Abstract
We obtain a condition describing when the quasimodular forms given by the Bloch–Okounkov theorem as q-brackets of certain functions on partitions are actually modular. This condition involves the kernel of an operator \(\Delta \). We describe an explicit basis for this kernel, which is very similar to the space of classical harmonic polynomials.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Given a family of quasimodular forms, the question which of its members are modular often has an interesting answer. For example, consider the family of theta series
given by all homogeneous polynomials \(P\in \mathbb {Z}[x_1,\ldots , x_r]\). The quasimodular form \(\theta _P\) is modular if and only if P is harmonic (i.e. \(P\in \ker \sum _{i=1}^r \frac{\partial ^2}{\partial x_i^2}\)) [10]. (As quasimodular forms were not yet defined, Schoeneberg only showed that \(\theta _P\) is modular if P is harmonic. However, for every polynomial P it follows that \(\theta _P\) is quasimodular by decomposing P as in Formula (1).) Also, for every two modular forms f, g, one can consider the linear combination of products of derivatives of f and g given by
This linear combination is a quasimodular form which is modular precisely if it is a multiple of the Rankin–Cohen bracket \([f,g]_n\) [4, 9]. In this paper, we provide a condition to decide which member of the family of quasimodular forms provided by the Bloch–Okounkov theorem is modular. Let \(\mathscr {P}\) denote the set of all partitions of integers and \(|\lambda |\) denote the integer that \(\lambda \) is a partition of. Given a function \(f:\mathscr {P}\rightarrow \mathbb {Q}\), define the q-bracket of f by
The celebrated Bloch–Okounkov theorem states that for a certain family of functions \(f:\mathscr {P}\rightarrow \mathbb {Q}\) (called shifted symmetric polynomials and defined in Sect. 2) the q-brackets \(\langle f \rangle _q\) are the q-expansions of quasimodular forms [2].
Besides being a wonderful result, the Bloch–Okounkov theorem has many applications in enumerative geometry. For example, a special case of the Bloch–Okounkov theorem was discovered by Dijkgraaf and provided with a mathematically rigorous proof by Kaneko and Zagier, implying that the generating series of simple Hurwitz numbers over a torus are quasimodular [5, 7]. Also, in the computation of asymptotics of geometrical invariants, such as volumes of moduli spaces of holomorphic differentials and Siegel–Veech constants, the Bloch–Okounkov theorem is applied [3, 6].
Zagier gave a surprisingly short and elementary proof of the Bloch–Okounkov theorem [13]. A corollary of his work, which we discuss in Sect. 3, is the following proposition:
Proposition 1
There exists actions of the Lie algebra \(\mathfrak {sl}_2\) on both the algebra of shifted symmetric polynomials \(\varLambda ^*\) and the algebra of quasimodular forms \(\widetilde{M}\) such that the q-bracket \(\langle \cdot \rangle _q: \varLambda ^*\rightarrow \widetilde{M}\) is \(\mathfrak {sl}_2\)-equivariant.
The answer to the question in the title is provided by one of the operators \(\Delta \) which defines this \(\mathfrak {sl}_2\)-action on \(\varLambda ^*\). Namely letting \(\mathcal {H}=\ker \Delta |_{\varLambda ^*}\), we prove the following theorem:
Theorem 1
Let \(f\in \varLambda ^*\). Then \(\langle f \rangle _q\) is modular if and only if \(f=h+k\) with \(h\in \mathcal {H}\) and \(k\in \ker \langle \cdot \rangle _q\).
