2}\, . \end{aligned}$$

Hence,

$$\begin{aligned} \left\langle \bigotimes \nolimits _{k\in \underline{k}} S_{k}^0\right\rangle _q \, =\, D^{n-1} G_{|\underline{k}|-2n+2}. \end{aligned}$$

By introducing distinct variables in  Eq. (28) for each repeated part of \(\underline{k}\), we obtain the same result if not all parts of \(\underline{k}\) are distinct.

Note that if \(n\ge 2\), by Lemma 2.4.2 both sides of the equation do not change if one replaces \(S_k^0\) by \(S_k\). In case \(n=1\) we have established (27) in Proposition 3.1.1 or in Proposition 3.1.2. Hence, (27) holds and (26) is then implied by Möbius inversion. \(\square \)

Denoting

$$\begin{aligned} p_k(z) \, =\, {\left\{ \begin{array}{ll} \frac{z^{k-2}}{(k-2)!} &{} k\ge 0, \\ \frac{z^{-2}}{2} &{} k=0\end{array}\right. } \end{aligned}$$

and setting \(S_0(\lambda ) \equiv 1\), one has the following expression for the generating series of the q-bracket of the generators of \(\mathcal {S}\):

Corollary 3.3.2

$$\begin{aligned} \sum _{k_1,\ldots ,k_n\ge 0} \langle S_{k_1}\cdots S_{k_n}\rangle _q \, p_{k_1}(z_1)\cdots p_{k_n}(z_n) \, =\, \sum _{\alpha \in \Pi (n)}\prod _{A\in \alpha } D^{|A|-1}\frac{P^{\text {even}}(\tau ;z_A)}{2}, \end{aligned}$$

where \(z_A=\sum _{a\in A}z_a\) and

$$\begin{aligned} P^{\text {even}}(\tau ;z_1,\ldots , z_n) \, =\, \frac{1}{2^n}\sum _{s\in \{-1,1\}^n} P(\tau ,s_1z_1+\ldots +s_nz_n) \end{aligned}$$

is the totally even part of the propagator in (7).

Intermezzo: surjectivity of the q-bracket

We deduce from Theorem 3.3.1 the surjectivity of the q-bracket: Every quasimodular form is the q-bracket of some \(f\in \mathcal {S}\).

Theorem 3.4.1

The q-bracket \(\langle \, \cdot \,\rangle _q:\mathcal {S}\rightarrow \widetilde{M}\) is surjective.

Note that this is not obvious since the q-bracket is not an algebra homomorphism. Denote by \(\vartheta _k:M_k\rightarrow M_{k+2}\) the Serre derivative, given by \(\vartheta _k=D+2kG_2\). Extend this notation by letting \(\vartheta _x:\widetilde{M}\rightarrow \widetilde{M}\) for \(x\in \mathbb {Q}\) be given by \(\vartheta _x=D+2xG_2\).

Proposition 3.4.2

Let \(x\in \mathbb {Q}\backslash 2\mathbb {Z}_{\ge 0}\). Then

$$\begin{aligned} \widetilde{M}_k^{(\le p)} \, =\, \bigoplus _{r=0}^p \vartheta _x^r M_{k-2r}\, . \end{aligned}$$

Proof

Let \(f\in M_k\) with \(f\ne 0\). Observe that \(\vartheta _xf\) is modular precisely if \(k=x\). By our assumption on x, this is not the case. Hence, \(\vartheta _x\) increases the depth strictly by one. The result follows by induction on p by the same argument as in [22, Proposition 20]. Namely, if \(\varphi \in \widetilde{M}_k^{\le p}\), then the last coefficient \(\varphi _p\) in the quasimodular transformation (5) is a modular form of weight \(k-2p\). Hence, \(\varphi \) is a linear combination of \(\vartheta _x^p \varphi _p\) and a quasimodular form of depth strictly smaller than p. \(\square \)

Proof of Theorem 3.4.1

First observe that \((D+G_2)\langle f\rangle _q = \langle S_2f\rangle _q\,\). As \(D+G_2\) is not a Serre derivative, by Proposition 3.4.2 it follows that it suffices to show that the q-bracket is surjective on modular forms. Every modular form can be written as a polynomial of degree at most 2 in Eisenstein series, see [19, Section 5]. Hence, we show that the q-bracket is surjective on polynomials of degree at most 2 in all Eisenstein series, possibly involving the quasimodular Eisenstein series \(G_2\).

Eisenstein series are in the image of the q-bracket by Theorem 3.3.1. Note that \(DG_{k}\) can be written a polynomial of degree 2 in Eisenstein series, explicitly:

$$\begin{aligned} DG_k \, =\, \frac{k+3}{2(k+1)}G_{k+2}-\sum _{\begin{array}{c} 0<j<k \\ j\equiv 1\, (2) \end{array}}\left( {\begin{array}{c}k\\ j\end{array}}\right) G_{j+1}G_{k+1-j}\, . \end{aligned}$$

Also, we have an explicit formula for the q-bracket of \(S_kS_l\):

$$\begin{aligned} \langle S_{k}S_{l}\rangle _q \, =\, G_kG_l+DG_{k+l-2} \end{aligned}$$
(30)

so that this q-bracket is expressible as a polynomial of degree at most 2 in the Eisenstein series.

Now fix an integer \(m\ge 4\). We consider the Eqs. (30) for all \(k+l=m\). It suffices to show that we can invert these equations, i.e., write \(G_kG_l\) as a linear combination of q-brackets of products of at most two \(S_i\). A direct computation shows that the determinant of the matrix corresponding to the equations above equals

$$\begin{aligned} 1-\sum _{\begin{array}{c} 0<j<m \\ j\equiv 1\, (2) \end{array}}\left( {\begin{array}{c}m\\ j\end{array}}\right) \, =\, 1-2^{m-3}\;<\;0. \end{aligned}$$

Hence, the q-bracket is surjective. \(\square \)

Remark 3.4.3

Only the last step of above proof uses the explicit formula (30) for the derivative of Eisenstein series. The author expects one could conclude the proof by an abstract argument, but he is not aware of such an argument.

The connected product of moment functions

In the second approach we compute the connected product \(S_{k_1}\,|\,\ldots \,|\,S_{k_n}\), which by Proposition 3.2.5 yields the left-hand side of (26) after taking the q-bracket. The result is formulated in Theorem 3.5.4 and depends on two technical lemma’s which we state first.

In order to do so, we start by introducing the following notation. For a partition \(\lambda \) and a subset A of \(\mathbb {N}\), we write \(\lambda |_A\) for the partition where a part of size m occurs \(r_m(\lambda )\) times if \(m\in A\) and does not occur if \(m\not \in A\). For example, \((5,4,3,3,1,1,1)|_{\{4,1\}}=(4,1,1,1).\)

Definition 3.5.1

We say \(f:\mathscr {P}\rightarrow \mathbb {Q}\) is supported on A if \(f(\lambda )=f(\lambda |_A)\) for all partitions \(\lambda \).

The first lemma expresses the induced product of two functions F and G supported on disjoint sets as the pointwise product of these functions, and of two functions F and G supported on the same singleton set as a convolution product of functions.

Lemma 3.5.2

Suppose X and Y are subsets of \(\mathbb {N}\) and \(F,F',G,G':\mathscr {P}\rightarrow \mathbb {Q}\) are supported on XXY and Y, respectively. Then

(i):

\(F\odot F'\) is supported on X;

(ii):

If X and Y are disjoint, then

$$\begin{aligned} FG\odot F'G'\, =\, (F\odot F')(G\odot G'), \quad \quad \text {in particular} \quad \quad F\odot G \, =\, FG; \end{aligned}$$
(iii):

If \(X=Y=\{m\}\), then

$$\begin{aligned} (F\odot G)(\lambda )\, =\, \partial (f{{\,\mathrm{*}\,}}g)(r_m(\lambda )), \end{aligned}$$

where f and g are such that \(F(\lambda )=f(r_m(\lambda ))\)\(G(\lambda )=g(r_m(\lambda ))\).

Proof

By Proposition 3.2.3, we have

$$\begin{aligned} (F\odot F')(\lambda ) \, =\, \sum _{\alpha \cup \beta \cup \gamma =\lambda }(-1)^{\ell (\gamma )}\,F(\alpha )\,F'(\beta ), \end{aligned}$$

where it is understood that \(\gamma \) is a strict partition. We have that

$$\begin{aligned} (F\odot F')(\lambda )&\, =\, \Bigl (\,\sum _{\alpha \cup \beta \cup \gamma =\lambda |_{X}}(-1)^{\ell (\gamma )}\,F(\alpha )\,F'(\beta )\Bigr )\Bigl (\,\sum _{\alpha \cup \beta \cup \gamma =\lambda |_{X^c}}(-1)^{\ell (\gamma )}\Bigr ) \\&\, =\, (F\odot F')(\lambda |_{X})\cdot (1\odot 1)(\lambda |_{X^c}). \end{aligned}$$

Recall \(f\odot 1=f\) for all functions f, hence \((F\odot F')(\lambda ) = (F\odot F')(\lambda |_{X})\), which is the first statement.

Next, we have that

$$\begin{aligned} (FG\odot F'G')(\lambda ) \, =\, \sum _{\alpha \cup \beta \cup \gamma =\lambda } (-1)^{\ell (\gamma )}\, (FG)(\alpha )\, (F'G')(\beta ), \end{aligned}$$

where again it is understood that \(\gamma \) is a strict partition. Using the fact that \(F,F',G\) and \(G'\) are supported on XXY and Y, respectively, we obtain

$$\begin{aligned} (FG\odot F'G')(\lambda ) \, =\, \sum _{\alpha \cup \beta \cup \gamma =\lambda } (-1)^{\ell (\gamma |_X)+\ell (\gamma |_Y)+\ell (\gamma |_{Z})}\,F(\alpha |_X)\,G(\alpha |_Y)\,F'(\beta |_X)\,G'(\beta |_Y), \end{aligned}$$
(31)

where Z denotes the complement of \(X\cup Y\) in \(\mathbb {N}\). We factor the right-hand side of (31) as

$$\begin{aligned} \Bigl (\,\sum _{\alpha \cup \beta \cup \gamma =\lambda |_X} (-1)^{\ell (\gamma )}\,F(\alpha )\,F'(\beta )\Bigr )\Bigl (\,\sum _{\alpha \cup \beta \cup \gamma \!=\!\!\lambda |_Y} (-1)^{\ell (\gamma )}\,G(\alpha )\,G'(\beta )\Bigr )\Bigl (\sum _{\alpha \cup \beta \cup \gamma =\lambda |_{Z}} (-1)^{\ell (\gamma )}\Bigr ). \end{aligned}$$

By definition of the product \(\odot \), we conclude

$$\begin{aligned} (FG\odot F'G')(\lambda ) \!=\! (F\odot F')(\lambda |_X)\,(G\odot G')(\lambda |_Y)\,(1\odot 1)(\lambda |_{Z})\!=\!(F\odot F')(\lambda )\,(G\odot G')(\lambda ). \end{aligned}$$

By taking \(F'\) and G to be the constant function 1 (which is supported on every X and Y), we see that \(F\odot G' =FG'\) is implied by \(FG\odot F'G'=(F\odot F')(G\odot G')\).

Next, for iii we have

$$\begin{aligned} (F\odot G)(\lambda )&= \sum _{\alpha \cup \beta \cup \gamma =\lambda } (-1)^{\ell (\gamma )} f(r_m(\alpha ))\,g(r_m(\beta ))\\&\, =\, \sum _{\alpha \cup \beta \cup \gamma =\lambda |_{\{m\}}} (-1)^{\ell (\gamma )} f(r_m(\alpha ))\,g(r_m(\beta )) \end{aligned}$$

Letting \(i=r_m(\alpha )\) and \(j=r_m(\beta )\), we have

$$\begin{aligned} (F\odot G)(\lambda )&= \sum _{i+j=r_m(\lambda )} f(i)\,g(j)-\sum _{i+j+1=r_m(\lambda )}f(i)\,g(j)\\&\, =\, (f{{\,\mathrm{*}\,}}g)(r_m(\lambda ))-(f{{\,\mathrm{*}\,}}g)(r_m(\lambda )-1)\\&\, =\, \partial (f{{\,\mathrm{*}\,}}g)(r_m(\lambda )). \square \end{aligned}$$

The second lemma is concerned with the vanishing of certain sums of the Möbius functions of set partitions. Given \(\alpha \in \Pi (n)\) and a subset Z of [n], we let

$$\begin{aligned} \alpha |_Z=\{A\cap Z \mid A\in \alpha \text { s.t.~} A\cap Z\ne \emptyset \} \in \Pi (Z), \end{aligned}$$

where \(\Pi (Z)\) denotes the set of all partitions of the set Z. Observe that

$$\begin{aligned} \ell (\alpha )\, =\, \ell (\alpha |_Z)+|\{A\in \alpha \mid A\cap Z=\emptyset \}|, \end{aligned}$$

in particular \(\ell (\alpha |_Z)\le \ell (\alpha )\). Given \(Z\subset [n]\), define an equivalence relation on \(\Pi (n)\) by writing \(\alpha \sim \beta \) if

$$\begin{aligned} \alpha |_Z=\beta |_Z \quad \text {and} \quad \alpha |_{Z^c} = \beta |_{Z^c}. \end{aligned}$$
(32)

