Hilbert series for twisted commutative algebras

Suppose that for each n>= 0 we have a representation $M_n$ of the symmetric group S_n. Such sequences arise in a wide variety of contexts, and often exhibit uniformity in some way. We prove a number of general results along these lines in this paper: our prototypical theorem states that if $M_n$ can be given a suitable module structure over a twisted commutative algebra then the sequence $M_n$ follows a predictable pattern. We phrase these results precisely in the language of Hilbert series (or Poincar\'e series, or formal characters) of modules over tca's.


Introduction
Suppose that for each n ≥ 0 we have a complex representation M n of the symmetric group S n . Such sequences of symmetric group representations arise in a variety of contexts, and naturally occurring examples tend to exhibit some kind of uniformity. For example: (a) Fix a manifold X of dimension ≥ 2 and a non-negative integer i. Let M n be the ith cohomology group of the configuration space of n labeled points on X. This example was studied in [CEF]. One of the theorems in loc. cit. states that, under mild assumptions on X, there is a single character polynomial that gives the character of M n for all sufficiently large n. (b) Fix p and r. Let M n be the space of p-syzygies of the Segre embedding (P r ) n → P rn+n−1 . This case was studied in [Sno]. One of the theorems in loc. cit. states that the generating function of the sequence {dim(M n )} is rational. (c) Suppose that N is a functor associating to each finite dimensional vector space V a module N(V ) over the ring Sym(C d ⊗ V ), for some fixed d (and that satisfies some technical conditions). For example, one could take N(V ) to be the coordinate ring of the rth determinantal variety for some 0 ≤ r ≤ d. Now let M n be the Schur-Weyl dual of the degree n piece of N. This example is studied in [Sno] and, in much greater detail, [SS5]. Again, it is known that the generating function of the dimension sequence is rational. (d) Let E be a representation of a reductive group G, and take M n = (E ⊗n ) G . The generating function of {dim(M n )} is D-finite (and typically not rational, or even algebraic). This is presumably well-known, though we do not know a reference. See §7.4 for a proof. In this paper, we study the uniformity properties of sequences {M n } with the aim of strengthening and generalizing results like those mentioned above. We prove a number of results of the form "if {M n } admits a suitable module structure over a twisted commutative algebra then the sequence {M n } follows a predictable pattern." The specific results give various precise meanings to "suitable" and "predictable pattern." 1.1. Statement of results. We now state some of our results in a little more detail. Let k be a field of characteristic 0. A twisted commutative algebra (tca) over k is an associative unital graded k-algebra A = n≥0 A n equipped with an action of the symmetric group S n on A n , such that the multiplication is commutative up to a "twist" by the symmetric group. A module over a tca A is a graded vector space M = n≥0 M n equipped with an action of S n on M n and a multiplication A n × M m → M n+m satisfying suitable axioms. Note that a module M gives rise to a sequence of representations {M n } as discussed above. TCA's and their modules have been a central object of study in the developing field of representation stability. For example, k[t] can be regarded as a tca (with t of degree 1, and all S n actions trivial), and modules over it are equivalent to the FI-modules of Church-Ellenberg-Farb [CEF].
Suppose that M is a module over a tca. We define its Hilbert series by H M (t) = n≥0 dim(M n ) t n n! .
The following theorem was proved in [Sno], and later reproved in [SS4]: Theorem 1.1. Suppose M is a finitely generated module over a tca finitely generated in degree 1. Then H M (t) is a polynomial in t and e t .
This theorem implies that the generating function for {dim M n } is rational, but is even stronger than this. The results of this paper generalize and strengthen this theorem in various ways.
We now informally describe some of the main results of this paper. In what follows, A denotes a tca generated in degree 1 and M denotes a finitely generated A-module.
(a) We introduce the formal character Θ M of M. This records the character of each representation M n , and thus contains much more information than the Hilbert series.
Using the structure theory of A-modules developed in [SS5], we give a very precise description of Θ M . Our result shows that there is a finite expression for Θ M , and moreover that the pieces in this expression reflect the structure of M. As a corollary, we see that if M and N are finitely generated A-modules such that M n and N n have the same character for all n, then M and N represent the same class in the Grothendieck group of A-modules.