The last section of this article is devoted to describing the graded algebra \(\mathcal {H}\). We call \(\mathcal {H}\) the space of shifted symmetric harmonic polynomials, as the description of this space turns out to be very similar to the space of classical harmonic polynomials. Let \(\mathcal {P}_d\) be the space of polynomials of degree d in \(m\ge 3\) variables \(x_1,\ldots ,x_m\), let \(||x||^2=\sum _{i}x_i^2\), and recall that the space \(\mathscr {H}_d\) of degree d harmonic polynomials is given by \(\ker \sum _{i=1}^r \frac{\partial ^2}{\partial x_i^2}\). The main theorem of harmonic polynomials states that every polynomial \(P\in \mathcal {P}_d\) can uniquely be written in the form
with \(h_i\in \mathscr {H}_{d-2i}\) and \(d'=\lfloor d/2\rfloor \). Define \(K\), the Kelvin transform, and \(D^\alpha \) for \(\alpha \) an m-tuple of non-negative integers by
An explicit basis for \(\mathscr {H}_d\) is given by
see for example [1]. We prove the following analogous results for the space of shifted symmetric polynomials:
Theorem 2
For every \(f\in \varLambda ^*_n\) there exists unique \(h_{i}\in \mathcal {H}_{n-2i}\) (\(i=0,1,\ldots , n'\) and \(n'={\lfloor \tfrac{n}{2}\rfloor }\)) such that
where \(Q_2\) is an element of \(\varLambda ^*_2\) given by \(Q_2(\lambda )=|\lambda |-\frac{1}{24}\). \(\square \)
Theorem 3
The set
is a vector space basis of \(\mathcal {H}_n\), where \(\mathrm {pr}, K\), and \(\Delta _\lambda \) are defined by (4), Definition 4, respectively, Definition 6.
The action of \(\mathfrak {sl}_2\) given by Proposition 1 makes \(\varLambda ^*\) into an infinite-dimensional \(\mathfrak {sl}_2\)-representation for which the elements of \(\mathcal {H}\) are the lowest weight vectors. Theorem 2 is equivalent to the statement that \(\varLambda ^*\) is a direct sum of the (not necessarily irreducible) lowest weight modules
2 Shifted symmetric polynomials
Shifted symmetric polynomials were introduced by Okounkov and Olshanski as the following analogue of symmetric polynomials [8]. Let \(\varLambda ^*(m)\) be the space of rational polynomials in m variables \(x_1,\ldots , x_m\) which are shifted symmetric, i.e. invariant under the action of all \(\sigma \in \mathfrak {S}_m\) given by \(x_i\mapsto x_{\sigma (i)}+i-\sigma (i)\) (or more symmetrically \(x_i-i\mapsto x_{\sigma (i)}-\sigma (i)\)). Note that \(\varLambda ^*(m)\) is filtered by the degree of the polynomials. We have forgetful maps \(\varLambda ^*(m)\rightarrow \varLambda ^*({m-1})\) given by \(x_m\mapsto 0\), so that we can define the space of shifted symmetric polynomials \(\varLambda ^*\) as \(\displaystyle \varprojlim _m \varLambda ^*(m)\) in the category of filtered algebras. Considering a partition \(\lambda \) as a non-increasing sequence \((\lambda _1,\lambda _2,\ldots )\) of non-negative integers \(\lambda _i\), we can interpret \(\varLambda ^*\) as being a subspace of all functions \(\mathscr {P}\rightarrow \mathbb {Q}\).
One can find a concrete basis for this abstractly defined space by considering the generating series
for every \(\lambda \in \mathscr {P}\) (the constant \(\tfrac{1}{2}\) turns out to be convenient for defining a grading on \(\varLambda ^*\)). As \(w_\lambda (T)\) converges for \(T>1\) and equals
one can define shifted symmetric polynomials \(Q_i(\lambda )\) for \(i\ge 0\) by
The first few shifted symmetric polynomials \(Q_i\) are given by
The \(Q_i\) freely generate the algebra of shifted symmetric polynomials, i.e. \(\varLambda ^*=\mathbb {Q}[Q_2,Q_3,\ldots ]\). It is believed that \(\varLambda ^*\) is maximal in the sense that for all \(Q:\mathscr {P}\rightarrow \mathbb {Q}\) with \(Q\not \in \varLambda ^*\) it holds that \(\langle \varLambda ^*[Q]\rangle _q \not \subseteq \widetilde{M}\).