Lemma 3.5.3

Let \(Z\subseteq [n]\). If \(Z\ne \emptyset \) and \(Z\ne [n]\), then for all \(\beta \in \Pi (n)\) we have

$$\begin{aligned} \sum _{\alpha \sim \beta }\mu (\alpha ,\mathbf {1})=0. \end{aligned}$$

Proof

Observe that \(\alpha \sim \beta \) precisely if for all \(A\in \alpha \) we have (\(A\cap Z=\emptyset \) or \(A\cap Z\in \beta |_Z\)) and similarly we have (\(A\cap Z^c=\emptyset \) or \(A\cap Z^c\in \beta |_{Z^c}\)). Hence, every \(A\in \alpha \) is the union of some \(A_1\in \alpha |_Z\cup \{\emptyset \}\) and \(A_2\in \alpha |_{Z^c}\cup \{\emptyset \}\) with not both \(A_1=\emptyset \) and \(A_2=\emptyset \). Write \(a=\ell (\beta |_{Z})\)\(b=\ell (\beta |_{Z^c})\), and assume without loss of generality that \(a\le b\). Write k for the number of \(A\in \alpha \) for which both \(A_1\ne \emptyset \) and \(A_2\ne \emptyset \). Now, \(\ell (\alpha )=a+b-k\). Moreover, given kZ and \(\beta \), there are

$$\begin{aligned} \left( {\begin{array}{c}a\\ k\end{array}}\right) \left( {\begin{array}{c}b\\ k\end{array}}\right) k! \end{aligned}$$

ways to choose \(\alpha \sim \beta \) with \(\ell (\alpha )=a+b-k\). Hence, we find

$$\begin{aligned} \sum _{\alpha \sim \beta }\mu (\alpha ,\mathbf {1})&\, =\, \sum _{k=0}^a (-1)^{a+b-k-1}(a+b-k-1)!\, \left( {\begin{array}{c}a\\ k\end{array}}\right) \left( {\begin{array}{c}b\\ k\end{array}}\right) k!\\&\, =\, (-1)^{a+b-1}(a+b-1)!\sum _{k=0}^a \frac{(-a)_k(-b)_k}{(-a-b+1)_k(1)_k}, \end{aligned}$$

where \((d)_k=\prod _{i=0}^{k-1} (d+i)\) is the rising Pochhammer symbol. This expression equals up to the constant \((-1)^{a+b-1}(a+b-1)!\) the special value \({}_2 F_1(-a,-b,-a-b+1;1)\) of the hypergeometric function \({}_2 F_1(-a,-b,-a-b+1;z)\), which vanishes by Gauss’s theorem subject to \(a,b>0\). As \(Z\ne \emptyset \), we have \(a>0\). Also, \(b>0\) as \(Z\ne [n]\). \(\square \)

The following result not only computes the connected product of the moment functions \(S_k\), but also is one of the main technical results needed to prove Theorem 1.1.

Theorem 3.5.4

Let \(k_i,f_i\) for \(i=1,\ldots , n\) be such that (18) defines \(S_{k_i,f_i}\). Then,

(i):

There exists a function \(g:\mathbb {N}\rightarrow \mathbb {Q}\) such that

$$\begin{aligned} S_{k_1,f_1}\,|\,\ldots \,|\,S_{k_n,f_n}\, =\, S_{|\underline{k}|,g}. \end{aligned}$$

In fact,

where \(f_A = \prod _{a\in A} f_{a}\) and  denotes the convolution product (11).

(ii):

If \(f_1(x)=x\), then \(\partial g = f_1\, \partial \tilde{g}\) with \(\tilde{g}\) given by \(S_{k_2,f_2}\,|\,\ldots \,|\,S_{k_n,f_n}=S_{|\underline{k}|,\tilde{g}}.\)

Remark 3.5.5

We extend g by \(g(0)=0\). Here and later in this work, we usually omit the dependence of g on \(f_1,\ldots , f_n\) in the notation.

Proof

For the first part, we let \(\underline{m}^{\underline{k_A}} \, \underline{f_A} \circ r_{\underline{m}}\) denote \(\prod _{i}m_i^{k_{A_i}} \cdot f_{A_i}\circ r_{m_i},\) where \(r_{m_i}\) is considered as a function \(\mathscr {P}\rightarrow \mathbb {Q}\). In case \(n=1\) the result (i) is trivially true, so we assume \(n\ge 2\). By definition of the connected product and \(S_{k,f}\) (see (23) and (18) respectively), we have

$$\begin{aligned} S_{k_1,f_1}\,|\,\ldots \,|\,S_{k_n,f_n}&\, =\, \sum _{\alpha \in \Pi (n)} \mu (\alpha ,\mathbf {1})\bigodot _{A\in \alpha } \,\Bigl (\, \sum _{\underline{m}\in \mathbb {N}^{\ell (A)}} \underline{m}^{\underline{k_A}} \, \underline{f_A} \circ r_{\underline{m}}\Bigr )\nonumber \\&\, =\, \sum _{\underline{m}\in \mathbb {N}^n}\sum _{\alpha \in \Pi (n)} \mu (\alpha ,\mathbf {1})\bigodot _{A\in \alpha } \underline{m_A}^{\underline{k_A}}\, \underline{f_A} \circ r_{\underline{m}}\, . \end{aligned}$$
(33)

For all \(m\ge 0\), the function \(r_{m}:\mathscr {P}\rightarrow \mathbb {Q}\) is supported on \(\{m\}\). Having Lemma 3.5.2 in mind, we aim to factor the functions in (33) as a product of functions supported on a singleton set. Given \(\underline{m}\in \mathbb {N}^n\), we start by all functions supported on \(\{m_1\}\), that is, we let \(Z(\underline{m})=\{i \mid m_i=m_1\}\subset [n]\). Note that \(Z(\underline{m})\) determines all i for which the support of \(r_{m_i}\) contains \(m_1\). Denote by \(E(\underline{m})\) the set of equivalence classes of \(\Pi (n)\) for this choice of \(Z=Z(\underline{m})\). We split the sum over \(\alpha \in \Pi (n)\) in (33) as a sum over the elements of \(E(\underline{m})\), i.e.,

$$\begin{aligned} S_{k_1,f_1}\,|\,\ldots \,|\,S_{k_n,f_n}&\, =\, \sum _{\underline{m}\in \mathbb {N}^n}\sum _{[\beta ]\in E(\underline{m})}\sum _{\alpha \in [\beta ]} \mu (\alpha ,\mathbf {1})\bigodot _{A\in \alpha } \underline{m_A}^{\underline{k_A}} \, \underline{f_A}\circ r_{\underline{m_A}}. \end{aligned}$$
(34)

Then, given \(\underline{m}\in \mathbb {N}^n\)\(Z=Z(\underline{m})\) and \(A\in \alpha |_Z\), the function \(\lambda \mapsto \underline{m_A}^{\underline{k_A}}\, \underline{f_A}(r_{\underline{m_A}}(\lambda ))\) is supported on \(\{m_1\}\), whereas for \(A\in \alpha |_{Z^c}\) the function \(\lambda \mapsto \underline{m_A}^{\underline{k_A}} \underline{f_A}(r_{\underline{m_A}(\lambda )})\) is supported on \(\mathbb {N}\backslash \{m_1\}\). Hence, by Lemma 3.5.2(ii) we find that (34) equals

$$\begin{aligned} \sum _{\underline{m}\in \mathbb {N}^n}\sum _{[\beta ]\in E(\underline{m})}\sum _{\alpha \in [\beta ]} \mu (\alpha ,\mathbf {1})\biggl (\, \bigodot _{A\in \alpha |_Z} \underline{m_A}^{\underline{k_A}} \, \underline{f_A} \circ r_{\underline{m_A}}\biggr ) \biggl (\,\bigodot _{A\in \alpha |_{Z^c}} \underline{m_A}^{\underline{k_A}} \, \underline{f_A}\circ r_{\underline{m_A}}\biggr ). \end{aligned}$$
(35)

Instead of writing the second factor as a product of functions which are all supported on a singleton set, we make the following observation.

As \(\alpha |_Z=\beta |_Z\) and \(\alpha |_{Z^{c}}=\beta |_{Z^c}\), the only dependence on \(\alpha \) in the above equation is in \(\mu (\alpha ,\mathbf {1})\). By construction \(Z(\underline{m})\) is non-empty. Hence, by Lemma 3.5.3 we have that if \(Z\ne [n]\) then for all \(\beta \in E(\underline{m})\) we have \(\sum _{\alpha \in [\beta ]}\mu (\alpha ,\mathbf {1})=0.\) This implies that we can restrict the first sum in (35) to \(\underline{m}\in \mathbb {N}^n\) for which \(m_i=m_j\) for all ij, that is,

$$\begin{aligned} S_{k_1,f_1}\,|\,\ldots \,|\,S_{k_n,f_n}&\, =\, \sum _{m\in \mathbb {N}}\sum _{\alpha \in \Pi (n)} \mu (\alpha ,\mathbf {1})\bigodot _{A\in \alpha } \prod _{a\in A}m^{k_a}\cdot {f_{a}}\circ r_{m}\, . \end{aligned}$$

Applying Lemma 3.5.2iii \(\ell (\alpha )-1\) times and using (12), we obtain the desired result.

For the second part, let \(Z=\{1\}\) and consider an equivalence class \([\beta ]\) for the equivalence relation (32) determined by Z. We split the sum

over all conjugacy classes. Write \(A_1\) for the element of \(\alpha \) for which \(1\in A_1\). Denote \(\hat{A}_1=A_1\backslash \{1\}\) and \(\gamma =\beta |_{\{2,\ldots ,n\}}\). In case \(A_1=\{1\}\) one has by (15) that

(36)

In case \(A_1\ne \{1\}\) (i.e., \(|A_1|\ge 2\)), one finds by (13) that

(37)

As \([\beta ]\) contains one element for which (36) holds and \(\ell (\gamma )\) elements for which (37) holds, one finds

By (12) and (14), this equals

Hence, summing over all conjugacy classes, we obtain

The case when \(f_1(x)=\ldots =f_n(x)=x\) is the easiest example (for arbitrary \(n\in \mathbb {N}\)) of the above result. In this case one generalizes Theorem 3.3.1 by a result which, in accordance with our main principle of establishing identities before the q-bracket, yields this theorem after taking the q-bracket.

Corollary 3.5.6

For all positive even \(k_1,\ldots , k_n\), one has

$$\begin{aligned} S_{k_1}\,|\,\ldots \,|\,S_{k_n}\, =\, S_{|\underline{k}|-n,\mathcal {F}_n}\, . \end{aligned}$$

Proof

Recall \(S_k=S_{k-1,\mathrm {id}}\) and apply Theorem 3.5.4(ii) \({n-1}\) times. \(\square \)

Later we will use Theorem 3.3.1 when the \(f_i\) are Faulhaber polynomials. This is the situation in which we prove the main result of this paper, in which case the following lemma is useful.

Lemma 3.5.7

If \(f_1,\ldots ,f_n\) are Faulhaber polynomials of degrees \(d_1,\ldots , d_n\), respectively, and \({g:\mathbb {N}\rightarrow \mathbb {Q}}\) is as in Theorem 3.5.4, then there exists a polynomial p such that \(\partial g(m)=p(m)\) for all \(m\in \mathbb {N}\). Moreover, p is strictly of degree \(|\underline{d}|-1\), is even or odd and \(p(0)=0\).

Proof

By Theorem 3.5.4(ii) we can assume w.l.o.g. that none of the degrees \(d_i\) equals 1. Now, consider a monomial  in \(\partial g\). Note that both \({{\,\mathrm{*}\,}}\) and \(\partial \) are operators on the space of polynomials, more precisely:

$$\begin{aligned} {{\,\mathrm{*}\,}}:\mathbb {Q}[x]_{\le k}\times \mathbb {Q}[x]_{\le l} \rightarrow \mathbb {Q}[x]_{\le k+l+1} \quad \quad \text {and}\quad \quad \partial :\mathbb {Q}[x]_{\le k} \rightarrow \mathbb {Q}[x]_{\le k-1} \end{aligned}$$

as

$$\begin{aligned} x^k*x^l\, =\, \frac{k!l!}{(k+l+1)!}x^{k+l+1}+O(x^{k+l})\quad \quad \text {and}\quad \quad \partial (x^k)\, =\, kx^{k-1}+O(x^{k-2}). \end{aligned}$$

Hence, the degree of such a monomial is \(|\underline{d}|-1\). Now observe that by the symmetry (16) one has

$$\begin{aligned} \partial f_A(x) \, =\, f_A(x)-f_A(x-1) \, =\, f_A(x)-(-1)^{|A|}f_A(-x). \end{aligned}$$

Therefore, we see that \(\partial f_A\) is even or odd and as the convolution product preserves this property, every monomial is even or odd. By the same arguments \(\partial f_A(0)=0\) and hence the constant term of every monomial vanishes. Therefore, every monomial  in g satisfies the desired properties, so that it remains to show that the leading coefficient does not vanish.