(b) Using the method of [Sno], we give a different proof of a rationality result for Θ M . Actually, we work with the enhanced Hilbert series introduced in [SS1], which is equivalent to the formal character. This is less precise than the proof described in (a), but does not rely on the theory from [SS5]. (c) We show how the Fourier transform from [SS5] affects (enhanced) Hilbert series. (d) We explain how all of the results on (enhanced) Hilbert series carry over to the more subtle Poincaré series. (e) Theorem 1.1 can be stated equivalently as: there exists a polynomial p(T ) with nonnegative integer roots such that p( d dt )H M (t) = 0. We prove a categorification of this result, in which d dt is replaced with the Schur derivative. Additionally, we prove a result for tca's not necessarily generated in degree 1: (f) If A is a finitely generated tca with A 0 = C and M is a finitely generated and bounded A-module, then H M (t) is a D-finite power series.
1.2. Open problems. We now list some questions and open problems related to the work in this paper.
• To what extent do the results here carry over to positive characteristic? Theorem 1.1 is known to hold in positive characteristic by [SS4,Corollary 7.1.7], but the other results of this paper are not known. • Result (a) above gives a finite expression for the formal character Θ M in the case where M is a finitely generated module over a tca A finitely generated in degree 1. Is there a more general result if one assumes that A is bounded instead of generated in degree 1? • Result (e) categorifies Theorem 1.1. Can the rationality theorem for the formal character be similarly categorified? Presumably, such a categorification should make use of the more general Schur derivatives D λ . • The discussion in §5.4 raises a number of questions about the categorified rationality.
For example: in the tensor category of differential operators, what can one say about the ideal annihilating a given module? Result (e) ensures that it is non-zero. • What can be said about the Hilbert series of a finitely generated module over an arbitrary finitely generated tca? For example, is it D-finite? • There are interesting sequences {M n } of symmetric group representations that do not come from tca's, but related structures, such as FS op -modules (see [SS4,§8]). To what extent do the results here extend to those sequences?
1.3. Outline. In §2 we review some background material that we will require. In §3 we introduce the formal character and prove our "rationality" theorem for it. In §4.1 we translate our results about formal characters to Hilbert series (and Poincaré series), and also give an elementary proof of the rationality theorem in this setting. In §5 we prove a result categorifying the rationality theorem. In §6 we prove D-finiteness for Hilbert series over bounded tca's not necessarily generated in degree 1. Finally, in §7 we give some examples and applications of our results.

Preliminaries
This paper is a continuation of [SS5] and hence will use the same notation.
Throughout, X denotes a separated, noetherian C-scheme of finite Krull dimension, and E is a vector bundle over X. We will primarily be interested in the case when X is a point, but in order to simplify some of the arguments, it will be necessary to allow X to be a Grassmannian. Any extra generality beyond that will not be used, but does not present any additional difficulties. The reader unfamiliar with the language of sheaf theory can assume X is a point for most of the paper, in which case quasi-coherent sheaves specialize to complex vector spaces, and coherent sheaves specialize to finite-dimensional vector spaces.
Throughout, we make use of the category V X , which has several equivalent models. The two that we use here are the category of sequences of symmetric group representations on quasi-coherent O X -modules, and the category of polynomial functors from the category of vector spaces to quasi-coherent O X -modules. See [SS2,§5] for more details in the case X = Spec(C). In the first model, every object V X admits a decomposition λ M λ ⊗ F λ where M λ is the usual Specht module over C and F λ is a quasi-coherent sheaf on X; in the second model, we get a similar decomposition, but with M λ replaced by the Schur functor S λ . Given F ∈ V X in the second model, we use ℓ(F ) to denote the maximum number of parts of any λ such that S λ has a nonzero multiplicity space. We say that F is bounded if ℓ(F ) < ∞. We let V dfg X denote the subcategory of V X where the multiplicity spaces F λ are coherent. (The "dfg" superscript means "degreewise finitely generated.") We let V fg X be the subcategory of V dfg X where all but finitely many of the F λ vanish. We also let V = C ∞ = n≥1 C n be the standard representation of GL ∞ ; evaluating a Schur functor on V gives an equivalence between the category of polynomial functors and the category of polynomial representations of GL ∞ .
Since we are working over a field of characteristic 0, we can use an alternative description of tca's: they are polynomial functors from the category of vector spaces to the category of (sheaves of) commutative algebras. We let A(E) denote the tca which sends a vector space V to the commutative algebra Sym(E ⊗ V ).
2.1. K-theory of relative Grassmannians. For a non-negative integer r, Gr r (E) denotes the relative Grassmannian of rank r quotients of E. Let π r : Gr r (E) → X be the structure map, and let Q = Q r be the tautological quotient bundle on Gr r (E) and let R = R r denote the tautological subbundle, so that we have a short exact sequence Let Λ denote the ring of symmetric functions, and let Λ d be the space of symmetric functions which are homogeneous of degree d.