Remark 1
The space \(\varLambda ^*\) can equally well be defined in terms of the Frobenius coordinates. Given a partition with Frobenius coordinates \((a_1,\ldots , a_r,b_1,\ldots , b_r)\), where \(a_i\) and \(b_i\) are the arm and leg lengths of the cells on the main diagonal, let
Then
where \(\beta _k\) is the constant given by
We extend \(\varLambda ^*\) to an algebra where \(Q_1\not \equiv 0\). Observe that a non-increasing sequence \((\lambda _1,\lambda _2,\ldots )\) of integers corresponds to a partition precisely if it converges to 0. If, however, it converges to an integer n, Eqs. (2) and (3) still define \(Q_k(\lambda )\). In fact, in this case
by [13, Proposition 1] where \(\varvec{\partial } Q_0=0\), \(\varvec{\partial } Q_k = Q_{k-1}\) for \(k\ge 1\), and \(\lambda -n=(\lambda _1-n,\lambda _2-n,\ldots )\) corresponds to a partition (i.e. converges to 0). In particular, \(Q_1(\lambda )=n\) equals the number the sequence \(\lambda \) converges to. We now define the Bloch–Okounkov ring \(\mathcal {R}\) to be \(\varLambda ^*[Q_1]\), considered as a subspace of all functions from non-increasing eventually constant sequences of integers to \(\mathbb {Q}\). It is convenient to work with \(\mathcal {R}\) instead of \(\varLambda ^*\) to define the differential operators \(\Delta \) and more generally \(\Delta _\lambda \) later. Both on \(\varLambda ^*\) and \(\mathcal {R}\), we define a weight grading by assigning to \(Q_i\) weight i. Denote the projection map by
We extend \(\langle \cdot \rangle _q\) to \(\mathcal {R}\).
The operator \(E=\sum _{m=0}^\infty Q_m \frac{\partial }{\partial Q_{m}}\) on \(\mathcal {R}\) multiplies an element of \(\mathcal {R}\) by its weight. Moreover, we consider the differential operators
Let \(\Delta =\tfrac{1}{2}(\mathscr {D}-\varvec{\partial }^2)\), i.e.
In the following (antisymmetric) table, the entry in the row of operator A and column of operator B denotes the commutator [A, B], for proofs see [13, Lemma 3].
Definition 1
A triple (X, Y, H) of operators is called an \(\mathfrak {sl}_2\)-triple if
Let \(\hat{Q}_2:=Q_2-\tfrac{1}{2}Q_1^2\) and \(\hat{E}:=E-Q_1\varvec{\partial }-\frac{1}{2}.\) The following result follows by a direct computation using the above table:
Proposition 2
The operators \((\hat{Q}_2,\Delta ,\hat{E})\) form an \(\mathfrak {sl}_2\)-triple. \(\square \)
For later reference, we compute \([\Delta ,Q_2^n]\). This could be done inductively by noting that \([\Delta ,Q_2^n] = Q_2^{n-1}[\Delta ,Q_2]+[\Delta ,Q_2^{n-1}]Q_2\) and using the commutation relations in the above table. The proof below is a direct computation from the definition of \(\Delta \).
Lemma 1
For all \(n\in \mathbb {N}\), the following relation holds
Proof
Let \(f\in \mathbb {Q}[Q_1,Q_2]\), \(g\in \mathscr {R}\), and \(n\in \mathbb {N}\). Then
\(\square \)
3 An \(\mathfrak {sl}_2\)-equivariant mapping
The space of quasimodular forms for \(\mathrm {SL}_2(\mathbb {Z})\) is given by \(\widetilde{M} =\mathbb {Q}[P,Q,R]\), where P, Q, and R are the Eisenstein series of weight 2, 4, and 6, respectively (in Ramanujan’s notation). We let \(\widetilde{M}^{(\le p)}_k\) be the space of quasimodular forms of weight k and depth \(\le p\) (the depth of a quasimodular form written as a polynomial in P, Q, and R is the degree of this polynomial in P). See [12, Section 5.3] or [13, Section 2] for an introduction into quasimodular forms.
The space of quasimodular forms is closed under differentiation, more precisely the operators \(D=q\frac{d}{dq}\), \(\mathfrak {d}=12\frac{\partial }{\partial P}\), and the weight operator W given by \(Wf=kf\) for \(f\in \widetilde{M}_k\) preserve \(\widetilde{M}\) and form an \(\mathfrak {sl}_2\)-triple. In order to compute the action of D in terms of the generators P, Q, and R, one uses the Ramanujan identities
In the context of the Bloch–Okounkov theorem, it is more natural to work with \(\hat{D} := D - \frac{P}{24}\), as for all \(f\in \varLambda ^*\) one has \(\langle Q_2 f\rangle _q = \hat{D} \langle f \rangle _q\). Moreover, \(\hat{D}\) has the property that it increases the depth of a quasimodular form by 1, in contrast to D for which \(D(1)=0\) does not have depth 1:
Lemma 2
Let \(f\in \widetilde{M}\) be of depth r. Then \(\hat{D}f\) is of depth \(r+1\).