As \(\mathcal {F}_l = \frac{1}{l}x^l+O(x^{l-1})\), the leading coefficient of a monomial as above equals

$$\begin{aligned}&\frac{|\underline{d}|}{\prod _{i}d_i}\frac{\prod _{i=1}^n d_{A_i}!}{|\underline{d}|!}, \end{aligned}$$

where for a set B we have set \(d_{B}=\sum _{b\in B}d_b\). Hence, the leading coefficient of \(\partial g\) equals

$$\begin{aligned} \frac{|\underline{d}|}{\prod _{i}d_i}\cdot \sum _{\alpha \in \Pi (n)} \mu (\alpha ,\mathbf {1})\left( {\begin{array}{c}|\underline{d}|\\ d_{A_1},\ldots ,d_{A_r}\end{array}}\right) ^{-1}, \end{aligned}$$
(38)

where \(\alpha =\{A_1,\ldots ,A_r\}\). Note that this number has the following combinatorial interpretation. Let n balls be given which are colored such that \(d_1\) balls are colored in the first color, \(d_2\) in the second color, etc. Suppose we use the same multiset of colors to additionally mark each ball with a dot (possibly of the same color), that is, \(d_1\) balls are marked with a dot of the first color, \(d_2\) with a dot of the second color, etc. Given a subset C of the set of all colors, it may happen that if we consider all balls colored by the colors of C, all the dots on these balls are colored by the same set of colors C. We then say that the balls are well-colored with respect to C. For example, both the empty set of colors and the set of all possible colors give rise to a well-coloring of balls. If we independently at random color and mark the balls as above, the probability that the balls colored by a subset C are well-colored is \(\left( {\begin{array}{c}|\underline{d}|\\ d_C\end{array}}\right) ^{-1}\). Hence, by applying Möbius inversion the number

$$\begin{aligned}\sum _{\alpha \in \Pi (n)} \mu (\alpha ,\mathbf {1})\left( {\begin{array}{c}|\underline{d}|\\ d_{A_1},\ldots ,d_{A_r}\end{array}}\right) ^{-1}\end{aligned}$$

equals the probability that if we independently at random color and mark the balls as above, there does not exist a proper non-empty subset C of the colors such that the balls colored by C are well-colored. If we mark at least one ball of every color i with color \(i+1\) (modulo n), such a set C cannot exist. Hence, the number (38) is positive, so the polynomial p is strictly of degree \(|\underline{d}|-1\). \(\square \)

Three quasimodular algebras

Introduction

Given integers kl with \(k\ge 0\) and \(l\ge 1\) recall the definition of the double moment functions in (4) by

$$\begin{aligned} T_{k,l}(\lambda ) \, =\, -\frac{B_{k+l}}{2(k+l)}(\delta _{l,1}+\delta _{k,0}) + \sum _{m=1}^\infty m^k \mathcal {F}_l(r_m(\lambda )).~ \end{aligned}$$

Unless stated explicitly, we always assume that

$$\begin{aligned} k\in \mathbb {Z}_{\ge 0}, l\in \mathbb {Z}_{\ge 1}, k+l\in 2\mathbb {Z}. \end{aligned}$$
(39)

Moreover, it turns out to be useful to define \(T_{0,0}\equiv T_{-1,1} \equiv -1\) and \(T_{k,l}\equiv 0\) for other pairs (kl) with \(k<0\) or \(l<1\).

Remark 4.1.1

The double moment functions specialize to the moment functions studied in the previous section whenever \(l=1\), i.e., \(T_{k,1}=S_{k+1}\). Also, as \(\mathcal {F}_l(1)=1\), for a strict partition \(\lambda \) one has \(T_{k,l}(\lambda )=S_k(\lambda )\). Hence, our functions \(T_{k,l}\) can be seen as an extension of the algebra of supersymmetric polynomials, mentioned in the introduction, to functions on all partitions (and not only on strict partitions).

Remark 4.1.2

In case \(k+l\) is odd, the q-bracket of \(T_{k,l}\) does not vanish—in contrast to the shifted symmetric functions for which the q-bracket vanishes for all odd weights. However, the q-bracket of a polynomial involving the double moment functions in both even and odd weights also is a polynomial in the so-called combinatorial Eisenstein series, defined in Definition 7.2.4.

These double moment functions give rise to three different graded algebras, which turn out to be quasimodular (see page 1).

Definition 4.1.3

Define the \(\mathbb {Q}\)-algebras \(\mathcal {S}, \mathrm {Sym}^\odot (\mathcal {S})\) and \(\mathcal {T}\) by the condition that

  • \(\mathcal {S}\) is generated by the moment functions \(S_k\) under the pointwise product;

  • \(\mathrm {Sym}^\odot (\mathcal {S})\) is generated by the elements of \(\mathcal {S}\) under the induced product;

  • \(\mathcal {T}\) is generated by the double moment functions under the pointwise product.

Our main result Theorem 1.1 is slightly refined by the following statement.

Theorem 4.1.4

Let X be any of the algebras \(\mathcal {S},\mathrm {Sym}^\odot (\mathcal {S})\) and \(\mathcal {T}\). Then, X is

  • quasimodular;

  • closed under the pointwise product;

  • closed under the induced product if \(X\ne \mathcal {S}\).

Moreover, the three algebras are related by \(\displaystyle \mathcal {S}\subsetneq \mathrm {Sym}^\odot (\mathcal {S}) \subsetneq \mathcal {T}\).

Remark 4.1.5

Observe that being closed under the pointwise product is not implied by being quasimodular. For example, the algebra \(\mathcal {R}=\langle \, \cdot \,\rangle _q^{-1}(\widetilde{M})\) in Remark 3.2.6 is quasimodular, closed under the induced product and \(\mathcal {T}\subset \mathcal {R}\), but \(\mathcal {R}\) is not closed under the pointwise product [23, Section 9].

In the next section we provide different bases for these algebras: in this way we obtain many examples of functions with a quasimodular q-bracket, and moreover, the study of these bases leads to a proof of Theorem 4.1.4.

Remark 4.1.6

The algebras \(\mathcal {T}\) and \(\Lambda ^*\) are different algebras, as follows from the observation that \(f(\lambda )=(-1)^k f(\lambda ')\) for all \(f\in \Lambda _k^*\), which follows by writing a shifted symmetric polynomial as a symmetric polynomial in the Frobenius coordinates. This does not hold for all \(f\in \mathcal {T}\), as can easily be checked numerically. On the other hand, it is not true that \(f(\lambda )\ne \pm f(\lambda ')\) for all \(f\in \mathcal {T}\), as \(Q_2=T_{1,1}\) with \(Q_k\) defined by Eq. (2). More precisely, one has

$$\begin{aligned} \mathcal {T}\cap \Lambda ^* = \mathbb {Q}[Q_2]. \end{aligned}$$

Namely, if \(f\in \mathcal {T}\cap \Lambda ^*\), consider a strict partition \(\lambda \) (i.e., a partition for which \(r_m(\lambda )\le 1\) for all m). Then, we have that \(f(\lambda )\) is symmetric polynomial in the parts \(\lambda _1,\lambda _2,\ldots \). On the other hand, as \(f\in \Lambda ^*\), it follows that \(f(\lambda )\) is a shifted symmetric polynomial in the parts \(\lambda _1,\lambda _2,\ldots \). The only polynomials of degree d in the variables \(x_i\) that are both symmetric and shifted symmetric are up to a constant given by \(\left( \sum _{i} x_i\right) ^d\), hence \(f\in \mathbb {Q}[Q_2]\).

The basis given by double moment functions

In this section we show that \(\mathcal {T}\) is closed under the induced product. Moreover, we show that \(\mathcal {S}\) and \(\mathrm {Sym}^\odot (\mathcal {S})\) are subalgebras of \(\mathcal {T}\). In the next section, we use these results to define a weight grading on \(\mathcal {T}\). Observe that as a vector space \(\mathcal {T}\) is spanned by the functions \(T_{\underline{k},\underline{l}}\), defined by \(T_{\underline{k},\underline{l}}=\prod _{i} T_{k_i,l_i}\), for all \(\underline{k}, \underline{l}\in \mathbb {Z}^n\) satisfying the conditions (39) for all pairs \((k,l)=(k_i,l_i)\).

Theorem 4.2.1

The algebra \(\mathcal {T}\) is closed under the induced product.

Proof

Observe that

$$\begin{aligned} T_{\underline{k},\underline{l}}\odot T_{\underline{k}',\underline{l}'} \, =\, T_{\underline{k},\underline{l}} T_{\underline{k}',\underline{l}'} -T_{\underline{k},\underline{l}}\,|\,T_{\underline{k}',\underline{l}'}\, . \end{aligned}$$

Hence, it suffices to show that \(T_{\underline{k},\underline{l}}\,|\,T_{\underline{k}',\underline{l}'}\) can be expressed in terms of elements of \(\mathcal {T}\).

By Theorem 3.5.4 and Lemma 3.5.7, we have that an expression of the form:

$$\begin{aligned} T_{k_1,l_1}\,|\,\cdots \,|\,T_{k_n,l_n} \end{aligned}$$

is an element of \(\mathcal {T}\). Proposition 3.2.7 implies that \(f_1f_2\,|\,f_3\,|\,f_4\,|\,\ldots \,|\,f_n\) equals

$$\begin{aligned}&(f_1\,|\,f_2\,|\,\ldots \,|\,f_n) \,+\, \sum _{A\sqcup B=\{3,\ldots , n\}}\bigl ( (f_1\,|\,f_{A_1}\,|\,f_{A_2}\,|\,\ldots )\cdot (f_2\,|\,f_{B_1}\,|\,f_{B_2}\,|\,\ldots )\\&\quad -(f_1\,|\,f_{A_1}\,|\,f_{A_2}\,|\,\ldots )\,\big |\, (f_2\,|\,f_{B_1}\,|\,f_{B_2}\,|\,\ldots )\bigr ). \end{aligned}$$

Hence, by using this proposition recursively, we can replace the pointwise products in \(T_{\underline{k},\underline{l}}\) and \(T_{\underline{k}',\underline{l}'}\) by a linear combination of connected products of double moment functions \(T_{k,l}\), showing that \(T_{\underline{k},\underline{l}}\,|\,T_{\underline{k}',\underline{l}'}\) is an element of \(\mathcal {T}\). \(\square \)

Now, we determine a basis for the three algebras. Let \(\mathcal {T}^\mathrm {mon}\) be the set of all monomials for the pointwise product in \(\mathcal {T}\). Two elements of \(\mathcal {T}^\mathrm {mon}\) are considered to be the same if one can reorder the products so that they agree, for example \(T_{1,1}T_{3,5}\) and \(T_{3,5}T_{1,1}\) are the same function. In other words, every elements of \(\mathcal {T}^\mathrm {mon}\) can be written as \(T_{\underline{k},\underline{l}}\) in a unique way up to commutativity of the (pointwise) product.

Theorem 4.2.2

We have

$$\begin{aligned} \mathcal {S}\subsetneq \mathrm {Sym}^\odot (\mathcal {S}) \subsetneq \mathcal {T}. \end{aligned}$$
(40)

Moreover, a basis for

  • \(\mathcal {T}\) is given by \(\mathcal {T}^\mathrm {mon}\);

  • \(\mathrm {Sym}^\odot (\mathcal {S})\) is given by all \(T_{\underline{k},\underline{l}}\in \mathcal {T}^\mathrm {mon}\) satisfying \(k_i\ge l_i\) for all i;

  • \(\mathcal {S}\) is given by all \(T_{\underline{k},\underline{l}}\in \mathcal {T}^\mathrm {mon}\) satisfying \(l_i=1\) for all i.

Proof

It suffices to prove the second part, as from the stated bases statement (40) follows immediately.

By definition the elements of \(\mathcal {T}^\mathrm {mon}\) generate \(\mathcal {T}\) as a vector space. Hence, it suffices to show that they are linearly independent, i.e., that if

$$\begin{aligned} \sum _{\alpha \in I} c_\alpha T_\alpha (\lambda ) \, =\, 0 \end{aligned}$$
(41)

for all \(\lambda \in \mathscr {P}\), where I is the set of all pairs \((\underline{k},\underline{l})\) up to simultaneous reordering and \(c_\alpha \in \mathbb {Q}\), we have that \(c_\alpha =0\) for all \(\alpha \).

First of all, let \(\lambda =(N_1,N_2)\) and consider (41) as \(N_1\rightarrow \infty \). Note that \(T_{\underline{k},\underline{l}}(\lambda )\) grows as 

$$\begin{aligned} N_1^{|\underline{k}|}+N_2^{k_{\mathrm {min}}}N_1^{|\underline{k}\backslash k_{\mathrm {min}}|} \end{aligned}$$

plus lower-order terms, where \(k_{\mathrm {min}}\) is the smallest of the \(k_i\) in \(\underline{k}\). Hence, \(|\underline{k}|\) should be constant among all \(T_\alpha \) in (41). Moreover, we conclude that \(k_{\mathrm {min}}\) should be constant among all \(T_\alpha \) in (41). Continuing by considering the lower-order terms, we conclude that \(\underline{k}\) is constant among all \(T_\alpha \). Similarly, by instead considering partitions consisting of \(N_1\) times the part 1 and \(N_2\) times the part 2, we conclude that \(\underline{l}\) is constant among all \(T_{\alpha }\). Hence, there is at most one \(\alpha \) with nonzero coefficient \(c_\alpha \). We conclude that \(c_\alpha =0\) for all \(\alpha \in I\).