The following can be found in [SS5,Theorem 6.19]: Theorem 2.1. The maps i r induce an isomorphism of Λ-modules rank E r=0 Λ ⊗ K(Gr r (E)) → K(A(E)).

2.2.
Integration on the torus. Let T be the diagonal torus in GL(d). Denote by α 1 , . . . , α d the standard characters T → G m , and identify the representation ring K(T ) with the ring of Laurent polynomials Q[α 1 , α −1 1 , . . . , α d , α −1 d ]. We let f → T f dα be the projection map K(T ) → Q which takes the constant term of f . We let f → f be the ring homomorphism Weyl's integration formula (see [FH,§26.2]) can then be stated as follows: if f, g ∈ K(T ) are the characters of irreducible representations of GL(d), then 2.3. Symmetric groups and symmetric functions. Fix a nonnegative integer n. Let λ be an integer partition of n and let c λ be the conjugacy class of permutations with cycle type λ. Also, partitions parametrize the irreducible complex representations M λ . We let tr(c µ |M λ ) denote the trace of any element of c µ acting on M λ . We refer to [SS2,§2] for basic properties and further references. Given an integer partition λ = (λ 1 , . . . , λ r ), let m i (λ) denote the number of λ j that are equal to i, and set λ! = i m i (λ)! and z λ = λ! i i m i (λ) .
For an integer n, let p n = p n (α) denote the power sum symmetric polynomial d i=1 α n i . For a partition λ = (λ 1 , . . . , λ r ), let p λ denote the polynomial p λ 1 · · · p λr . We allow d to be infinite. The notation s λ is reserved for the Schur function. By [Sta,7.17.5,7.18.5], we have

Formal characters
In this section, we will assume all schemes X have an ample line bundle L. 1 Let V be an object of V dfg X . Recall that V decomposes as λ V λ ⊗S λ (V) where V λ ∈ Mod fg X . We define the formal character of V as where, as usual, [V λ ] is the class of V λ in K(X) and s λ ∈ Λ is the Schur function. Thus Θ V is a potentially infinite series whose terms belong to Λ ⊗ K(X). Since Λ is the polynomial ring in the complete homogeneous symmetric functions {s n } n≥1 , one can also think of Θ V as a power series in these variables with coefficients in K(X). When X is a point, dim(V λ ) is the class of the vector space V λ in K(X) ∼ = Z.
We write K ′ (X) for the Grothendieck group of vector bundles on X. The group K(X) is a module over K ′ (X), and there is a natural map K ′ (X) → K(X).
Remark 3.1 (Splitting principle). Given a symmetric polynomial f (x 1 , . . . , x n ) and a vector bundle E of rank n on X, then we define f ([E]) ∈ K ′ (X) as follows: if E has a filtration whose associated graded is L 1 ⊕ · · · ⊕ L n , where the L i are line bundles, then f ([E]) = f ([L 1 ], . . . , [L n ]). In general, consider the relative flag variety π : Flag(E) → X. In this case, π * E has a filtration by line bundles, and we define f ([E]) := Rπ * (f ([π * E])). (See [Fu,§3.2]; note that when E is a line bundle, Flag(E) = X and π * and π * are both identity maps.) We now introduce some notation needed to state our main result on formal characters. For k ≥ 0, let σ k = n≥k n k s n . (In the definition of σ 0 we use the convention s 0 = 1.) For a partition λ, we put where m n (λ) is the multiplicity of n in λ.
Since σ 0 − 1 is a nonzerodivisor, it suffices to handle the case d 0 = 0. But now we can simply observe that the σ k 's with k ≥ 1 are related to the s n 's with n ≥ 1 by an upper triangular change of variables, and the s n 's are algebraically independent.
Let Λ = Λ ⊗ Z[σ 0 , σ 1 , . . . ] be the polynomial ring in the s n 's and σ n 's. Let m (r) λ be the monomial symmetric polynomial in r variables associated to the partition λ, and define For a proper map π : Y → X, we define a bilinear pairing We define a map θ r : Λ ⊗ K(Gr r (E)) → Λ ⊗ K(X), Here Q is the tautological quotient bundle on Gr r (E). We will see below that this sum is finite: the maximum value of λ 1 for which the term can be non-zero is bounded by a function of dim(X) and rank(E). We can now state our main result: In particular, Θ M is a polynomial in the s n 's and σ n 's with coefficients in K(X).