Proof
Consider a monomial \(P^aQ^bR^c\) with \(a,b,c\in \mathbb {Z}_{\ge 0}\). By the Ramanujan identities, we find
where \(O(P^{a})\) denotes a quasimodular form of depth at most a. The lemma follows by noting that \(\frac{a}{12}+\frac{b}{3}+\frac{c}{2}-\frac{1}{24}\) is non-zero for \(a,b,c\in \mathbb {Z}\). \(\square \)
Moreover, letting \(\hat{W}=W-\frac{1}{2}\), the triple (\(\hat{D}, \mathfrak {d}, \hat{W})\) forms an \(\mathfrak {sl}_2\)-triple as well. With respect to these operators, the q-bracket becomes \(\mathfrak {sl}_2\)-equivariant. The following proposition is a detailed version of Proposition 1:
Proposition 3
(The \(\mathfrak {sl}_2\)-equivariant Bloch–Okounkov theorem) The mapping \(\langle \cdot \rangle _q: \mathcal {R} \rightarrow \widetilde{M}\) is \(\mathfrak {sl}_2\)-equivariant with respect to the \(\mathfrak {sl}_2\)-triple \((\hat{Q}_2, \Delta ,\hat{E})\) on \(\mathcal {R}\) and the \(\mathfrak {sl}_2\)-triple \((\hat{D}, \mathfrak {d}, \hat{W})\) on \(\widetilde{M}\), i.e. for all \(f\in \mathcal {R}\), one has
Proof
This follows directly from [13, Equation (37)] and the fact that for all \(f\in \mathcal {R}\) one has \(\langle Q_1 f \rangle _q=0\). \(\square \)
4 Describing the space of shifted symmetric harmonic polynomials
In this section, we study the kernel of \(\Delta \). As \([\Delta ,Q_1]=0\), we restrict ourselves without loss of generality to \(\varLambda ^*\). Note, however, that \(\Delta \) does not act on \(\varLambda ^*\) as, for example, \(\Delta (Q_3)=-\tfrac{1}{2}Q_1\). However, \(\mathrm {pr}\Delta \) does act on \(\varLambda ^*\).
Definition 2
Let
be the space of shifted symmetric harmonic polynomials.
Proposition 4
If \(f\in Q_2\varLambda ^*\) is non-zero, then \(f \not \in \mathcal {H}\).
Proof
Write \(f=Q_2^n f'\) with \(f'\in \varLambda ^*\) and \(f'\not \in Q_2\varLambda ^*\). Then
by Lemma 1. As \(f'\) is not divisible by \(Q_2\), it follows that \(\mathrm {pr}\Delta (f) =0\) precisely if \(f'=0\). \(\square \)
Proposition 5
For all \(n\in \mathbb {Z}\), one has
Proof
For uniqueness, suppose \(f=Q_2g+h\) and \(f=Q_2g'+h'\) with \(g,g'\in \varLambda _{n-2}^*\) and \(h,h' \in \mathcal {H}_n\). Then, \(Q_2(g-g')=h'-h\in \mathcal {H}\). By Proposition 4 we find \(g=g'\) and hence \(h=h'\).
Now, define the linear map \(T:\varLambda _n^*\rightarrow \varLambda _n^*\) by \(f\mapsto \mathrm {pr}\Delta (Q_2f).\) By Proposition 4 we find that T is injective, which by finite dimensionality of \(\varLambda _n^*\) implies that T is surjective. Hence, given \(f\in \varLambda _{n}^*\) let \(g\in \varLambda _{n-2}^*\) be such that \(T(g)=\mathrm {pr}\Delta (f) \in \varLambda _{n-2}^*\). Let \(h=f-Q_2g\). As \(f=Q_2g+h\), it suffices to show that \(h\in \mathcal {H}\). That holds true because \(\mathrm {pr}\Delta (h)=\mathrm {pr}\Delta (f)-\mathrm {pr}\Delta (Q_2g)=0\). \(\square \)
Proposition 5 implies Theorem 2 and the following corollary. Denote by p(n) the number of partitions of n.