For \(\mathrm {Sym}^\odot (\mathcal {S})\) we show, first of all, that indeed \(T_{\underline{k},\underline{l}}\in \mathrm {Sym}^\odot (\mathcal {S})\) if \(k_i\ge l_i\) for all i. Let \(k\ge l\) of the same parity be given. By Corollary 3.5.6 we find that

$$\begin{aligned} \underbrace{T_{1,1}\,|\,T_{1,1}\,|\,\ldots \,|\,T_{1,1}}_{l-1}\,|\,T_{k-l+1,1}\, =\, \underbrace{S_2\,|\,S_2\,|\,\ldots \,|\,S_2}_{l-1}\,|\,S_{k-l+2}\, =\, T_{k,l}\, . \end{aligned}$$

Therefore, \(T_{k,l}\in \mathrm {Sym}^\odot (\mathcal {S})\) for all \(k\ge l\). Moreover, by applying Möbius inversion on Eq. (23), which defines the connected product, we find

$$\begin{aligned} T_{\underline{k},\underline{l}} \, =\, \sum _{\alpha \in \Pi (n)}\bigodot _{A\in \alpha }(T_{k_{A_1},l_{A_1}}\,|\,T_{k_{A_2},l_{A_1}}\,|\,\ldots ). \end{aligned}$$
(42)

As we already showed that \(T_{k,l}\in \mathrm {Sym}^\odot (\mathcal {S})\) if \(k\ge l\), we find \(T_{\underline{k},\underline{l}}\in \mathrm {Sym}^\odot (\mathcal {S})\) if \(k_i\ge l_i\) for all i.

Next, we show that all elements in \(\mathrm {Sym}^\odot (\mathcal {S})\) are a linear combination of the \(T_{\underline{k},\underline{l}}\) satisfying \({k_i\ge l_i}\). As \(\mathcal {S}\) clearly is contained in the space generated by the \(T_{\underline{k},\underline{l}}\) for which \(k_i\ge l_i\), it suffices to show that the latter space is closed under \(\odot \). For this we follow the proof of Theorem 4.2.1 observing that in each step \(k_i\ge l_i\), so that indeed the \(T_{\underline{k},\underline{l}}\) for which \(k_i\ge l_i\) form a generating set for \(\mathrm {Sym}^\odot (\mathcal {S})\).

As we already showed that the \(T_{\underline{k},\underline{l}}\) are linearly independent, we conclude that the \(T_{\underline{k},\underline{l}}\in \mathcal {T}^\mathrm {mon}\) satisfying \(k_i\ge l_i\) for all i form a basis for \(\mathrm {Sym}^\odot (\mathcal {S})\).

The last part of the statement follows directly, as by definition all \(T_{\underline{k},\underline{l}}\in \mathcal {T}^\mathrm {mon}\) satisfying \(l_i=1\) for all i generate \(\mathcal {S}\), and by the above they are linearly independent. \(\square \)

The basis defining the weight grading

By definition, the double moment functions generate \(\mathcal {T}\) under the pointwise product. In this section we show that we can replace the pointwise product in the latter statement by the induced product. Again we will consider every reordering of the factors in \(T_{k_1,l_1}\odot \cdots \odot T_{k_n,l_n}\) due to commutativity of the products to be the same element. Then, we have:

Theorem 4.3.1

The elements \(T_{k_1,l_1}\odot \cdots \odot T_{k_n,l_n}\) form a basis for \(\mathcal {T}\). A basis for the subspace \(\mathrm {Sym}^\odot (\mathcal {S})\) is given by the subset of elements for which \(k_i\ge l_i\) for all i.

Proof

Assign to \(T_{k,l}\) weight \({k+l}\). This defines a weight filtering on \(\mathcal {T}\) with respect to the pointwise product. Consider the subspace of elements of weight at most w in \(\mathcal {T}\). The number of basis elements in the basis given by the pointwise product in the previous section equals the number of induced products of the \(T_{k,l}\). Hence, it suffice that the induced products of the \(T_{k,l}\) generate \(\mathcal {T}\). For this we proceed by induction first on the weight and then on the depth. Here, by depth we mean the unique filtering under the pointwise product for which every \(T_{k,l}\) has depth 1, usually called the total depth.

Trivially, every element of weight 0 or depth 0 is generated by (empty) induced products of the \(T_{k,l}\). Next, consider \({T_{\underline{k},\underline{l}}\in \mathcal {T}}\) and assume all elements of lower weight and of the same weight and lower depth are generated by induced product of the \(T_{k,l}\). Let \(T_{\underline{k},\underline{l}}\in \mathcal {T}\) of weight w be given and write \(\underline{k}',\underline{l}'\) for \(\underline{k},\underline{l}\) after omitting the last (nth) entry. Then

$$\begin{aligned} T_{\underline{k},\underline{l}}\, =\, T_{\underline{k}',\underline{l}'}\odot T_{k_n,l_n} - T_{\underline{k}',\underline{l}'}\,|\,T_{k_n,l_n}\, . \end{aligned}$$

Note that \(T_{\underline{k}',\underline{l}'}~\) is of weight strictly less than w, hence is generated by induced products of the \(T_{k,l}\). Moreover, by Proposition 3.2.7 and Theorem 3.5.4 it follows that the depth of  \(T_{\underline{k}',\underline{l}'}\,|\,T_{k_n,l_n}\) is at most \({n-1}\). Hence, by our induction hypothesis, it is generated by induced products of the \(T_{k,l}\). We conclude that \(T_{\underline{k},\underline{l}}\) is generated by induced products of the \(T_{k,l}\), which proves the first part of the theorem.

The second part follows by the same proof, everywhere restricting to those \(T_{k,l}\) for which \({k\ge l}\). \(\square \)

By the above theorem, we can define a weight grading on \(\mathcal {T}\).

Definition 4.3.2

Define a weight grading on \(\mathcal {T}\) by assigning to \(T_{k,l}\) weight \(k+l\) and extending under the induced product.

Note that both the grading on \(\mathcal {T}\) and the grading on \(\mathcal {S}\) correspond to the grading on quasimodular forms after taking the q-bracket. Hence, the grading on \(\mathcal {S}\) is the restriction of the grading on \(\mathcal {T}\).

The weight grading defines a weight operator. In Sect. 5 we extend this weight operator to an \(\mathfrak {sl}_2\)-triple acting on \(\mathcal {T}\), so that \(\mathcal {T}\) becomes an \(\mathfrak {sl}_2\)-algebra.

The n-point functions

As induced products of the \(T_{k,l}\) form a basis for \(\mathcal {T}\), knowing \(\langle f\rangle _q\) for all \(f\in \mathcal {T}\) is equivalent to knowing the following generating function, called the n-point function

$$\begin{aligned} F_n(u_1,\ldots u_n, v_1,\ldots v_n) \, =\, \sum _{\underline{k},\underline{l}} \langle T_{k_1,l_1}\odot \cdots \odot T_{k_n,l_n}\rangle _q \frac{u_1^{k_1}\cdots u_n^{k_n} v_1^{l_1-1} \cdots v_n^{l_n-1}}{k_1!\cdots k_n!(l_1-1)!\cdots (l_n-1)!} \end{aligned}$$

for all \(n\ge 0\). Here the sum is over all \(k_i,l_i\) such that \(k_i+l_i\) is even and m! is consider to be 1 for \(m<0\). As the q-bracket is a homomorphism with respect to the induced product, we directly conclude that

$$\begin{aligned} F_n(\underline{u},\underline{v})\, =\, \prod _{i=1}^n F_1(u_i,v_i). \end{aligned}$$
(43)

We also define the partition function by

$$\begin{aligned} \Phi (\underline{t}) \, =\, \sum _{n=0}^\infty \frac{1}{n!} \sum _{\underline{k},\underline{l}}\langle T_{k_1,l_1}\odot \cdots \odot T_{k_n,l_n}\rangle _q t_{k_1,l_1} \cdots t_{k_n,l_n}\, . \end{aligned}$$

The following result (together with (43)) expresses these functions in terms of the Jacobi theta series (see (8)).

Theorem 4.4.1

For all \(n\ge 0\) one has

$$\begin{aligned} F_1(u,v) \, =\, -\frac{1}{2}\frac{\theta '(0)\theta (u+v)}{\theta (u)\theta (v)}, \quad \quad \Phi (\underline{t}) \, =\, \exp \Bigl ([x^0y^0]F_1\left( \frac{\partial }{\partial x},\frac{\partial }{\partial y}\right) \sum _{k,l}t_{k,l} x^ky^l\Bigr ), \end{aligned}$$

where \([x^0y^0]\) denotes taking the constant coefficient.

Proof

We have that

$$\begin{aligned} F_1(u,v) \, =\, \prod _{i=1}^n \biggl (-\frac{1}{2u}-\frac{1}{2v}+\biggl (\sum _{k,l} D^{l-1} G_{k-l+2}+\sum _{k,l} D^{k} G_{l-k}\biggr )\frac{u^{k}v^{l-1}}{k!(l-1)!}\biggr ), \end{aligned}$$

where in the sum it is understood that \(k+l\) is even, \(k\ge 0, l\ge 1\). The expression for \(F_1(u,v)\) in the statement now follows from [20, \(\S \S 3\)]. The expression for \(\Phi \) follows immediately from this result. \(\square \)

Differential operators

The derivative of a function on partitions

Note that for all \(f\in \mathbb {Q}^{\mathscr {P}}\) one has

$$\begin{aligned} D\langle f \rangle _q \, =\, \langle S_2 f\rangle _q - \langle S_2\rangle _q\langle f \rangle _q\, . \end{aligned}$$
(44)

Hence, by letting \(Df:=S_2\,|\,f=S_2f-S_2\odot f\) for \(f\in \mathbb {Q}^{\mathscr {P}}\), we have that \(D\langle f \rangle _q = \langle Df \rangle _q\, .\) Moreover, D acts as a derivation:

Proposition 5.1.1

The map \(D:\mathbb {Q}^\mathscr {P}\rightarrow \mathbb {Q}^\mathscr {P}\) is an equivariant derivation, i.e., D is linear, satisfies the Leibniz rule and

$$\begin{aligned} D\langle f \rangle _q \, =\, \langle Df\rangle _q. \end{aligned}$$

In fact, for all \(k\ge 1\), the mapping \(f\mapsto S_k\,|\,f\) is a derivation. Recall the definition of the Möbius function \(\mu \) defined in Proposition 3.1.3 and denote \(S_k^0=S_k-S_k(\emptyset )\).

Lemma 5.1.2

For all even \(m\ge 2\) one has

(i):

\(\displaystyle S_m^0\odot \mu = -S_m^0\,\mu \);

(ii):

The mapping \((\mathbb {Q}^\mathscr {P},\odot )\rightarrow (\mathbb {Q}^\mathscr {P},\odot ), f\mapsto S_m\,|\,f\) is a derivation, uniquely determined on \(\mathcal {T}\) by

$$\begin{aligned} \displaystyle S_m\,|\,T_{k,l} \, =\, T_{k+m-1,l+1}\,. \end{aligned}$$

Remark 5.1.3

In case \(m\ge 4\), the derivation \(f\mapsto S_m\,|\,f\) does not correspond to a derivation on \(\widetilde{M}\), i.e., a derivation \(\mathfrak {d}_m\) such that \(\mathfrak {d}_m\langle f\rangle _q = \langle S_m\,|\,f\rangle _q\) for all \(f\in \mathcal {T}\). For instance, although the q-brackets of \(T_{m,m}\) and \(T_{m-1,m+1}\) are the same, the q-brackets of \(S_{m}\,|\,T_{m,m} = T_{2m-1,m+1}\) and \(S_{m}\,|\,T_{m-1,m+1} = T_{2m-2,m+2}\) are different.

Proof

First of all, by Proposition 3.1.4, one has

$$\begin{aligned} \Bigl (\,\sum _{\lambda \in \mathscr {P}}u_\lambda \Bigr )\langle S_k^0\odot \mu \rangle _{\underline{u}} \, =\, \Bigl (\,\sum _{m,r\ge 1} m^{k-1}u_m^r\Bigr )\Bigl (\,\sum _{\lambda \in \mathscr {P}}\mu (\lambda )\,u_{\lambda }\Bigr ). \end{aligned}$$
(45)

Let \(\mathscr {S}_m\) be the set of strict partitions not containing m as a part. Then, we can rewrite (45) as

$$\begin{aligned} \sum _{m}\sum _{\lambda \in \mathscr {S}_m} m^{k-1}\mu (\lambda )\,u_mu_{\lambda }\, =\, -\sum _{\lambda \in \mathscr {P}} S_k^0(\lambda )\,\mu (\lambda )\,u_{\lambda }\, , \end{aligned}$$

since \(\mu (\lambda \cup (m))=-\mu (\lambda )\) for \(\lambda \in \mathscr {S}_m\), so that for \(r\ge 2\) the coefficient of \(u_m^ru_\lambda \) cancels in pairs. We conclude that \(S_k^0\odot \mu = -S_k^0\,\mu \).