Before proving the theorem, we note a few corollaries. First, in Lemma 3.8 below, we show that θ r is injective. Since everything in the image of θ r has total degree r in the σ's, it follows that the images of the θ r 's are linearly independent. We conclude: Let F • K(Gr r (E)) be the filtration by codimension of support. We also consider the topological filtration F • top K ′ (Gr r (E)) on K ′ (Gr r (E)) which is a ring filtration (see [FL,§V.3]). In particular, M can be represented by a complex whose homology is 0 along that subvariety. It follows that M For the following, let Mod A,≤r be the subcategory of A-modules which are locally annihilated by powers of the rth determinantal ideal a r , and let T >r : Mod A → Mod A / Mod A,≤r be the localization functor. It admits a section functor (right adjoint) S >r : Mod A / Mod A,≤r → Mod A and we set Σ >r = S >r • T >r . This is a left exact functor, and induces a functor RΓ >r on D(A). Next, we define Γ ≤r to be the functor that assigns a module M to the maximal submodule which is locally annihilated by a r . This is also left exact and induces a functor RΓ ≤r on D(A). Finally, we let D b fg (A) r be the full subcategory of D b fg (A) of objects which are sent to 0 by RΓ ≤r−1 and RΣ >r . See [SS5, §6.1] for more details.
The point of the corollary is that, if one gives σ n degree n, then the degree of Θ M is related to the support dimension of M on Gr r (E). Of course, if M is not in D b fg (A(E)) r then one can apply the corollary separately to each projection RΣ ≥r RΓ ≤r (M).
We now begin with the proof of the theorem. We start with a few lemmas.
where the multiplication on the right side uses the K ′ (X)-module structure on K(X).
Proof. By [FL,Corollary V.3.10], ℓ d+1 = 0. We have Lemma 3.7. Let Q be a locally free coherent sheaf of rank r on Y . Then Proof. By the splitting principle, we may assume Since ℓ d+1 i = 0, the bound λ 1 ≤ d in the index for the sum is superfluous. We note that m (r) Proof of Theorem 3.3. Fix r, let Y = Gr r (E), let π : Y → X be the structure map, and let Q be the tautological bundle on Y . By [SS5,Corollary 6.16], it suffices to prove the result Since formation of formal characters is compatible with derived pushforward, we find . Thus the above is exactly equal to rank E r=0 θ r (c r ), which proves the theorem.
Lemma 3.8. The map θ r is injective.
In [SS5,§7], we define an equivalence of categories ) called the "Fourier transform." To close this section, we examine how F E affects formal characters. In what follows, we write F in place of F E .
The definition of F depends on the choice of a dualizing complex ω X on X, so we fix a choice now. We let D be the induced duality on D b fg (X) and K(X) and K ′ (X). Let (−) ‡ : Λ → Λ be the ring homomorphism defined by s ‡ λ = (−1) |λ| s λ † . We extend (−) ‡ to infinite formal linear combinations of s λ 's in the obvious manner.
Since Λ is graded with s n and s 1 n of degree n, the identity continues to hold if we replace s n by (1 + t) n s n and s 1 n by (1 + t) n s 1 n .
Our main result on the Fourier transform is: O X , and let T (M) i denote the degree i piece of this object, regarded as a complex in V X . By [SS5,Prop 7.1], we have and so We define F (M) by applying D and (−) † to K (M). Note that the above sum is Θ T (M ) with the terms of odd degree multiplied by −1; when we apply (−) † to this, we get exactly Θ ‡ T (M ) . We thus have the identity Combining with the previous equation, we obtain the stated result.
Corollary 3.11. Suppose E is trivial of rank n.
Proof. In this case, Θ ′ A(E) = (Θ ′ A(C) ) n , and it is clear that Θ

The main theorems on Hilbert and Poincaré series
4.1. Standard Hilbert series. Pick V ∈ V dfg X , which we think of as a sequence of S nrepresentations on coherent O X -modules, and decompose it as V = λ V λ ⊗ M λ . We define its Hilbert series by is a power series with coefficients in the group K(X). In [Sno], the following theorem was proved: In this section, we generalize this theorem, and examine how the form of H M (t) relates to the structure of M. To a large extent, we answer [Sno,Question 3].
For a vector bundle E on X, define We define a map In particular, we have and hence ex(Θ M ) = H M (t). From the above discussion and the definitions of θ r and µ r , we have ex • θ r = µ r . Our main theorem on Hilbert series is then: Proof. By Theorem 3.3, we have Θ M = d r=0 θ r (c r ). Now apply ex to both sides. One consequence of this theorem is that the coefficient of e rt in H M (t) depends only on the projection RΠ r (M) of M to D b fg (A) r . We now examine how the Fourier transform interacts with Hilbert series.
is obtained from p r (t) by applying D to its coefficients. Proof. Apply ex to Proposition 3.10. 4.2. Enhanced Hilbert series. Let λ be a partition. Recall that m i (λ) is the number of times i occurs in λ. Define Thus X λ (t) is a polynomial in the variables t i that encodes the character of M λ .