Corollary 1
The dimension of \(\mathcal {H}_n\) equals the number of partitions of n in parts of size at least 3, i.e.
Proof
Observe that \(\dim \varLambda _n^*\) equals the number of partitions of n in parts of size at least 2. Hence, \(\dim \varLambda _n^*=p(n)-p(n-1)\) and the Corollary follows from Proposition 5. \(\square \)
Proof of Theorem 1
If \(\langle f \rangle _q\) is modular, then \(\langle \Delta f \rangle _q=\mathfrak {d}\langle f \rangle _q =0\). Write \(f=\sum _{r=0}^{n'} Q_2^rh_{r}\) as in Theorem 2 with \(n'=\lfloor \tfrac{n}{2}\rfloor \). Then by Lemma 1 it follows that \(\mathrm {pr}\Delta f = \sum _{r=0}^{n'} r(n-r-\frac{3}{2})Q_2^{r-1}h_r.\) Hence,
As \(\langle h_r\rangle _q\) is modular, either it is equal to 0 or it has depth 0. Suppose the maximum m of all \(r\ge 1\) such that \(\langle h_r\rangle _q\) is non-zero exists. Then, by Lemma 2 it follows that the left-hand side of (7) has depth \(m-1\), in particular is not equal to 0. So, \(h_1,\ldots , h_{n'}\in \ker \langle \cdot \rangle _q\). Note that \(f\in \ker \langle \cdot \rangle _q\) implies that \(Q_2f\in \ker \langle \cdot \rangle _q\). Therefore, \(k:=\sum _{r=1}^{n'} Q_2^rh_{r} \in \ker \langle \cdot \rangle _q\) and \(f=h+k\) with \(h=h_0\) harmonic.
The converse follows directly as \(\mathfrak {d}\langle h+k\rangle _q = \mathfrak {d}\langle h \rangle _q=\langle \Delta h \rangle _q = 0.\)\(\square \)
Remark 2
A description of the kernel of \(\langle \cdot \rangle _q\) is not known.
Another corollary of Proposition 5 is the notion of depth of shifted symmetric polynomials which corresponds to the depth of quasimodular forms:
Definition 3
The space \(\varLambda _k^{*(\le p)}\) of shifted symmetric polynomials of depth \(\le p\) is the space of \(f\in \varLambda _k^*\) such that one can write
with \(h_r\in \mathcal {H}_{k-2r}\).
Theorem 4
If \(f\in \varLambda _k^{*(\le p)}\), then \(\langle f \rangle _q \in \widetilde{M}_k^{(\le p)}\).
Proof
Expanding f as in Definition 3 we find
By Lemma 2, we find that the depth of \(\langle f\rangle _q\) is at most p. \(\square \)
Next, we set up notation to determine the basis of \(\mathcal {H}\) given by Theorem 3. Let \(\tilde{\mathcal {R}}=\mathcal {R}[Q_2^{-1/2}]\) and \(\tilde{\varLambda }=\varLambda ^*[Q_2^{-1/2}]\) be the formal polynomial algebras graded by assigning to \(Q_k\) weight k (note that the weights are—possibly negative—integers). Extend \(\Delta \) to \(\tilde{\varLambda }\) and observe that \(\Delta (\tilde{\varLambda })\subset \tilde{\varLambda }\). Also extend \(\mathcal {H}\) by setting
Definition 4
Define the partition-Kelvin transform\(K:\tilde{\varLambda }_n\rightarrow \tilde{\varLambda }_{3-n}\) by
Note that \(K\) is an involution. Moreover, f is harmonic if and only if K(f) is harmonic, which follows directly from the computation
Example 1
As \(K(1)=Q_2^{3/2}\), it follows that \(Q_2^{3/2}\in \tilde{\mathcal {H}}\).
Definition 5
Given \(\underline{i}\in \mathbb {Z}_{\ge 0}^n\), let
Define the nth order differential operators \(\mathscr {D}_n\) on \(\tilde{\mathcal {R}}\) by
where the coefficient is a multinomial coefficient.
This definition generalises the operators \(\varvec{\partial }\) and \(\mathscr {D}\) to higher weights: \(\mathscr {D}_1=\varvec{\partial }\), \(\mathscr {D}_2=\mathscr {D}\), and \(\mathscr {D}_n\) reduces the weight by n.