For the second part, note that (i) implies that

$$\begin{aligned} S_k\odot \mu \, =\, -\Bigl (S_k+\frac{B_k}{k}\Bigr )\,\mu . \end{aligned}$$

Let \(f,g\in \mathbb {Q}^\mathscr {P}\) be given. Then

$$\begin{aligned} S_k\,|\,(f\odot g)&\, =\, S_k(f\odot g) - S_k\odot f\odot g. \end{aligned}$$

If \(\alpha \cup \beta \cup \gamma =\lambda \) then \(S_k(\lambda ) = S_k(\alpha )+S_k(\beta )+S_k(\gamma )+\frac{B_k}{k}\), hence

$$\begin{aligned} S_k(\lambda )\,(f\odot g)(\lambda )&\, =\, \sum _{\alpha \cup \beta \cup \gamma =\lambda } \Bigl (S_k(\alpha )+S_k(\beta )+S_k(\gamma )+\frac{B_k}{k}\Bigr )f(\alpha )\,g(\beta )\,\mu (\gamma ) \\&\, =\, (S_kf)\odot g \,+\, f\odot (S_kg) \,+\,\sum _{\alpha \cup \beta \cup \gamma =\lambda } \Bigl (S_k(\gamma )+\frac{B_k}{k}\Bigr )f(\alpha )\,g(\beta )\,\mu (\gamma ) \\&\, =\, (S_kf)\odot g \,+\, f\odot (S_kg) \,-\,\sum _{\alpha \cup \beta \cup \gamma =\lambda }(S_k\odot \mu )(\gamma )\,f(\alpha )\,g(\beta ) \\&\, =\, (S_kf)\odot g+ f\odot (S_kg) \,-\,S_k\odot f\odot g. \end{aligned}$$

Therefore,

$$\begin{aligned} S_k\,|\,(f\odot g) \, =\, (S_kf)\odot g+ f\odot (S_kg)-2\,S_k\odot f\odot g \, =\, (S_k\,|\,f)\odot g + f\odot (S_k\,|\,g), \end{aligned}$$

i.e., the mapping \(f\mapsto S_k\,|\,f\) is a derivation. The formula \(S_m\,|\,T_{k,l} = T_{k+m-1,l+1}\) follows directly from Theorem 3.5.4. \(\square \)

Proof of Proposition 5.1.1

As \(S_2\,|\,f = S_2f-S_2\odot f\) is derivation by the above lemma, the results follows directly from (44). \(\square \)

The equivariant q-bracket

In this section we extend the action by the \(\mathfrak {sl}_2\)-triple \((D, \mathfrak {d},W)\) on quasimodular forms to \(\mathcal {T}\). As the derivation \(\mathfrak {d}\) does not act on all power series in q, but only on quasimodular forms, we cannot hope to define \(\mathfrak {d}\) on all functions on partitions as we did with D. On the algebra \(\mathcal {T}\), however, this is possible. We define an \(\mathfrak {sl}_2\)-action on this space and we show that the q-bracket restricted to \(\mathcal {T}\) is an equivariant map of \(\mathfrak {sl}_2\)-algebras.

Note that the following definition agrees with the definition of D in the previous section:

Definition 5.2.1

Define the derivations \(D,W,\mathfrak {d}\) on \(\mathcal {T}\) by

$$\begin{aligned} D\, T_{k,l}&\, =\, T_{k+1,l+1}, \\ W\, T_{k,l}&\, =\, (k+l)T_{k,l}, \\ \mathfrak {d}\,T_{k,l}&\, =\, k(l-1)T_{k-1,l-1}-\tfrac{1}{2}\delta _{k+l-2}. \end{aligned}$$

One immediately checks that DW and \(\mathfrak {d}\) satisfy the commutation relation of an \(\mathfrak {sl}_2\)-triple on \(\mathcal {T}\). The corresponding acting of \(\mathfrak {sl}_2\) on \(\mathcal {T}\) makes the q-bracket equivariant, so that a refined version of Theorem 1.2 is:

Theorem 5.2.2

(The \(\mathfrak {sl}_2\)-equivariant symmetric Bloch–Okounkov theorem) The algebra \(\mathcal {T}\) is an \(\mathfrak {sl}_2\)-algebra with respect to the above action of \(\mathfrak {sl}_2\) on \(\mathcal {T}\). Moreover, the q-bracket becomes an equivariant map of \(\mathfrak {sl}_2\)-algebras, i.e., for \(f \in \mathcal {T}\) one has

$$\begin{aligned} D\langle f \rangle _q = \langle Df\rangle _q, \quad W\langle f \rangle _q = \langle Wf\rangle _q, \quad \mathfrak {d}\langle f \rangle _q = \langle \mathfrak {d}f\rangle _q\,. \end{aligned}$$

Proof

We already observed that the first of the three equality holds and the second is the homogeneity statement. Hence, it suffices to prove the last statement. Using (9) we find that for \(a\ge 0, b\ge 2\) one has

$$\begin{aligned} \mathfrak {d}(D^aG_b) \, =\, a(a+b-1)D^{a-1}G_b-\tfrac{1}{2}\delta _{a+b-2}. \end{aligned}$$

Hence,

$$\begin{aligned} \mathfrak {d}\langle T_{k,l}\rangle _q \, =\, k(l-1)\langle T_{k-1,l-1}\rangle _q-\delta _{k+l-2} \, =\, \langle \mathfrak {d}\,T_{k,l}\rangle _q \end{aligned}$$

and the last statement follows from the Leibniz rule. \(\square \)

Rankin–Cohen brackets

The \(\mathfrak {sl}_2\)-action allows us to define Rankin–Cohen brackets on \(\mathcal {T}\).

Definition 5.3.1

For two elements \(f,g\in \mathcal {T}\) and \(n\ge 0\) the nth Rankin–Cohen bracket is given by

$$\begin{aligned} {[}f,g]_n \, =\, \sum _{\begin{array}{c} r,s\ge 0\\ r+s=n \end{array}}(-1)^r\left( {\begin{array}{c}k+n-1\\ s\end{array}}\right) \left( {\begin{array}{c}l+n-1\\ r\end{array}}\right) D^rf\odot D^sg. \end{aligned}$$
(46)

Note that the formula (46) would have defined the Rankin–Cohen brackets on \(\widetilde{M}\) if D acts by \(q\frac{\partial }{\partial q}\) and the induced product is replaced by the usual product, whereas in this line D acts on \(\mathcal {T}\) as explained in the previous sections.

If \(f,g\in \ker \mathfrak {d}\), then \(\langle f\rangle _q\) and \(\langle g \rangle _q\) are modular forms. The Rankin–Cohen bracket of two modular forms is a modular form; analogously, we have:

Proposition 5.3.2

If \(f,g\in \ker \mathfrak {d}\), then \([f,g]_n\in \ker \mathfrak {d}\).

Proof

Using (9), we find that

$$\begin{aligned} \mathfrak {d}[f,g]_n&\, =\, \sum _{\begin{array}{c} r,s\ge 0\\ r+s=n \end{array}}(-1)^r\frac{(k+n-1)!}{s!(k+r-2)!}\frac{(l+n-1)!}{(r-1)!(k+s-1)!}D^{r-1}f\odot D^sg\\&\quad \,+\, (-1)^r\frac{(k+n-1)!}{(s-1)!(k+r-1)!}\frac{(l+n-1)!}{r!(l+s-2)!}D^rf\odot D^{s-1}g, \end{aligned}$$

where \(\frac{1}{(-1)!}\) should taken to be 0. This is a telescoping sum, vanishing identically. \(\square \)

Remark 5.3.3

The above bracket makes the algebra \(\mathcal {T}\) into a Rankin–Cohen algebra, meaning the following. Let \(A_*=\oplus _{k\ge 0} A_k\) be a graded K-vector space with \(A_0=K\) and \(\dim A_k<\infty \) (for us \(A=\mathcal {T}\)). We say A is a Rankin–Cohen algebra if there are bilinear operations \([\,,\,]_n:A_k \otimes A_l \rightarrow A_{k+l+2n}\ (k, l, n> 0)\) which satisfy all the algebraic identities satisfied by the Rankin–Cohen brackets on \(\widetilde{M}\) [21].

A restricted \(\mathfrak {sl}_2\)-action

Theorem 5.2.2 does not make \(\mathcal {S}\) into an \(\mathfrak {sl}_2\)-algebra. Namely, D does not preserve \(\mathcal {S}\). However, if we allow ourselves to deform the \(\mathfrak {sl}_2\)-triple \((D, \mathfrak {d}, W)\) as in [18], we can define an \(\mathfrak {sl}_2\)-action on \(\mathcal {S}\). This action, however, does not make \(\mathcal {S}\) into an \(\mathfrak {sl}_2\)-algebra, as the deformed operators are not derivations.

The operator taking the role of \(\mathfrak {d}\) is the operator \(\mathfrak {s}:\mathcal {S}_k \rightarrow \mathcal {S}_{k-2}\) defined by

$$\begin{aligned} \mathfrak {s} \, =\, \frac{1}{2}\sum _{k,l\ge 0}(k+l) S_{k+l}\frac{\partial ^2}{\partial S_{k+1}\,\partial S_{l+1}}-\frac{1}{2}\frac{\partial }{\partial S_2}. \end{aligned}$$

The operator D is replaced by multiplication with \(S_2\).

Lemma 5.4.1

The triple \((S_2,\mathfrak {s} ,W-\tfrac{1}{2})\) forms an \(\mathfrak {sl}_2\)-triple of operators acting on \(\mathcal {S}\).

Proof

Observe that

$$\begin{aligned} {[}\mathfrak {s} ,S_2]f\, =\, \sum _{k}(k+1)S_{k+1}\frac{\partial }{\partial S_{k+1}}f-\tfrac{1}{2}f\, =\, (W-\tfrac{1}{2})f. \end{aligned}$$

As \(\mathfrak {s}\) and \(S_2\) decrease, respectively, increasing the weight by 2, the claim follows. \(\square \)

Theorem 5.4.2

The q-bracket \(\langle \, \cdot \, \rangle _q: \mathcal {S}\rightarrow \widetilde{M}\) is an equivariant mapping with respect to the \(\mathfrak {sl}_2\)-triple \({(S_2, \mathfrak {s},W-\tfrac{1}{2})}\) on \(\mathcal {S}\) and the \(\mathfrak {sl}_2\)-triple \((D+G_2, \mathfrak {d}, W-\tfrac{1}{2})\) on \(\widetilde{M}\), i.e., for all \(f\in \mathcal {S}\) one has

$$\begin{aligned} (D+G_2)\langle f\rangle _q \, =\, \langle S_2f\rangle _q, \quad (W-\tfrac{1}{2})\langle f\rangle _q\, =\, \langle (E-\tfrac{1}{2})f\rangle _q, \quad \mathfrak {d}\langle f\rangle _q\, =\, \langle \mathfrak {s} f\rangle _q\,. \end{aligned}$$
(47)

Proof

The first of the three equalities in (47) follows from the definition of the q-bracket; the second is the homogeneity statement of Theorem 5.2.2. Hence, it remains to prove the last equation \(\mathfrak {d}\langle f\rangle _q=\langle \mathfrak {s} f\rangle _q\, .\)

Given \(\underline{k}\in \mathbb {N}^n\), let \(\underline{k}^{i}\in \mathbb {N}^{n-1}\) be given by \(\underline{k}^i:=(k_1,\ldots , k_{i-1},k_{i+1},\ldots , k_n)\) omitting \(k_i\). Similarly, define \(\underline{k}^{i,j}\in \mathbb {N}^{n-2}\) by omitting \(k_i\) and \(k_j\). Then

$$\begin{aligned} \mathfrak {s} S_{\underline{k}} \, =\, \sum _{i\ne j}(k_i+k_j-2) S_{k_i+k_j-2} S_{\underline{k}^{i,j}} \,-\,\frac{1}{2}\sum _{i:\, k_i=2} S_{\underline{k}^{i}}\, . \end{aligned}$$

By Theorem 3.3.1, one finds

For \(I\in \beta \) and \(\underline{l}\in \mathbb {N}^I\), let

$$\begin{aligned} C(I,\underline{l})\, :=\, \sum _{i,j\in I, i\ne j} (l_i+l_j-2) \, =\, (\ell (I)-1)(|\underline{l}|-\ell (I)). \end{aligned}$$

It follows that \(\sum _{i\ne j} (k_i+k_j-2)\left\langle S_{k_i+k_j-2} S_{\underline{k}^{i,j}} \right\rangle _q~\) equals

$$\begin{aligned} \sum _{\beta \in \Pi (n)}\sum _{I\in \beta }2C(I,\underline{k})D^{\ell (I)-2}G_{|k_{I}|-2\ell (I)+2}\prod _{B\ne I} D^{\ell (B)-1}G_{|k_B|-2\ell (B)+2}\,. \end{aligned}$$

On the other hand, observe that if f is of weight \(|\underline{l}|-2\ell (I)+2\), Eq. (9) yields

$$\begin{aligned} {[}\mathfrak {d},D^{\ell (I)-1}]f \, =\, C(I,\underline{l})D^{\ell (I)-2}f. \end{aligned}$$

Hence, using \(\mathfrak {d}G_k \, =\, -\tfrac{1}{2}\delta _{k,2}\), we obtain

$$\begin{aligned} {[}\mathfrak {d},D^{\ell (I)-1}]G_{|k_{I}|-2\ell (I)+2} \, =\, C(B,\underline{k}_I)D^{\ell (I)-2}G_{|k_{I}|-2\ell (I)+2} - \frac{1}{2}\delta _{\underline{k}_I,(2)}\,. \end{aligned}$$

Therefore,

$$\begin{aligned} \mathfrak {d}\langle S_{\underline{k}} \rangle _q&\, =\, \sum _{\beta \in \Pi (n)} \sum _{I\in \beta } C(I,\underline{k}_I)D^{\ell (I)-2}G_{|k_{I}|-2\ell (I)+2}\prod _{B\ne I} D^{\ell (B)-1}G_{|k_B|-2\ell (B)+2} \\&\quad -\frac{1}{2}\sum _{i:\, k_i=2} \sum _{\beta \in \Pi ([n]\backslash \{i\})} \prod _{B\in \beta } D^{\ell (B)-1}G_{|k_B|-2\ell (B)+2}\,, \end{aligned}$$

which by the above reasoning is exactly equal to \(\langle \mathfrak {s} S_{\underline{k}}\rangle _q\,\). \(\square \)

Relating the two products

The structure constants

In Theorem 3.5.4, we deduced that

In the particular case that \(f_1=\ldots =f_n\) is the identity function, we saw in Corollary 3.5.6 that \(g=\mathcal {F}_n\). If \(f_1,\ldots ,f_n\) are Faulhaber polynomials, the function g is not necessarily equal Faulhaber polynomial on all \(m\in \mathbb {N}\), but, by Lemma 3.5.7\(\partial g\) equals some polynomial. Also, using g is uniquely determined by \(\partial g\), the function g equals some polynomial. We expand g as a linear combination of Faulhaber polynomials.