Lemma 4.4. The map is an isomorphism of rings.
Proof. The p n are algebraically independent generators for Λ ⊗ Q, so as can be seen from (2.3), ϕ can be described as p n → nt n , which is evidently a ring isomorphism.
We can also extend ϕ to a map Now let V ∈ V dfg X . Thinking of V in the symmetric group model, we can decompose it as V = λ V λ ⊗ M λ . We define the enhanced Hilbert series of V by This is a power series in the variables t i with coefficients in K(X). This series was introduced in [SS1] as an improvement of the Hilbert series, the idea being that it records the character of the S n -representation of V n rather than just its dimension. Clearly, H V (t) is obtained from the formal character Θ V by applying the ring isomorphism ϕ. Thus the two invariants contain the same information and are just packaged in a different manner. Our results on the formal character can be easily translated to the language of enhanced Hilbert series. We just state the rationality theorem here. For k ≥ 0, put T k = n≥k n k t k . It is a simple exercise to show that ϕ(σ n ) has the form p(T ) exp(T 0 ) where p(T ) is a polynomial in the T k with k ≥ 1. Applying this to Theorem 3.3 yields:  where p i (t, T ) is a polynomial in t 1 , t 2 , . . . and T 1 , T 2 , . . . with coefficients in K(X).
Remark 4.6. One obtains the usual Hilbert series H M (t) from the enhanced Hilbert series H M (t) by setting t i = 0 for i ≥ 2. Thus the above theorem recovers our earlier rationality result for the non-enhanced Hilbert series.

4.3.
Elementary proof of Theorem 4.5. We now give an elementary proof (i.e., not using the results of [SS5]) of our main theorem on enhanced Hilbert series. The proof follows the proof of rationality of the usual Hilbert series given in [Sno,§3.1]. Namely, we express H M (t) in terms of the T -equivariant Hilbert series of M(C d ), where T is the standard maximal torus in GL(d) for d sufficiently large. Once this expression is obtained, a formal manipulation gives the theorem.
Let M be a finitely generated A-module, where A is a tca finitely generated in degree 1. We assume (without loss of generality) that M is a finitely generated module over A = A(U) for some finite dimensional vector space U. Fix an integer d ≥ max(ℓ(A), ℓ(M)).
Let V be a vector space (possibly infinite dimensional) on which T acts. Write V = V α , where the sum is over the characters α of T and V α denotes the α-weight space. Assuming each V α is finite dimensional, we define the T -equivariant Hilbert series of V by Each character α is a monomial in the α ±1 i , and so H V,T (α) can be regarded as a formal series in these variables. Note that if V comes from a polynomial representation of GL(d) then no α −1 i 's appear in H V,T (α). Regard A and M as Schur functors and evaluate on the vector space C d . We obtain a finitely generated C-algebra A(C d ) and a finitely generated A(C d )-module M(C d ), equipped with compatible actions of GL(d). We can form the T -equivariant Hilbert series of M(C d ), which we denote by H M (C d ),T (α). By the comments of the last paragraph, this is a formal series in the α i . A simple argument with equivariant resolutions over A(C d ) (see [Sno,Lemma 3.3]), shows that where h(α) is a polynomial in the α i 's and n = dim(U). Define where the sum is taken over all partitions λ. The following lemma gives a different expression for K(t, α).
Lemma 4.7. For a partition λ, let s λ (α) ∈ K(T ) denote the character of S λ (C d ). Then where the sum is over all partitions λ with ℓ(λ) ≤ d.
Proof. By [Sta,Corollary 7.17.4], we have p µ = λ tr(c µ |M λ )s λ as elements of K(T ), where the sum is over partitions λ of |µ| with ℓ(λ) ≤ d. Multiplying by t µ /µ! and summing over µ, we find The following is the key lemma, relating H M (t) to the more accessible object H M (C d ),T .
Lemma 4.8. We have On the other hand, We therefore have To go from the first line to the second, we used Weyl's integration formula (2.2). Both sums in the second equation are over partitions with ℓ ≤ d; in particular, both s λ and s µ are characters of (non-zero) irreducible representations of GL(d). To go from the second line to the third, we used Lemma 4.7.
We need one more lemma before proving the theorem.
