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 {\Delta}. 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 θ P (τ ) = x∈Z r P (x)q x 2 1 +...+x 2 r (q = e 2πiτ ) 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 ) [Sch39] 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 n r=1 a r f (r) g (n−r) (a r ∈ C).
This linear combination is a quasimodular form which is modular precisely if it is a multiple of the Rankin-Cohen bracket [f, g] n [Ran56,Coh75]. 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| denotes the integer that λ is a partition of. Given a function f : P → Q, define the q-bracket of f by f q := λ∈P f (λ)q |λ| λ∈P q |λ| .
The celebrated Bloch-Okounkov theorem states that for a certain family of functions f : P → Q (called shifted symmetric polynomials and defined in Section 2) the q-brackets f q are the qexpansions of quasimodular forms [BO00].
Besides being a wonderful result, the Bloch-Okounkov theorem has many application 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 [Dij95,KZ95]. Also, in computation of the asymptotics of geometrical invariants, such as volumes of moduli spaces of holomorphic differentials and Siegel-Veech constants the Bloch-Okounkov theorem is applied [EO01,CMZ18].
Zagier gave a surprisingly short and elementary proof of the Bloch-Okounkov theorem [Zag16]. A corollary of his work, which we discuss in Section 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 r i=1 ∂ 2 ∂x 2 i . 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 [ABR01]. We prove the following analogous results for the space of shifted symmetric polynomials: Theorem 3. For every f ∈ Λ * n there exists unique h i ∈ H n−2i (i = 0, 1, . . . , n ′ and n ′ = ⌊ n 2 ⌋) such that f = h 0 + Q 2 h 1 + . . . + Q n ′ 2 h n ′ , where Q 2 is an element of Λ * 2 given by Q 2 (λ) = |λ| − 1 24 .
Theorem 4. The set is a vector space basis of H n , where pr, K and ∆ λ are defined by (4), Definition 16 respectively Definition 20.
The action of sl 2 given by Proposition 1 makes Λ * into an infinite-dimensional sl 2 -representation for which the elements of H are the lowest weight vectors. Theorem 3 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 [OO97]. 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 . Note that Λ * (m) 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 functions, i.e. Λ * = Q[Q 2 , Q 3 , . . .]. It is believed that Λ * is maximal in the sense that for all Q : Remark. 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-lenghts of the cells on the main diagonal, let 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, Equations (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 pr : R → Λ * . (4) We extend · q to R.

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 [Zag08, Section 5.3] or [Zag16, 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 = kf 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 8. Let f ∈ M be of depth r. ThenDf 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 [Zag16, 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 can 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 * .
be the space of shifted symmetric harmonic polynomials.
Proof. Write f = Q n 2 f ′ with f ′ ∈ Λ * and f ′ ∈ Q 2 Λ * . Then Proposition 12. 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 Lemma 11 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 Lemma 11 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 T (g) = pr∆(f ) ∈ Λ n−2 . Let h = f − gQ 2 . As f = Q 2 g + h, it suffices to show that h ∈ H. That holds true because pr∆(h) = pr∆(f ) − pr∆(Q 2 g) = 0.
Proposition 12 implies Theorem 3 and the following corollary. Denote by p(n) the number of partitions of n.
Corollary 13. The dimension of H n equals the number of partitions of n in at least 3 parts, i.e.
Proof. Observe that dim Λ n equals the number of partitions of n in at least 2 parts. Hence, dim Λ n = p(n) − p(n − 1) and the Corollary follows from Proposition 12.
Proof of Theorem 2. 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 3 with n ′ = ⌊ n 2 ⌋. Then by Lemma 7 it follows that pr∆f = n ′ r=0 r(n − r − 3 2 )Q r−1 2 h r . Hence, 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 8 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.
A description of the kernel of · q is not known. △ Another corollary of Theorem 12 is the notion of depth of shifted symmetric polynomials which corresponds to the depth of quasimodular forms: Definition 14. The space Λ * (≤p) k of shifted symmetric polynomials of depth ≤ p is the space of f ∈ Λ k such that one can write Theorem 15.
Proof. Expanding f as in Definition 14 we find By Lemma 8, we find that the depth of f q is at most p.
Definition 18. Given i ∈ Z n ≥0 , let 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 heighter weights: D 1 = ∂, D 2 = D and D n reduces the weight by n.
Lemma 19. The operators {D n } n∈N commute pairwise.
Proof. Set I = |i| = i 1 +i 2 +. . .+i n and J = |j| = j 1 +j 2 +. . .+j m . Let ak = (a 1 , . . . , a k−1 , a k+1 , . . . , a n ). Then δ j l ,I−1 I J i 1 , i 2 , . . . , i n , j 1 , j 2 , . . . ,ĵ l , . . . , j m Hence, [D n , D m ] is a linear combination of terms of the form Q |a|+1 . 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. By Möbius inversion
The first three operators are given by Proposition 21. 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 19. For property (c), let f ∈Λ be given. Then Observe that by the identity (n − i) n i = i n i + 1 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, K 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 ∂ ∂Q n−i 2 up to a coefficient. There is only one such term in ∆ λ with coefficient |λ| λ 1 ,...,λr λ 1 ! . . . λ r !Q λ .
For f ∈ R, we let f ∨ be the operator where every occurence of Q i in f is replaced by ∆ i . We get the following unusual identity: Proof. By Lemma 22 we know that the statement holds true up to adding Q 2 f on the right-hand side for some f ∈Λ 1−n . However, as both sides of (9) are harmonic and the shifted symmetric polynomial Q 2 f is harmonic precisely if f = 0 by Lemma 11, it follows that f = 0 and (9) holds true.
Proof of Theorem 4. Let B n = {h λ | λ ∈ P n all parts are ≥ 3}. First of all, observe that by Corollary 13 the number of elements in B n is precisely the dimension of H n . Moreover, the weight of an element in B equals |λ| = n. By Lemma 22 it follows that the elements of B n are linearly independent harmonic shifted symmetric polynomials.