Lemma 3
The operators \(\{\mathscr {D}_n\}_{n\in \mathbb {N}}\) commute pairwise.
Proof
Set \(I=|\underline{i}|\) and \(J=|\underline{j}|\). Let \(\underline{a}^{\hat{k}}=(a_1,\ldots ,a_{k-1},a_{k+1},\ldots , a_n)\). Then
Hence, \([\mathscr {D}_n,\mathscr {D}_m]\) is a linear combination of terms of the form \(Q_{|\underline{a}|+1} \partial _{\underline{a}},\) where \( \underline{a}\in \mathbb {Z}_{\ge 0}^{n+m-1}\). We collect all terms for different vectors \(\underline{a}\) which consists of the same parts (i.e. we group all vectors \(\underline{a}\) which correspond to the same partition). Then, the coefficient of such a term equals
Hence, \([\mathscr {D}_n,\mathscr {D}_m]=0\). \(\square \)
It does not hold true that \([\mathscr {D}_n,Q_1]=0\) for all \(n\in \mathbb {N}\). Therefore, we introduce the following operators:
Definition 6
Let
For \(\lambda \in \mathscr {P}\) let
(Note that \(\Delta _0=\mathscr {D}_0=1\), so this is in fact a finite product.)
Remark 3
By Möbius inversion
The first three operators are given by
Proposition 6
The operators \(\Delta _\lambda \) satisfy the following properties: for all partitions \(\lambda ,\lambda '\)
-
(a)
the order of \(\Delta _{|\lambda |}\) is \(|\lambda |;\)
-
(b)
\([\Delta _\lambda ,\Delta _{\lambda '}]=0;\)
-
(c)
\([\Delta _\lambda ,Q_1]=0\).
Proof
Property (a) follows by construction and (b) is a direct consequence of Lemma 3. For property (c), let \(f\in \tilde{\varLambda }\) be given. Then
Observe that by the identity
the sum in the last line is a telescoping sum, equal to zero. Hence \(\Delta _n(Q_1f)=Q_1\Delta _n(f)\) as desired. \(\square \)
In particular, the above proposition yields \([\Delta _\lambda , \Delta ]=0\) and \([\Delta _\lambda ,\mathrm {pr}]=0\).
Denote by \((x)_{n}\) the falling factorial power\((x)_n = \prod _{i=0}^{n-1} (x-i)\) and for \(\lambda \in \mathscr {P}_n\) define \(Q_\lambda =\prod _{i=1}^\infty Q_{\lambda _i}.\) Let
Observe that \(h_\lambda \) is harmonic, as \(\mathrm {pr}\Delta \) commutes with \(\mathrm {pr}\) and \(\Delta _\lambda \).
Proposition 7
For all \(\lambda \in \mathscr {P}_n\) there exists an \(f\in \varLambda ^*_{n-2}\) such that
Proof
Note that the left-hand side is an element of \(\varLambda ^*\) of which the monomials divisible by \(Q_2^i\) correspond precisely to terms in \(\Delta _\lambda \) involving precisely \(n-i\) derivatives of \(K(1)\) to \(Q_2\). Hence, as \(\Delta _\lambda \) has order n all terms not divisible by \(Q_2\) correspond to terms in \(\Delta _\lambda \) which equal \(\frac{\partial ^{n} }{\partial Q_2^{n}}\) up to a coefficient. There is only one such term in \(\Delta _{\lambda }\) with coefficient \(\left( {\begin{array}{c}|\lambda |\\ \lambda _1,\ldots ,\lambda _r\end{array}}\right) \lambda _1!\ldots \lambda _r!Q_{\lambda }\). \(\square \)
For \(f\in \mathcal {R}\), we let \(f^\vee \) be the operator where every occurrence of \(Q_i\) in f is replaced by \(\Delta _i\). We get the following unusual identity:
Corollary 2
If \(h\in \mathcal {H}_n\), then
Proof