Lemma 4.9. Let u 1 , . . . , u d be indeterminates. We then have Proof. For x ∈ Z d ≥0 , let α x denote α x 1 1 · · · α x d d , and similarly define u x . Then we have Here x varies over Z d ≥0 , while λ varies over all partitions. Now, a simple computation shows that the coefficient of α x in p λ is given by where the sum is taken over partitions µ 1 , . . . , µ d such that µ 1 ∪ · · · ∪ µ d = λ and |µ i | = x i . We thus find Combining the previous two equations gives the stated result.
Proof of Theorem 4.5. Put and, more generally, We have ϕ n (1, t) = T n and ϕ n (0, t) = n!t n (with t 0 interpreted as 0). Lemma 4.9 can be restated as t)). (4.10) Finally, suppose we are given an expression where p(α) is a polynomial. We claim that it is a polynomial in exp(T 0 ) and {T i , t i | i ≥ 1}. This claim proves the theorem, as we know that H M (t) can be expressed as such an integral by Lemma 4.8.
We now prove the claim. When n i = 0 for all i, the integral in question is a polynomial in the t i , so there is nothing to show in that case. By linearity and applying polynomial long division in each variable separately, we may assume that p(α) is a sum of monomials α m 1 1 · · · α m d d where m i < n i for all i. Now apply ( ∂ ∂u 1 ) n 1 −1 · · · ( ∂ ∂u d ) n d −1 to (4.10) and then set u i = 1 for all i. The result is a polynomial in exp(T 0 ) and {T i | i ≥ 1}, which can be seen by using the right side of (4.10). The left side becomes a scalar multiple of For any m i ≤ n i , we then also know that is a basis for the span of monomials in α k 1 1 · · · α k d d such that 0 ≤ k i < n i . Hence by linearity and combining our earlier reduction, we conclude that our original expression is a polynomial in exp(T 0 ) and {T i , t i | i ≥ 1}.

Poincaré series. Given an A-module M, define its Poincaré series by
Setting q = 1 and multiplying by H A (t) = e dt recovers the Hilbert series H M (t). Note that the Poincaré series has non-trivial information about the A-module structure of M, whereas the Hilbert series only knows about the underlying object of V X . The Poincaré series does not factor through K-theory, so it is much harder to study than the Hilbert series. The following is our main result about it: Theorem 4.11. Let M be a finitely generated A-module. There exist f r (t, q) ∈ K(X)[t, q, q −1 ] such that Proof. Let N k = H −k (F (M)), where F (M) is the Fourier transform of M. A simple formal manipulation gives P M (t, q) = k≥0 (−q) −k H N k (−qt). The result now follows from the fact that each N k is finitely generated and only finitely many N k are non-zero [SS5,Theorem 7.7] and the structure theorem for H N k (Theorem 4.2).
There is much more one could say about Poincaré series and variants using the methods of this paper: for example, one could define an "enhanced Poincaré series" or a Poincaré-variant of the formal character, and prove results about them.

A categorification of rationality
5.1. Motivation. Let D : V → V be the Schur derivative. If F is a polynomial functor, then DF is the functor assigning to a vector space V the subspace of F (V ⊕ C) on which G m acts through its standard character (this copy of G m acts by multiplication on C only). In terms of sequences of symmetric group representations, the Schur derivative takes a sequence M = (M n ) n≥0 to the sequence DM given by (DM) n = M n+1 | Sn . See [SS2,§6.4] for more details. (There the Schur derivative is denoted D, but this conflicts with our notation for duality functors.) Let A = A(E) and suppose M is a finitely generated A-module. One observes that H The discussion of the previous paragraph thus suggests a stronger result, namely, that there exists subspaces V 1 , . . . , V n such that ∂ V 1 · · · ∂ Vn annihilates M. We will show that this is indeed the case. This is a categorification of the rationality theorem for H M (t).
One may view this as an analogue of the existence of a system of parameters for a finitely generated module. For some discussion of this analogy, see [SS1,Remark 5.4.2]. (To actually get a functor, one should work with the homotopy category of injective objects.) We define a differential operator on D b fg (A) to be a finite composition of endofunctors of the form ∂ V . We say that M satisfies a differential equation if there exists a differential operator ∂ such that ∂(M) = 0.

The main theorem. Let
Theorem 5.1. Every object of D b fg (A) satisfies a differential equation. Lemma 5.2. Any two differential operators commute (up to isomorphism). In particular, if ∂ is a differential operator and M satisfies a differential equation then ∂(M) also satisfies a differential equation.
Lemma 5.3. In an exact triangle in D(A), if two terms satisfy differential equations then so does the third.