Definition 6.1.1

Given integers \(l_1,\ldots , l_n\), we define the structure constants \(C^{\underline{l}}_i\) by

$$\begin{aligned} g(\mathcal {F}_{l_1},\ldots , \mathcal {F}_{l_n}) \, =\, \sum _{i=0}^{|\underline{l}|-1} C_{i}^{\underline{l}}\, \mathcal {F}_{|\underline{l}|-i}\, . \end{aligned}$$

Observe that \(C_{i}^{\underline{l}}=0\) for odd i, as \(\partial g\) is even or odd. Corollary 3.5.6 is the statement

$$\begin{aligned} C_{i}^{(1,\ldots ,1)} \, =\, {\left\{ \begin{array}{ll} 1 &{} i=0 \\ 0 &{} \text {else.} \end{array}\right. } \end{aligned}$$

More generally, by Theorem 3.5.4(ii) one has \(C^{1,\underline{l}}_i=C^{\underline{l}}_i\), so that w.l.o.g. we can assume \(l_i>1\). In this section, we give an explicit, but involved, formula for these coefficients in terms of Bernoulli numbers and binomial coefficients. In order to do so, for \(l_1,l_2\ge 1\) and \(i\in \mathbb {Z}_{\ge 0}\), we introduce the following numbers:

$$\begin{aligned} \mathcal {B}_i^{l_1,l_2} \, :=\, {\left\{ \begin{array}{ll} \frac{(l_1-1)!(l_2-1)!}{(l_1+l_2-1)!} &{} i=0, \\ \zeta (1-i)\left( (-1)^{l_2}\left( {\begin{array}{c}l_1-1\\ i-l_2\end{array}}\right) +(-1)^{l_1}\left( {\begin{array}{c}l_2-1\\ i-l_1\end{array}}\right) \right)&i>0,\end{array}\right. } \end{aligned}$$

which by [2, Proposition A.10] satisfy

$$\begin{aligned} \sum _{i=0}^{l_1+l_2-2}\mathcal {B}_i^{l_1,l_2}\frac{B_{l_1+l_2-i}}{l_1+l_2-i} = (-1)^{l_1l_2}\frac{B_{l_1+l_2}-B_{l_1}B_{l_2}}{l_1 l_2}. \end{aligned}$$

Note that \(\zeta (1-i)=(-1)^{i+1}\frac{B_i}{i}\) for \(i\ge 1\). The following polynomials can be expressed in terms of these coefficients:

Lemma 6.1.2

For all \(l_1,l_2,\ldots ,l_r\ge 2\) one has the following identities:

(i):

\(\displaystyle \mathcal {F}_{l_1}(x) \, =\, \sum _{i=0}^{\infty } \mathcal {B}_i^{l_1,1} x^{l_1-i}\);

(ii):

\(\displaystyle (\partial \mathcal {F}_{l_1}*\partial \mathcal {F}_{l_2})(x) \, =\, \sum _{i=0}^\infty \mathcal {B}_i^{l_1,l_2}x^{l_1+l_2-i-1}\);

(iii):

\(\displaystyle \partial (\mathcal {F}_{l_1}\cdots \mathcal {F}_{l_r})(x) \, =\, 2\sum _{|\underline{i}|\equiv 1\, (2)} \mathcal {B}_{i_1}^{l_1,1}\cdots \mathcal {B}_{i_r}^{l_r,1}x^{|\underline{l}|-|\underline{i}|}\).

Proof

The first two equations, of which the former is the well-known expansion of the Faulhaber polynomials, follow by considering the corresponding generating series. In order to prove (ii), we let \(n\in \mathbb {N}\) and consider

$$\begin{aligned} \mathcal {G}(n)\, :=\,&\sum _{l_1,l_2=1}^\infty (\partial \mathcal {F}_{l_1}*\partial \mathcal {F}_{l_2})(n) \frac{z_1^{l_1-1}}{(l_1-1)!} \frac{z_2^{l_2-1}}{(l_2-1)!} \\ \, =\,&\sum _{m_1+m_2=n}e^{m_1z_1+m_2z_2}\\ \, =\,&\frac{e^{nz_1}}{e^{z_1-z_2}-1}+\frac{e^{nz_2}}{e^{z_2-z_1}-1}. \end{aligned}$$

As the generating series of the Bernoulli numbers \(\sum _{j=0}^\infty B_j \frac{z^j}{j!} = z(e^z-1)^{-1}\) implies that

$$\begin{aligned} \frac{1}{e^{z_1-z_2}-1} \, =\, \frac{1}{z_1-z_2} + \sum _{j=1}^\infty \sum _{i=0}^{j-1} \frac{B_j}{j}(-1)^{i} \frac{z_1^{j-i-1}z_2^{i}}{(j-1-i)!i!}, \end{aligned}$$

we find

$$\begin{aligned} \mathcal {G}(n)&\, =\, \sum _{k=1}^\infty \biggl (\,\sum _{i=0}^{k-1}\frac{z_1^iz_2^{k-i-1}}{i!(k-i-1)!}+\sum _{j=1}^\infty \sum _{i=0}^{j-1} \frac{B_j}{j}(-1)^{i} \biggl (\frac{z_1^{k+j-i-1}z_2^{i}}{(j-i-1)!i!}+\frac{z_1^iz_2^{k+j-i-1}}{i!(j-i-1)!}\biggr )\biggr )\frac{n^k}{k!} \\&\, =\, \sum _{l_1,l_2=1}^\infty \sum _{i=0}^\infty (-1)^i\mathcal {B}_{i}^{l_1,l_2} \frac{z_1^{l_1-1}}{(l_1-1)!} \frac{z_2^{l_2-1}}{(l_2-1)!} n^{l_1+l_2-i-1}. \end{aligned}$$

Since \(\mathcal {B}_{i}^{l_1,l_2}\) vanishes for odd i if \(l_1,l_2>1\), this proves the second equation. The third equation follows from the first by noting that

$$\begin{aligned} \partial (\mathcal {F}_{l_1}\cdots \mathcal {F}_{l_r})(x)&\, =\, (\mathcal {F}_{l_1}\cdots \mathcal {F}_{l_r})(x)-(-1)^{|\underline{l}|}(\mathcal {F}_{l_1}\cdots \mathcal {F}_{l_r})(-x). \end{aligned}$$

\(\square \)

Using these identities, one obtains.

$$\begin{aligned} C_i^l \, =\, \mathcal {B}^{1,1}_{i}=\delta _{i,0}, \quad C_i^{l_1,l_2} \, =\, \mathcal {B}_i^{l_1,1}+\mathcal {B}_i^{l_2,1}-\mathcal {B}_{i}^{l_1,l_2} \end{aligned}$$

These easy expressions for small n are misleading, as \(6C_i^{l_1,l_2,l_3}\) equals

$$\begin{aligned} \frac{1}{4}\delta _{i,2} \,+\, 3\sum _{\begin{array}{c} i_1,i_2 \equiv 0\, (2) \\ i_1+i_2=i \end{array}} \mathcal {B}_{i_1}^{l_{1},1}\mathcal {B}_{i_2}^{l_{2},1} \,-\,\sum _{\begin{array}{c} i_1 \equiv 1\, (2), j_1 \\ i_1+j_1=i \end{array}} \mathcal {B}_{i_1}^{l_{1},1}\mathcal {B}_{j_1}^{l_{1},l_{2}+l_{3}-i_1} \,+\, 2\sum _{j_1+j_2=i}\mathcal {B}_{j_1}^{l_{1},l_{2}}\mathcal {B}_{j_2}^{l_{1}+l_{2}-j_1,l_{3}} \end{aligned}$$

up to full symmetrization, i.e., summing over all \(\sigma \in S_3\) with \(l_i\) replaced by \(l_{\sigma (i)}\). In general, given \(\alpha \in \Pi (n)\), write \(\alpha =\{A_1,\ldots , A_r\}\) and denote \(A^j=\cup _{i=1}^j A_j\). Also, for a vector \(\underline{k}\) and a set B we let \(k_B=\sum _{b\in B} k_b\). Then, the above observations allows us to write down the following formula, which is very amenable to computer calculation:

Proposition 6.1.3

Let \(l_1,\ldots , l_n>1\). Then,

$$\begin{aligned} C_i^{\underline{l}} \, =\, \sum _{\alpha \in \Pi (n)}\!2^{r}\mu (\alpha ,\mathbf {1}) \!\sum _{\begin{array}{c} i_1,\ldots ,i_n\\ |\underline{i}_A|\equiv 1 \, (2) \end{array}}\left( \prod _{k=1}^n \mathcal {B}_{i_k}^{l_k,1}\right) \! \left( \sum _{\begin{array}{c} j_1,\ldots ,j_{\ell (\alpha )-1}\\ |\underline{i}|+|\underline{j}|=i+r \end{array}} \prod _{s=1}^{r-1} \mathcal {B}^{l_{A^s}-j_{s-1},l_{A_{s+1}}-i_{A_{s+1}}+1}_{j_s}\right) \end{aligned}$$

Here, \(j_0:=l_{A_0}-i_{A_0}\).

Note that the latter formula is written in an asymmetric way, but (by associativity of the convolution product) is symmetric in the \(l_i\).

From the pointwise product to the induced product

Suppose an element of \(\mathcal {T}\) is given, written in the basis with respect to the pointwise product. How do we determine its (possibly mixed) weight and its representation in terms of the basis with respect to the \(\odot \) product? A first answer is given by applying Möbius inversion to Eq. (23), as given by Eq. (42), i.e.,

$$\begin{aligned} T_{\underline{k},\underline{l}} \, =\, \sum _{\alpha \in \Pi (n)}\bigodot _{A\in \alpha }(T_{k_{A_1},l_{A_1}}\,|\,T_{k_{A_2},l_{A_2}}\,|\,\ldots ). \end{aligned}$$
(48)

However, as every factor \(T_{k_{A_1},l_{A_1}}\,|\,T_{k_{A_2},l_{A_2}}\,|\,\ldots \) in the above equation is a linear combination of generators of different weights, it is useful to have a recursive version of this result. For this, we write \(\frac{\partial }{\partial T_{k,l}}\) for the derivative of \(f\in \mathcal {T}\) in the former basis (with respect to the pointwise product) and \(\frac{\partial }{\partial T_{\underline{k},\underline{l}}}\) for \(\prod _i \frac{\partial }{\partial T_{k_i,l_i}}\).

Proposition 6.2.1

Let \(k,l\ge 1\). There exist differential operators \(\mathfrak {s}_{i,j}\) for all \(i,j\in \mathbb {Z}\) such that \(\mathfrak {s}_{i,j}=0\) if \(j<0\) and for all \(f\in \mathcal {T}\) one has

$$\begin{aligned} T_{k,l}f \, =\, \sum _{i\ge 0}\sum _{j\ge -l+1} T_{k+i,l+j} \odot \mathfrak {s}_{i,j}(f). \end{aligned}$$

Explicitly,

$$\begin{aligned} \mathfrak {s}_{i,j} \, =\, \sum _{|\underline{a}|=i} \mathfrak {t}_{\underline{a},j}, \quad \quad \mathfrak {t}_{\underline{a},j} \, =\, \sum _{\underline{b}} C^{l,\underline{b}}_{|\underline{b}|-j} \frac{\partial }{\partial T_{\underline{a},\underline{b}}}, \end{aligned}$$

where \(\underline{a}\) and \(\underline{b}\) are vectors of integers of the same length and with \(|\underline{a}|=i\), the structure constants \(C^{l,\underline{b}}_{|\underline{b}|-j}\) are as in Proposition 6.1.3 and \({l,\underline{b}}\) denotes the vector \((l,b_1,b_2,\ldots )\).

