When is the Bloch–Okounkov q-bracket modular?

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 Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}. We describe an explicit basis for this kernel, which is very similar to the space of classical harmonic polynomials.


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 ∈ Z[x 1 , . . . , x r ]. The quasimodular form θ P is modular if and only if P is harmonic (i.e. P ∈ ker r i=1 ∂ 2 ∂ x 2 i ) [10]. (As quasimodular forms were not yet defined, Schoeneberg only showed that θ P is modular if P is harmonic. However, for every polynomial P it follows that θ 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 P denote the set of all partitions of integers and |λ| denote the integer that λ is a partition of. Given a function f : P → Q, define the q-bracket of f by The celebrated Bloch-Okounkov theorem states that for a certain family of functions f : P → Q (called shifted symmetric polynomials and defined in Sect. 2) the q-brackets f 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 sl 2 on both the algebra of shifted symmetric polynomials Λ * and the algebra of quasimodular forms M such that the q-bracket · q : Λ * → M is sl 2 -equivariant.
The answer to the question in the title is provided by one of the operators which defines this sl 2 -action on Λ * . Namely letting H = ker | Λ * , we prove the following theorem: The last section of this article is devoted to describing the graded algebra H. We call 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 P d be the space of polynomials of degree d in m ≥ 3 variables x 1 , . . . , x m , let ||x|| 2 = i x 2 i , and recall that the space H d of degree d harmonic polynomials is given by ker . The main theorem of harmonic polynomials states that every polynomial P ∈ P d can uniquely be written in the form with h i ∈ H d−2i and d = d/2 . Define K , the Kelvin transform, and D α for α an m-tuple of non-negative integers by An explicit basis for 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 ∈ Λ * n there exists unique h i ∈ H n−2i (i = 0, 1, . . . , n and n = n 2 ) such that where Q 2 is an element of Λ * 2 given by Q 2 (λ) = |λ| − 1 24 .

Theorem 3
The set is a vector space basis of H n , where pr, K , and λ are defined by (4), Definition 4, respectively, Definition 6.
The action of sl 2 given by Proposition 1 makes Λ * into an infinite-dimensional sl 2representation for which the elements of H are the lowest weight vectors. Theorem 2 is equivalent to the statement that Λ * is a direct sum of the (not necessarily irreducible) lowest weight modules

Shifted symmetric polynomials
Shifted symmetric polynomials were introduced by Okounkov and Olshanski as the following analogue of symmetric polynomials [8]. Let Λ * (m) be the space of rational polynomials in m variables x 1 , . . . , x m which are shifted symmetric, i.e. invariant under the action of all σ ∈ S m given by is filtered by the degree of the polynomials. We have forgetful maps Λ * (m) → Λ * (m − 1) given by x m → 0, so that we can define the space of shifted symmetric polynomials Λ * as lim ← − m Λ * (m) in the category of filtered algebras. Considering a partition λ as a non-increasing sequence (λ 1 , λ 2 , . . .) of non-negative integers λ i , we can interpret Λ * as being a subspace of all functions P → Q. One can find a concrete basis for this abstractly defined space by considering the generating series for every λ ∈ P (the constant 1 2 turns out to be convenient for defining a grading on Λ * ). As w λ (T ) converges for T > 1 and equals The first few shifted symmetric polynomials Q i are given by The Q i freely generate the algebra of shifted symmetric polynomials, i.e.
It is believed that Λ * is maximal in the sense that for all Q :

Remark 1
The space Λ * can equally well be defined in terms of the Frobenius coordinates. Given a partition with Frobenius coordinates (a 1 , . . . , a r , b 1 , . . . , b r ), where a i and b i are the arm and leg lengths of the cells on the main diagonal, let Then where β k is the constant given by We extend Λ * to an algebra where Q 1 ≡ 0. Observe that a non-increasing sequence (λ 1 , λ 2 , . . .) 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 (λ). In fact, in this case corresponds to a partition (i.e. converges to 0). In particular, Q 1 (λ) = n equals the number the sequence λ converges to. We now define the Bloch-Okounkov ring R to be Λ * [Q 1 ], considered as a subspace of all functions from non-increasing eventually constant sequences of integers to Q. It is convenient to work with R instead of Λ * to define the differential operators and more generally λ later. Both on Λ * and R, we define a weight grading by assigning to Q i weight i. Denote the projection map by We extend · q to R.
The operator E = ∞ m=0 Q m ∂ ∂ Q m on R multiplies an element of R by its weight. Moreover, we consider the differential operators 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].
The following result follows by a direct computation using the above .