Let r ≥ 0 be an integer, let Y = Gr r (E), let π : Y → Spec(C) be the structure map, let Q be the tautological bundle on Y , and let B = A(Q). For V ⊂ E, we define the differential operator ∂ V on B-modules just as for A-modules, and we define a differential operator on D(B) to be a composition of ∂ V 's.
also satisfies a differential equation. Proof. By induction on Tor-dimension, we can assume F is O Y -flat. We have The map E ⊗ (F ⊗ M) → D(F ⊗ M) maps into the factor F ⊗ D(M). We thus see that We thus see that if ∂ is a differential operator annihilating M then ∂(F ⊗ M) is a direct sum of objects of the form F ′ ⊗ M ′ , where F ′ has strictly lower degree than F and M ′ is the result of applying some differential operator to M. Since each M ′ satisfies a differential equation, the result follows by induction on the degree of F.
Lemma 5.5. B satisfies a differential equation.
For appropriate choices of the V i 's, the above complex is acyclic: indeed, simply choose the V i 's so that for every y ∈ Y there is some i for which V i (y) → Q(y) is an isomorphism.
Proof of Theorem 5.1. It is clear that differential operators commute with Rπ * . We thus fg (A) satisfies a differential equation. These objects generate D b fg (A) by [SS5,Corollary 6.16] (note that by [SS5,Remark 6.15], one can take the F to be O Y -flat). Thus all objects of D b fg (A) satisfy a differential equation. 5.3. Differential operators and duality. Let M be a finitely generated A(E)-module. By Proposition 4.3, we have Hence, if ∂ is a differential operator that annihilates f (t), then ∂, obtained by applying the substitution d dt → d dt − d to ∂, is a differential operator that annihilates e dt f (−t). We can realize this identity using differential operators as follows. Pick a minimal free resolution F • of M. Note that

Furthermore, for any subspace
Proof. (We omit many of the details in this proof to keep it short.) By definition, we have where K = K(E) is the Koszul complex and (−) ∨ denotes vector space duality on multiplicity spaces. Using the fact that D satisfies the Leibniz rule and D(Sym (E[1] The above identity neglects the differentials: in fact, the complex inside the Fourier transform is actually the cone on the canonical map E ⊗ M → D(M), namely ∂ E (M). We thus find that ∂ (E/V ) * (F E (M)) is the cone on the natural map 5.4. The category of differential operators. We end this section with some additional thoughts on differential operators. It could be interesting to pursue these observations further in the future.
Write E for the endofunctor M → E ⊗ M of Mod A , and write D for the endofunctor given by the Schur derivative. There is then a natural transformation E → D, from which all of the differential operator structure derives. Let G ⊂ GL(E ⊕ V) be the subgroup fixing all elements of E, and let C be the category of polynomial representations of G. Then C is the universal tensor category having an object X and a morphism E → X (one takes X = E ⊕ V): that is, given any tensor category D and a morphism E → X in D, there is a unique left-exact tensor functor C → D mapping E → E ⊕ V to E → X. (The group G is an example of a "general affine group," and the universal property of C, in the d = 1 case, can be found in [SS3,§5.4]. Note that E is simply isomorphic to a direct sum of d copies of the unit object of C.) The universal property of C furnishes a functor C → End(Mod A ) taking E → E ⊕ V to the transformation E → D discussed above. Alternatively, we can think of this functor as an action C × Mod A → Mod A . This action induces one on derived categories D The tensor product structure on D b (C) corresponds to composition of differential operators.
Our main theorem on differential equations states that every object of D b fg (A) is annihilated by some object of D b fg (C). It would be interesting if this result could be made more precise. For example, given M ∈ D b fg (A), what does the annihilator in D b fg (C) look like? Is it principal, as a tensor ideal?
Another potentially interesting observation: C is equivalent to both the category Mod 0 A of A-modules supported at 0 and the category Mod gen A of "generic" A-modules (see [SS5,Propositions 5.4,5.6]). Is this more than a coincidence? Does the action of C on Mod A come from, or extend to, an interesting action of Mod A on itself? We have not fully investigated these questions.

D-finiteness of Hilbert series of bounded modules
Recall that a power series f ∈ C t is D-finite if it satisfies a differential equation of the form In this section, we prove the following theorem.
Theorem 6.1. Let A be a finitely generated tca with A 0 = C and let M be a finitely generated A-module which is bounded. Then H M (t) is D-finite.
We can assume that A is a polynomial tca. Recall the notation from §4.3.
Lemma 6.2. Suppose max(ℓ(A), ℓ(M)) < ∞. For d ≥ max(ℓ(A), ℓ(M)), we have Proof. H M (t) is obtained from the enhanced Hilbert series H M (t) by doing the substitution t i → 0 for i ≥ 2 and t 1 → t. Apply this substitution to Lemma 4.8. Then K(t, α) becomes where the first equality comes from Schur-Weyl duality (or see [Sta,Corollary 7.12.5]).