Proof

By linearity, it suffices to prove the statement for monomials \(T_{\underline{k},\underline{l}}\). Hence, assume \(f=T_{\underline{k},\underline{l}}\). Applying  (48), extracting the factor containing \(T_{k,l}\) and applying (48) again, yields

$$\begin{aligned} T_{k,l}f&\, =\, \sum _{A\subset [n]}(T_{k,l}\,|\,T_{k_{A_1},l_{A_1}}\,|\,T_{k_{A_2},l_{A_2}}\,|\,\ldots )\odot T_{\underline{k}_{[n]\backslash A},\underline{l}_{[n]\backslash A}} \\&\, =\, \sum _{\underline{a},\underline{b}}(T_{k,l}\,|\,T_{a_1,b_1}\,|\,T_{a_2,b_2}\,|\,\ldots )\odot \frac{\partial }{\partial T_{\underline{a},\underline{b}}} f. \end{aligned}$$

By Definition 6.1.1 this equals

$$\begin{aligned} T_{k,l}f&\, =\, \sum _{\underline{a},\underline{b}}\sum _{j\in \mathbb {Z}} C_{j}^{l,\underline{b}} T_{|\underline{a}|+k,|\underline{b}|+l-j} \odot \frac{\partial }{\partial T_{\underline{a},\underline{b}}} f \end{aligned}$$

Replacing j by \(-j+|\underline{b}|\) and writing \(i=|\underline{a}|\), one obtains

$$\begin{aligned} T_{k,l}f&\, =\, \sum _{i\ge 0}\sum _{j\in \mathbb {Z}} T_{k+i,l+j} \odot \sum _{|\underline{a}|=i}\sum _{\underline{b}}C_{|\underline{b}|-j}^{l,\underline{b}}\frac{\partial }{\partial T_{\underline{a},\underline{b}}} f, \end{aligned}$$

as desired. \(\square \)

Corollary 6.2.2

For all \(k,l\ge 1\) and \(f\in \mathcal {T}\) one has

$$\begin{aligned} \langle T_{k,l}f \rangle _q \, =\, \sum _{a\ge 0}\sum _{b\ge 2} D^{a}G_{b} \langle \mathfrak {T}_{k,l}^{a,b}f\rangle _q\, , \end{aligned}$$

where \(\mathfrak {T}_{k,l}^{a,b}=\mathfrak {s}_{a-l+1,a+b-k-1}+\mathfrak {s}_{a+b-l,a-k}\).

Proof

Distinguishing two cases in the previous result yields

$$\begin{aligned} \langle T_{k,l}f\rangle _q&\, =\, \sum _{j< k+i-l} D^{l+j-1}G_{k+i-l-j+2} \odot \mathfrak {s}_{i,j}(f) \,+\, \sum _{i\ge 0}\sum _{j\ge k+i-l} D^{k+i}G_{l+j-k-i} \odot \mathfrak {s}_{i,j}(f)\\&\, =\, \sum _{a\ge 0}\sum _{b\ge 2} D^{a}G_{b} \langle (\mathfrak {s}_{a+b-k-1,a-l+1}+\mathfrak {s}_{a-k,a+b-l})(f)\rangle _q\,. \end{aligned}$$

\(\square \)

Related functions on partitions

We apply our results to interesting functions on partitions.

Hook-length moments

First of all, we focus on the hook-length moments \(H_k\) [5, part III]. These functions form a bridge between the symmetric algebra studied in this note and the shifted symmetric functions: The \(H_k\) themselves are shifted symmetric as

$$\begin{aligned} H_k(\lambda ) \, =\, \frac{1}{2}\sum _{i=0}^{k} \left( {\begin{array}{c}k-2\\ i-1\end{array}}\right) (-1)^i Q_i(\lambda )Q_{k-i}(\lambda ) \end{aligned}$$
(49)

and they are also equal to the Möller transform of the symmetric \(S_k\), i.e., \(H_k=\mathcal {M}(S_k)\), meaning the following. Denote \(z_\nu = \frac{n!}{|C_\nu |}\) with \(|C_\nu |\) the size of the conjugacy class corresponding to \(\nu \). Recall that

$$\begin{aligned} z_\nu \, =\, \prod _{m=1}^\infty m^{r_m(\nu )}r_m(\nu )!. \end{aligned}$$

Given \(f\in \mathbb {Q}^{\mathscr {P}}\), the Möller transform of f at a partition \(\lambda \in \mathscr {P}(n)\) is given by [23, Eqn (45)]

$$\begin{aligned} \mathcal {M}(f)(\lambda ) \, =\, \sum _{\nu \vdash n} z_\nu ^{-1}\chi ^\lambda (\nu )^2 f(\nu ), \end{aligned}$$

where the sum \(\nu \vdash n\) is over all partitions of size n and \(\chi ^\lambda (\rho )\) denotes the character of the representation corresponding to the partition \(\lambda \) evaluated at the conjugacy class corresponding to \(\rho \). Then \(\langle \mathcal {M}(f)\rangle _q\) is a quasimodular form if and only if \(\langle f \rangle _q\) is a quasimodular form (which follows directly by the column orthogonality relations for the symmetric group). In the next section, we study the Möller transform of elements of \(\mathcal {T}\), but first, we explain the Murnaghan–Nakayama rule, used in [5, part III] to show equality between \(\mathcal {M}(S_k)\) and (49) and give two other expressions for the hook-length moments.

To start with the latter, the hook-length moments, as their name suggests, are defined as moments of the hook-lengths, i.e.,

$$\begin{aligned} H_k(\lambda ) \, =\, -\frac{B_k}{2k} + \sum _{\xi \in Y_\lambda } h(\xi )^{k-2}, \end{aligned}$$

where \(Y_\lambda \) denotes the Young diagram of a partition \(\lambda \) and \(h(\xi )\) denotes the hook-length of a cell \(\xi \in Y_\lambda \).

Next, the following constructions related to the Young diagram, give rise to the Murnaghan–Nakayama rule for the characters of the symmetric group. Given partitions \(\lambda ,\nu \) with \(\nu _i\le \lambda _i\) for all i, we define the skew Young diagram \(\lambda /\nu \) by removing the cells of \(Y_\nu \) from \(Y_\lambda \). Denote by \(|\lambda /\nu |=|\lambda |-|\nu |\) the number of cells of this diagram. We call \(\lambda /\nu \) a border strip of \(\lambda \) if it is connected (through edges, not only through vertices) and contains no \(2\times 2\)-block. If \(\gamma =\lambda /\nu \) we write \(\lambda \setminus \gamma \) for \(\nu \). The height of a border strip \(\gamma \) is defined to be one less than the number of columns and denoted by \({{\,\mathrm{ht}\,}}(\gamma )\). Given \(\underline{m}\in \mathbb {N}^s\), we let a border strip tableau \(\gamma \) of type \(\underline{m}\) be a sequence \(\gamma _1,\ldots , \gamma _s\) such that \(\gamma _i\) is a border strip of \(\lambda \smallsetminus \gamma _1\smallsetminus \cdots \smallsetminus \gamma _{i-1}\) and \(|\gamma _i|=m_i\). Write \(Y_\gamma \) for the skew Young diagram consisting of all boxes of all the \(\gamma _i\) and write \({{\,\mathrm{ht}\,}}(\gamma )={{\,\mathrm{ht}\,}}(\gamma _1)+\ldots +{{\,\mathrm{ht}\,}}(\gamma _s)\). Denote by \({{\,\mathrm{BST}\,}}(\lambda ,\underline{m})\) and \({{\,\mathrm{BST}\,}}(\lambda /\nu ,\underline{m})\) the set of all border strip tableau of type \(\underline{m}\) within \(\lambda \) and \(\lambda /\nu \), respectively (Fig. 1).

Fig. 1
figure1

The Young diagrams corresponding to the border strip tableaux of type (2, 1, 2) within \({\lambda =(4,2,1,1)}\)

The Murnaghan–Nakayama rule (recursively) expresses the characters of the symmetric groups in terms the heights of border strip tableau. Namely, if \(\rho '\subseteq \rho \) (both \(\rho '\) and \(\rho \) considered as multisets)

$$\begin{aligned} \chi ^\lambda (\rho ) \, =\, \sum _{\gamma \in {{\,\mathrm{BST}\,}}(\lambda ,\rho ')} (-1)^{{{\,\mathrm{ht}\,}}(\gamma )} \chi ^{\lambda \backslash \gamma }(\rho -\rho '), \end{aligned}$$

where \(\rho -\rho '\) denotes the difference of (multi)sets. Of particular interest are the cases \(\rho '=\rho \) and \(\rho '=(\rho _1)\), yielding a direct or recursive combinatorial formula for \(\chi ^\lambda (\rho )\), respectively:

$$\begin{aligned} \chi ^\lambda (\rho )\, =\, \sum _{\gamma \in {{\,\mathrm{BST}\,}}(\lambda ,\rho )} (-1)^{{{\,\mathrm{ht}\,}}(\gamma )} \qquad \text {and}\qquad \chi ^\lambda (\rho ) = \sum _{|\gamma |=\rho _1} (-1)^{{{\,\mathrm{ht}\,}}(\gamma )} \,\chi ^{\lambda \backslash \gamma }(\rho _2,\rho _3,\ldots ), \end{aligned}$$

where the latter sum is over all borders strips \(\gamma \) of \(\lambda \) of length \(\rho _1\). The skew character \(\chi ^{\lambda /\nu }(\rho ')\) is defined by (\(|\lambda /\nu |=|\rho '|\))

$$\begin{aligned} \chi ^{\lambda /\nu }(\rho ')\, =\, \sum _{\begin{array}{c} \gamma \in {{\,\mathrm{BST}\,}}(\lambda /\nu ,\rho ') \end{array}} (-1)^{{{\,\mathrm{ht}\,}}(\gamma )} \end{aligned}$$

so that

$$\begin{aligned} \chi ^\lambda (\rho ) \, =\, \sum _{|\nu |=|\rho '|}\chi ^{\lambda /\nu }(\rho ')\,\chi ^{\nu }(\rho -\rho '). \end{aligned}$$

To conclude, we have the following definitions of the hook-length moments:

Definition 7.1.1

The hook-length moments \(H_k\) (\(k\ge 2\) even) are defined by either of the following equivalent definitions [5, Section 13]:

  1. (i)

    \(\displaystyle H_k(\lambda ) \, =\, -\frac{B_k}{2k} + \sum _{\xi \in Y_\lambda } h(\xi )^{k-2}\,;\)

  2. (ii)

    \(\displaystyle H_k(\lambda ) \, =\, -\frac{B_k}{2k} + \sum _{m=1}^\infty |\!{{\,\mathrm{BST}\,}}(\lambda ,m)|\, m^{k-2}\,;\)

  3. (iii)

    \(\displaystyle H_k \, =\, \frac{1}{2}\sum _{i=0}^{k} \left( {\begin{array}{c}k-2\\ i-1\end{array}}\right) (-1)^i Q_i\,Q_{k-i}\,;\)

  4. (iv)

    \(\displaystyle H_k \, =\, \mathcal {M}(S_k).\)

Border strip moments

The hook-length moments are Möller transformations of the \(S_k\). In this section we study the Möller transformation of the algebra \(\mathcal {T}\), which contains the vector space spanned by all the \(S_k\). In order to do so, we express elements of \(\mathcal {T}\) in terms of functions \(U_{\underline{k},\underline{l}}\) for which the induced product and Möller transformation are easy to compute. However, these function do not admit the property that the q-bracket is quasimodular if \(k_i+l_i\) is even for all i: each \(U_{\underline{k},\underline{l}}\) lies in the space generated by all the \(T_{\underline{k},\underline{l}}\) (possibly with \(k_i+l_i\) odd).

Let

$$\begin{aligned} \mathbb {N}(\underline{l}) \, =\, \{(\underbrace{m_1,\ldots , m_1}_{l_1},\underbrace{m_2,\ldots ,m_2}_{l_2},\ldots ) \mid m_i\ge 1\} \end{aligned}$$

the set of tuples of \(n:=|\underline{l}|\) positive integers, where the first \(l_1\), the second \(l_2\), etc., integers agree. For \(\underline{k}\in \mathbb {Z}_{\ge 0}^n\), define

$$\begin{aligned} U_{\underline{k},\underline{l}} \, =\, \sum _{\underline{m}\in \mathbb {N}(\underline{l})} \underline{m}^{\underline{k}}\prod _{a=1}^\infty \left( {\begin{array}{c}r_a(\lambda )\\ r_a(\underline{m})\end{array}}\right) . \end{aligned}$$

Observe that this product converges since \(r_a(\underline{m})=0\) for all but finitely many values of a. Let \(\mathcal {U}\) be the algebra generated by the \(U_{\underline{k},\underline{l}}\).

Generalize the hook-length moments in Definition 7.1.1(ii) by the following notion:

Definition 7.2.1

The border strip moments are given by

$$\begin{aligned} X_{\underline{k},\underline{l}}(\lambda ) \, =\, \sum _{\underline{m}\in \mathbb {N}(\underline{l})}\sum _{\gamma \in {{\,\mathrm{BST}\,}}(\lambda ,\underline{m})} \frac{\chi ^{\gamma }(\underline{m})^2}{z_{\underline{m}}}\, \underline{m}^{\underline{k}}\, . \end{aligned}$$

Let \(\mathcal {X}\) be the vector space spanned by all the \(X_{\underline{k},\underline{l}}\). Define a filtration on \(\mathcal {X}\) by assigning to \(X_{\underline{k},\underline{l}}\) degree \(|\underline{k}|+|\underline{l}|\).

Remark 7.2.2

Observe that for \(n=1\) and \(l=1\), the sum restricts to a sum over all border strips \(\gamma \) of \(\lambda \) and for such a border strip \(\gamma \) the factor \(\chi ^{\gamma }(\underline{m})^2\) equals 1 and \(z_m\) equals m. As the set of hook-lengths is in bijection with the set of all border strip lengths, one has that \(-\frac{B_k}{2k}+X_{k,1} = H_{k+1}\).

Denote by  the Stirling numbers of the second kind (i.e., the number of elements in \(\Pi (n)\) of length j).