By Proposition 7, we know that the statement holds true up to adding \(Q_2 f\) on the right-hand side for some \(f\in \varLambda ^*_{n-2}\). However, as both sides of (9) are harmonic and the shifted symmetric polynomial \(Q_2f\) is harmonic precisely if \(f=0\) by Proposition 4, it follows that \(f=0\) and (9) holds true. \(\square \)
Proof of Theorem 3
Let \(\mathcal {B}_n=\{h_\lambda \mid \lambda \in \mathscr {P}_n \text { all parts are } \ge 3\}\). First of all, observe that by Corollary 1 the number of elements in \(\mathcal {B}_n\) is precisely the dimension of \(\mathcal {H}_n\). Moreover, the weight of an element in \(\mathcal {B}_n\) equals \(|\lambda |=n\). By Proposition 7 it follows that the elements of \(\mathcal {B}_n\) are linearly independent harmonic shifted symmetric polynomials. \(\square \)
References
Axler, S., Bourdon, P., Wade, R.: Harmonic Function Theory. Graduate Texts in Mathematics, vol. 137, 2nd edn. Springer, New York (2011)
Bloch, S., Okounkov, A.: The character of the infinite wedge representation. Adv. Math. 149(1), 1–60 (2000)
Chen, D., Möller, M., Zagier, D.: Quasimodularity and large genus limits of Siegel–Veech constants. J. Am. Math. Soc. 31(4), 1059–1163 (2018)
Cohen, H.: Sums involving the values at negative integers of \(L\)-functions of quadratic characters. Math. Ann. 217(3), 271–285 (1975)
Dijkgraaf, R.: Mirror symmetry and elliptic curves. In: Dijkgraaf, R., Faber, C., van der Geer, G. (eds.) The Moduli Space of Curves (Texel Island, 1994) , volume 129 of Progress-Mathematics, pp. 149–163. Birkha̋user Boston (1995)
Eskin, A., Okounkov, A.: Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math. 145(1), 59–103 (2001)
Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms. In: Dijkgraaf, R., Faber, C., van der Geer, G. (eds.) The Moduli Space of Curves (Texel Island, 1994), volume 129 of Progress-Mathematics, pp. 165–172. Birkha̋user Boston, Boston (1995)
Okounkov, A., Olshanski, G.: Shifted Schur functions. Algebra i Analiz 9(2), 73–146 (1997)
Rankin, R.A.: The construction of automorphic forms from the derivatives of a given form. J. Indian Math. Soc. 20, 103–116 (1956)
Schoeneberg, B.: Das verhalten von mehrfachen thetareihen bei modulsubstitutionen. Math. Ann. 116(1), 511–523 (1939)
The Sage Developers.: SageMath, the Sage Mathematics Software System (Version 8.0) (2017). http://www.sagemath.org
Zagier, D.: Elliptic modular forms and their applications. In: Ranestad, K. (ed.) The 1-2-3 of Modular Forms, Universitext, pp. 1–103. Springer, Berlin (2008)
Zagier, D.: Partitions, quasimodular forms, and the Bloch–Okounkov theorem. Ramanujan J. 41(1–3), 345–368 (2016)
Acknowledgements
I would like to thank Gunther Cornelissen and Don Zagier for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Tables of shifted symmetric harmonic polynomials up to weight 10
Appendix: Tables of shifted symmetric harmonic polynomials up to weight 10
We list all harmonic polynomials \(h_{\lambda }\) of even weight at most 10. The corresponding q-brackets \(\langle h_\lambda \rangle _q\) are computed by the algorithm prescribed by Zagier [13] using SageMath [11].