An sl 2 -equivariant mapping
The space of quasimodular forms for SL 2 (Z) is given by M = 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 M (≤ p) k be the space of quasimodular forms of weight k and depth ≤ 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 d dq , d = 12 ∂ ∂ P , and the weight operator W given by W f = k f for f ∈ M k preserve M and form an 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 withD := D − P 24 , as for all f ∈ Λ * one has Q 2 f q =D f q . Moreover,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 ∈ M be of depth r . ThenD f is of depth r + 1.
Proof Consider a monomial P a Q b R c with a, b, c ∈ Z ≥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 a 12 + b 3 + c 2 − 1 24 is non-zero for a, b, c ∈ Z.
Moreover, lettingŴ = W − 1 2 , the triple (D, d,Ŵ ) forms an sl 2 -triple as well. With respect to these operators, the q-bracket becomes sl 2 -equivariant. The following proposition is a detailed version of Proposition 1: Proof This follows directly from [13,Equation (37)] and the fact that for all f ∈ R one has Q 1 f q = 0.

Describing the space of shifted symmetric harmonic polynomials
In this section, we study the kernel of . As [ , Q 1 ] = 0, we restrict ourselves without loss of generality to Λ * . Note, however, that does not act on Λ * as, for example, (Q 3 ) = − 1 2 Q 1 . However, pr does act on Λ * .

Definition 2 Let
be the space of shifted symmetric harmonic polynomials. Then Proposition 5 For all n ∈ Z, one has Proof For uniqueness, suppose f = Q 2 g + h and f = Q 2 g + h with g, g ∈ Λ * n−2 and h, h ∈ H n . Then, Q 2 (g − g ) = h − h ∈ H. By Proposition 4 we find g = g and hence h = h . Now, define the linear map T : Λ * n → Λ * n by f → pr (Q 2 f ). By Proposition 4 we find that T is injective, which by finite dimensionality of Λ * n implies that T is surjective. Hence, given f ∈ Λ * n let g ∈ Λ * n−2 be such that Proof Observe that dim Λ * n equals the number of partitions of n in parts of size at least 2. Hence, dim Λ * n = p(n) − p(n − 1) and the Corollary follows from Proposition 5.

Proof of Theorem 1
If f q is modular, then f q = d f q = 0. Write f = n r =0 Q r 2 h r as in Theorem 2 with n = n 2 . Then by Lemma 1 it follows that As h r q is modular, either it is equal to 0 or it has depth 0. Suppose the maximum m of all r ≥ 1 such that h r 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 , . . . , h n ∈ ker · q . Note that f ∈ ker · q implies that Q 2 f ∈ ker · q . Therefore, k := n r =1 Q r 2 h r ∈ ker · q and f = h + k with h = h 0 harmonic. The converse follows directly as d h + k q = d h q = h q = 0.
Remark 2 A description of the kernel of · 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: of shifted symmetric polynomials of depth ≤ p is the space of f ∈ Λ * k such that one can write .
Proof Expanding f as in Definition 3 we find By Lemma 2, we find that the depth of f q is at most p.
Next, we set up notation to determine the basis of H given by Theorem 3.
] be the formal polynomial algebras graded by assigning to Q k weight k (note that the weights are-possibly negativeintegers). Extend toΛ and observe that (Λ) ⊂Λ. Also extend H by settingH Definition 4 Define the partition-Kelvin transform K :Λ n →Λ 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 Define the nth order differential operators D n onR by where the coefficient is a multinomial coefficient.
This definition generalises the operators ∂ and D to higher weights: D 1 = ∂, D 2 = D, and D n reduces the weight by n.

≥0
. We collect all terms for different vectors a which consists of the same parts (i.e. we group all vectors a which correspond to the same partition). Then, the coefficient of such a term equals It does not hold true that [D n , Q 1 ] = 0 for all n ∈ N. Therefore, we introduce the following operators: For λ ∈ P let (Note that 0 = 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 λ satisfy the following properties: for all partitions λ, λ (a) the order of |λ| is |λ|; Proof Property (a) follows by construction and (b) is a direct consequence of Lemma 3. For property (c), let f ∈Λ be given. Then Observe that by the identity the sum in the last line is a telescoping sum, equal to zero. Hence n (Q 1 f ) = Q 1 n ( f ) as desired.
Observe that h λ is harmonic, as pr commutes with pr and λ .
Proof Note that the left-hand side is an element of Λ * of which the monomials divisible by Q i 2 correspond precisely to terms in λ involving precisely n − i derivatives of K (1) to Q 2 . Hence, as λ has order n all terms not divisible by Q 2 correspond to terms in λ which equal ∂ n ∂ Q n 2 up to a coefficient. There is only one such term in λ with coefficient |λ| λ 1 ,...,λ r λ 1 ! . . . λ r !Q λ .
For f ∈ R, we let f ∨ be the operator where every occurrence of Q i in f is replaced by i . We get the following unusual identity: 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 ∈ Λ * n−2 . However, as both sides of (9) are harmonic and the shifted symmetric polynomial Q 2 f is harmonic precisely if f = 0 by Proposition 4, it follows that f = 0 and (9) holds true.

Proof of Theorem 3
Let B n = {h λ | λ ∈ P n all parts are ≥ 3}. First of all, observe that by Corollary 1 the number of elements in B n is precisely the dimension of H n . Moreover, the weight of an element in B n equals |λ| = n. By Proposition 7 it follows that the elements of B n are linearly independent harmonic shifted symmetric polynomials.