Proposition 7.2.3

For all \(k\ge 0, l\ge 1, \underline{k},\underline{k}'\in \mathbb {Z}_{\ge 0}^n,\) and integer vectors \(\underline{l},\underline{l}'\) with \(|\underline{l}|=|\underline{l}'|=n,\) one has

(i):
(ii):

\(\displaystyle U_{\underline{k},\underline{l}}\odot U_{\underline{k}',\underline{l}'} \, =\, U_{\underline{k}\cup \underline{k}',\underline{l}\cup \underline{l}'}\, ;\)

(iii):

\(\displaystyle \mathcal {M}(U_{\underline{k},\underline{l}}) \, =\, X_{\underline{k},\underline{l}}\, .\)

Proof

For the first property, we use the known identity

As \(\mathcal {F}_l(x)\) and \(\left( {\begin{array}{c}x\\ j\end{array}}\right) \) are the unique polynomials with constant term equal to zero and such that \(\partial \mathcal {F}_l(x)=x^{l-1}\) and \(\partial \left( {\begin{array}{c}x\\ j\end{array}}\right) = \left( {\begin{array}{c}x-1\\ j-1\end{array}}\right) \), respectively, we find

which yields property (i).

Next, we show that for all \(i,j\ge 0\) one has

$$\begin{aligned} \partial \left( {\begin{array}{c}x\\ i\end{array}}\right) {{\,\mathrm{*}\,}}\left( {\begin{array}{c}x\\ j\end{array}}\right) \, =\, \left( {\begin{array}{c}x\\ i+j\end{array}}\right) . \end{aligned}$$

Both

$$\begin{aligned} \left( {\begin{array}{c}x\\ i\end{array}}\right) {{\,\mathrm{*}\,}}\left( {\begin{array}{c}x\\ j\end{array}}\right) \, =\, \sum _{m=0}^{x}\left( {\begin{array}{c}m\\ i\end{array}}\right) \left( {\begin{array}{c}x-m\\ j\end{array}}\right) \quad \quad \text {and}\quad \quad \left( {\begin{array}{c}x+1\\ i+j+1\end{array}}\right) \end{aligned}$$

are polynomials of degree at most \(i+j+1\) taking the value 0 for \(x=0,1,\ldots ,i+j-1\) and the value 1 for \(x=i+j\); hence, they are equal. Therefore,

$$\begin{aligned} \partial \left( {\begin{array}{c}x\\ i\end{array}}\right) {{\,\mathrm{*}\,}}\left( {\begin{array}{c}x\\ j\end{array}}\right) \, =\, \partial \left( {\begin{array}{c}x+1\\ i+j+1\end{array}}\right) =\left( {\begin{array}{c}x\\ i+j\end{array}}\right) . \end{aligned}$$

By Lemma 3.5.2 property (ii) follows.

Finally, we have that

$$\begin{aligned} \mathcal {M}(U_{\underline{k},\underline{l}})(\lambda ) \, =\, \sum _{\underline{m}\in \mathbb {N}(\underline{l})}\underline{m}^{\underline{k}} \sum _{\nu \vdash n}z_\nu ^{-1}\chi ^\lambda (\nu )^2 \prod _{a=1}^\infty \left( {\begin{array}{c}r_a(\nu )\\ r_a(\underline{m})\end{array}}\right) . \end{aligned}$$

Observe that given \(\underline{m}\) and \(\nu \) the term

$$\begin{aligned} z_\nu ^{-1}\chi ^\lambda (\nu )^2 \prod _{a=1}^\infty \left( {\begin{array}{c}r_a(\nu )\\ r_a(\underline{m})\end{array}}\right) \end{aligned}$$
(50)

vanishes unless \(r_{a}(\nu )\ge r_a(\underline{m})\) for all positive a. Let \(\nu '\) be the partition obtained from \(\nu \) by removing \(r_a(\underline{m})\) parts of size a from \(\nu \) for all positive a. Denote by \(n'=n-|\underline{m}|\) the size of \(\nu '\). By the Murnaghan–Nakayama rule one has

$$\begin{aligned} \chi ^\lambda (\nu ) \, =\, \sum _{\xi \in \mathrm {BS}(\lambda ,\underline{m})} \chi ^{\xi }(\underline{m}) \, \chi ^{\lambda \backslash \xi }(\nu '). \end{aligned}$$

One has

$$\begin{aligned} z_\nu ^{-1}\prod _{a=1}^\infty \left( {\begin{array}{c}r_{a}(\nu )\\ r_{a}(\underline{m})\end{array}}\right) \, =\, \prod _{a=1}^\infty \frac{1}{a^{r_a(\nu )}r_a(\underline{m})!\,(r_a(\nu )-r_a(\underline{m})) !} \, =\, \frac{1}{z_{\nu '}z_{\underline{m}}}. \end{aligned}$$

Hence, (50) equals

$$\begin{aligned} \sum _{\xi \in \mathrm {BS}(\lambda ,\underline{m})}\sum _{\rho \in \mathrm {BS}(\lambda ,\underline{m})} \frac{\chi ^{\xi }(\underline{m})\,\chi ^{\rho }(\underline{m})}{z_{\underline{m}}} \sum _{\nu '\vdash n'} z_{\nu '} \chi ^{\lambda \backslash \xi }(\nu ')\,\chi ^{\lambda \backslash \rho }(\nu '). \end{aligned}$$

The orthogonality relation for the symmetric group is the statement

$$\begin{aligned} \sum _{\nu '\vdash n'} z_{\nu '} \,\chi ^{\lambda \backslash \xi }(\nu ')\,\chi ^{\lambda \backslash \rho }(\nu ')\, =\, \delta _{\lambda \backslash \xi ,\lambda \backslash \rho }\, . \end{aligned}$$

Hence, we obtain the desired result. \(\square \)

The q-bracket of an element in \(\mathcal {X}\) is not necessarily a quasimodular form. However, it always lies in the following space of q-analogues of zeta values, see [11].

Definition 7.2.4

Let \(\mathcal {C}_{\le \ell }\) be the \(\mathbb {Q}\)-vector space consisting of all polynomials in the combinatorial Eisenstein series

$$\begin{aligned} G_k(\tau ) \, =\, -\frac{B_k}{2k}+\sum _{r=1}^\infty \sum _{m=1}^\infty m^{k-1} q^{mr} \quad \quad (k\ge 1, \text { not necessarily even}) \end{aligned}$$

and their derivatives of weight \(\le \ell \), where to \(D^rG_k\) we assign the weight \(k+2r\).

Now, Proposition 7.2.3 implies the following result:

Theorem 7.2.5

For all \(f\in \mathcal {X}_{\le k}\), one has \(\langle f \rangle _q \in \mathcal {C}_{\le k}.\)

Proof

By Proposition 7.2.3f equals the Möller transform of some polynomial in the \(T_{k,l}\) with respect to the product \(\odot \). Here, however, it may happen that \(k+l\) is odd. Mutatis mutandis in either of three approaches in Sect. 3.1, we find that the q-bracket of \(T_{k,l}\) lies in \(\mathcal {C}_{k+l}\), which proves the result. \(\square \)

Theorem 7.2.6

For all weights k one has \(\mathcal {M}(\mathcal {T}_k)\subset \mathcal {X}_{\le k}\). More precisely,

$$\begin{aligned} \frac{\mathcal {M}(T_{k_1,l_1}\odot \cdots \odot T_{k_n,l_n})}{(l_1-1)!\cdots (l_n-1)!} \, =\, X_{\underline{k},\underline{l}} + \text {elements in } \mathcal {X} \text { of lower degree}. \end{aligned}$$
(51)

Proof

Observe that Proposition 7.2.3 implies that \(\mathcal {M}(\mathcal {T}_k)\subset \mathcal {X}_{\le k}\). Equation (51) follows from this proposition after noting that the Möller transformation of \(T_{k,l} - (l-1)! U_{k,l}\) has degree strictly smaller than \(k+l\). \(\square \)

Example 7.2.7

The following two equations provide examples of linear combinations of elements of \(\mathcal {X}\) with a quasimodular q-bracket whenever \({k+l}\) and \(k_i\) are even integers.

See Appendix A for a table of elements in \(\mathcal {X}\) with quasimodular q-bracket and of small degree.

Remark 7.2.8

In many examples the \(X_{\underline{k},\underline{l}}\) are not shifted symmetric functions or generated by shifted symmetric functions under the induced product. For example, \(\mathcal {M}(T_{0,2}) \ne \mathcal {M}(S_2)\) and besides \(Q_2=\mathcal {M}(S_2)=S_2\) there are no other non-trivial functions generated by \(\Lambda ^*\) under the pointwise product. It remains an open question whether the elements of \(\mathcal {X}\) are in some sense related to shifted symmetric functions.

Moments of other partition invariants

So far we provided many examples of functions on partitions in \(\Lambda ^*\) and \(\mathcal {T}\) related to the representation theory of the symmetric group. Now, we see that many purely combinatorial notions lead to different bases for \(\mathcal {S}\). We compare these bases to corresponding bases of \(\Lambda ^*\). Most of these bases take the following form. Suppose an index set I and a sequence \(\{s_i\}_{i\in I}^\infty \) of elements of \(\mathbb {Q}^\mathscr {P}\) are given. Then, we define the kth moment of \(\underline{s}\) by (whenever this sum converges)

$$\begin{aligned} M_k(\underline{s})(\lambda ) \, =\, \sum _{i\in I} \Bigl (s_i(\lambda )^k-s_i(\emptyset )^k\Bigr ). \end{aligned}$$

For example, let the functions \(\underline{p},\underline{q}\) for the index set \(\mathbb {N}\) be given by

$$\begin{aligned} p_i(\lambda )=\lambda _i\, ,\quad \quad q_i(\lambda )=\lambda _i-i. \end{aligned}$$

Then, by definition,

$$\begin{aligned} S_k \, =\, S_k(\emptyset )+M_{k-1}(\underline{p}),\quad \quad Q_k \, =\, Q_k(\emptyset )+M_{k-1}(\underline{q}). \end{aligned}$$

Note that by definition \(M_k(\underline{s})(\emptyset )=0\). As the functions below will not respect the weight grading anyway, we will not include a constant term.

The sequences \(\underline{a},\underline{c},\underline{h},\underline{x}\) of functions on partitions are of further interest. Define these sequence, indexed by \(\xi =(i,j)\in \mathbb {Z}_{\ge 0}^2\), by 0 if \(\xi \not \in Y_\lambda \) and

if \(\xi \in Y_\lambda \). For \(\underline{h}\) and \(\underline{c}\) it is known that the corresponding moment functions are shifted symmetric, for the latter see [15, Theorem 4]. The moment functions corresponding to \(\underline{a}\) and \(\underline{x}\) turn out to be equal and to be elements of \(\mathcal {S}\).

Theorem 7.3.1

$$\begin{aligned} \mathcal {S}\, =\, \mathbb {Q}[M_k(\underline{a}) \mid k\ge 0 \text { even}] \, =\, \mathbb {Q}[M_k(\underline{x}) \mid k\ge 0 \text { even}]. \end{aligned}$$

Proof

As the Faulhaber polynomials \(\mathcal {F}_k\) with k odd form a basis for the space of all odd polynomials, the functions

$$\begin{aligned} \sum _{i=1}^\infty \mathcal {F}_k(\lambda _i) \, =\, \sum _{i=1}^\infty \sum _{a=1}^{\lambda _i} a^{k-1} \end{aligned}$$

generate \(\mathcal {S}\), which corresponds to the first equality in the statement. By interchanging the sums one obtains

$$\begin{aligned} \sum _{i=1}^\infty \mathcal {F}_k(\lambda _i) \, =\, \sum _{a=1}^\infty a^{k-1}\sum _{m=a}^\infty r_m(\lambda ) \, =\, \sum _{(i,j)\in Y_\lambda } i^{k-1}. \end{aligned}$$
(52)

Hence, the result is also true for \(\underline{s}=\underline{x}\). \(\square \)

Remark 7.3.2

Note that for a given i the number of \((i,j)\in Y_\lambda \) equals \(\lambda _i'\), where \(\lambda '\) is the conjugate partition of \(\lambda \). Hence, (52) can be written as

$$\begin{aligned} \sum _{i=1}^\infty i^{k-1}\lambda _i' \end{aligned}$$

and consequently these functions for k odd generate \(\mathcal {S}\). Note that these functions are different from the \(S_k(\lambda ')\). In fact, the algebra generated by the \(S_k(\lambda ')\) is distinct from the algebra \(\mathcal {S}\), in contrast to the algebra of shifted symmetric functions, for which \(Q_k(\lambda ')=(-1)^kQ_k(\lambda )\).

Notes

  1. 1.

    In that work the hook-length moment \(H_p\) (see also Sect. 7.1) was denoted by \(T_{p-1}\).

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Acknowledgements

I am very grateful for the conversations with and feedback received from both my supervisors Gunther Cornelissen and Don Zagier.

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Correspondence to Jan-Willem M. van Ittersum.

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Table of double moment functions up to weight 4

Table of double moment functions up to weight 4

For all basis elements \(f\in \mathcal {T}_{\le 4}\) in the basis provided by Theorem 4.3.1, we compute its representation in the basis consisting of double moment function and the quantities \(\langle f\rangle _{\underline{u}}, \langle f \rangle _q, D(f), \mathfrak {d}(f)\) and \(\mathcal {M}(f)\).

Weight at most 2

Weight 4

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van Ittersum, JW.M. A symmetric Bloch–Okounkov theorem. Res Math Sci 8, 19 (2021). https://doi.org/10.1007/s40687-021-00253-8

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Keywords

  • Modular forms
  • Eisenstein series
  • Partitions
  • Symmetric functions
  • Möbius inversion
  • Border strip tableaux