\(\lambda \) | \( h_{\lambda } \) | \(\langle h_{\lambda }\rangle _q \) |
---|---|---|
() | 1 | 1 |
(4) | \( \frac{27}{4} \left( Q_2^2+2 Q_4\right) \) | \(\frac{9}{320}Q\) |
(6) | \( \frac{225}{4} \left( 63 Q_6+9 Q_2 Q_4+Q_2^3\right) \) | \(-\frac{55}{384}R\) |
(3,3) | \( \frac{225}{4} \left( 63 Q_3^2-108 Q_2 Q_4+2 Q_2^3\right) \) | \(\frac{115}{384}R \) |
(8) | \( \frac{19845}{16} \left( 3960 Q_8+360 Q_2 Q_6+20 Q_2^2 Q_4+Q_2^4\right) \) | \( \frac{19173}{4096} Q^2\) |
(5,3) | \( \frac{19845}{2} \left( 495 Q_3Q_5 + 45 Q_2Q_3^2-1350 Q_2 Q_6 -50 Q_2^2Q_4 +2 Q_2^4\right) \) | \(-\frac{2415}{128} Q^2 \) |
(4,4) | \( \frac{297675}{8} \left( 132 Q_4^2 + 24 Q_2 Q_3^2-440Q_2 Q_6-28 Q_2^2Q_4 +Q_2^4\right) \) | \(-\frac{38241}{2048} Q^2\) |
(10) | \( \frac{382725}{8} \left( 450450 Q_{10} + 30030 Q_2 Q_8 + 1155 Q_2^2 Q_6 + 35 Q_2^3 Q_4 + Q_2^5\right) \) | \(-\frac{2053485}{4096} QR \) |
(7,3) | \( \frac{1913625}{8} \Big (90090 Q_3 Q_7 + 6006 Q_2 Q_3 Q_5 - 336336 Q_2 Q_8 + 231 Q_2 Q_3^2 +\) | |
\(\quad \quad \quad \quad - 12936 Q_2^2Q_6 -112 Q_2^3Q_4 + 10 Q_2^5\big )\) | \(\frac{11975985}{4096} QR \) | |
(6,4) | \(\frac{13395375}{8} \Big ( 12870 Q_4 Q_6 + 1716 Q_2 Q_3 Q_5 + 858 Q_2 Q_4^2 -96096 Q_2 Q_8 + \) | |
\( \quad \quad \quad \quad + 132 Q_2^2Q_3^2 - 6501 Q_2^2 Q_6 -89 Q_2^3 Q_4+5 Q_2^4\Big ) \) | \(\frac{21255885}{4096} QR\) | |
(5,5) | \( \frac{8037225}{4} \Big (10725 Q_5^2+1430 Q_2Q_3Q_5 + 1430 Q_2 Q_4^2-10010 Q_2 Q_8 +\) | |
\(\quad \quad \quad \quad + 165 Q_2^2Q_3^2 - 7700 Q_2^2 Q_6 -120 Q_2^3Q_4+6 Q_2^5\Big )\) | \(\frac{7759395}{1024} QR \) | |
(4,3,3) | \(\frac{13395375}{8} \Big (12870 Q_3^2Q_4-34320 Q_2Q_3Q_5+10296Q_2 Q_4^2+363Q_2^2Q_3^2+ \) | |
\( \quad \quad \quad \quad +55440Q_2^2 Q_6-376Q_2^3Q_4+10Q_2^5\Big ) \) | \(-\frac{16583805}{4096} QR \) |
In case \(|\lambda |\) is odd, the harmonic polynomials \(h_{\lambda }\) up to weight 9 are given in the following table. The q-bracket of odd degree (harmonic) polynomials is zero, hence trivially modular.
\(\lambda \) | \( h_{\lambda } \) |
---|---|
(3) | \( -\frac{9}{4}Q_3 \) |
(5) | \( -\frac{135}{4}\left( 5Q_5+Q_2Q_3\right) \) |
(7) | \( -\frac{14175}{16}\left( 126Q_7+14Q_2Q_5+Q_2^2Q_3\right) \) |
(4, 3) | \( -\frac{99225}{16}\left( 18Q_3Q_4 - 40 Q_2 Q_5+Q_2^2Q_3\right) \) |
(9) | \( -\frac{297675}{8}\left( 7722 Q_9 + 594Q_2 Q_7 +27 Q_2^2Q_5+Q_2^3Q_3\right) \) |
(6, 3) | \( -\frac{893025}{4}\left( 1287 Q_3 Q_6 +99 Q_2Q_3Q_4-4158 Q_2Q_7-162 Q_2^2Q_5+5Q_2^3Q_3\right) \) |
(5, 4) | \( - \frac{8037225}{8}\left( 286 Q_4Q_5+66 Q_2Q_3Q_4-1540 Q_2Q_7 -117 Q_2^2Q_5+3Q_2^3Q_3\right) \) |
(3, 3, 3) | \( -\frac{893025}{4}\left( 1287 Q_3^3-3564Q_2Q_3Q_4+3240Q_2^2Q_5+10Q_2^3Q_3\right) \) |
Rights and permissions
OpenAccess This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
van Ittersum, JW.M. When is the Bloch–Okounkov q-bracket modular?. Ramanujan J 52, 669–682 (2020). https://doi.org/10.1007/s11139-019-00144-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-019-00144-1