We have A(C d ) = Sym(V ) (we warn the reader that this is evaluation of A on the vector space C d , not to be confused with A(C d )) for some polynomial representation V of GL(d).
for some polynomial p. It suffices to treat the case where p is a monomial, say p(α) = α b , with b ∈ Z d . We then have and hence where for y ∈ Z d ≥0 we write |y| = i y i and y! = i y i !. Put Then F is a rational function of the u i . Indeed, let R be the ring C[ξ 1 , . . . , ξ n ] and give R a Z d ≥0 grading by deg(ξ i ) = A i . Then F is the T -equivariant Hilbert series of the free R-module with one generator of degree b, and is therefore rational [MS,Theorem 8.20]. The theorem then results from the following general result: Proof. Define the Hadamard product * on multivariate generating functions by the formula  Furthermore, we define a multivariate generating function F to be D-finite if the vector space (over the field of rational functions) spanned by all partial derivatives of F is finitedimensional. By [Lip,Remark 2], the Hadamard product of two D-finite generating functions is D-finite. It is clear that a rational function is D-finite and the exponential function exp(u 1 + · · · + u d ) is also D-finite. Hence we conclude that their Hadamard product F (u 1 , . . . , u d ) = y∈Z d ≥0 a y u y y! is also D-finite. Now do the substitution u i → t for i = 1, . . . , d. By the chain rule, so the result is also D-finite.

Examples and applications
7.1. Hilbert series of polynomial tca's. For an integer n ≥ 1, define a linear map , by taking t λ to t nλ , where nλ = (nλ 1 , nλ 2 , . . .). We extend this map to series in the obvious manner. The following is the main result of this section: Proposition 7.1. Let V be a finite length object of V concentrated in degree d = 0 and let A = Sym(V ). Then Proof. Both sides convert direct sums in V to products, so it suffices to treat the case where V = S λ (C ∞ ). Recall from (2.3) that The GL-character of A is T (1 − x T ) −1 where the sum is over all semistandard Young tableaux T of shape λ and x T is the product of x where m i (T ) is the multiplicity of i in T . Apply log to this expression: Making the substitution p i → it i , we obtain the stated result (see Lemma 4.4).
Example 7.2. Take V = (C ∞ ) ⊗d . Then H V (t) = t d 1 , and so H V (t [n] ) = t d n . We thus obtain Note that for d = 1 this is exp(T 0 ).
In particular, H A (t) = exp( 1 2 t 2 ). The case V = 2 (C ∞ ) is similar: just change the + sign to a − sign.
Remark 7.4. In the notation of the proposition, the tca A is generated in degree d, and so Theorem 4.5 cannot be applied to it for d > 1; in fact, the above calculations show that the conclusion of the theorem is false in this case. Nonetheless, the values of H A computed above are sufficiently nice that one might hope for a good generalization of Theorem 4.5 which includes these cases.

Formal characters of determinantal rings. Let A be the tca A(E)
, where E is a d-dimensional vector space. Let a r ⊂ A be the rth determinantal ideal, generated by the representation r+1 (C ∞ ) ⊗ r+1 (E). In this section, we derive a formula for the formal character of A/a r .
Recall that we have an isomorphism of Grothendieck groups Proof. Let Q r be the tautological bundle on Y r = Gr r (E), and let π r : Y r → Spec(C) be the structure map. Let i r : K(Gr r (E)) → K(A) be the map defined by i r ([V ]) = [R(π r ) * (V ⊗ A(Q r ))]. According to [SS5,Theorem 6.19], the maps i r induce the isomorphism (7.5). Now, The vector bundle S λ (Q r ) has no higher cohomology, and its space of sections is S λ (E). We thus find that R(π r ) * (A(Q r )) = A/a r , which proves the lemma.
For the rest of the section, we fix r and put Y = Gr r (E). We let Q be the tautological bundle on Y and π : Y → Spec(C) the structure map. From the above lemma and Theorem 3.3, we see that the formal character Θ A/ar is given by θ r ([O Y ]). We now compute θ r ([O Y ]), at least to some extent. To do this, we must compute the inner product M Lemma 7.7. If ℓ(λ) ≤ r, then S (r) Proof. We first verify this when λ = (n). In that case, it follows from the definitions (or the general formula [Mac,p. 47,Example I.3.10]) that i (x 1 , . . . , x r ), and so , and so we see Here we used the fact that π * (Sym i (Q)) has dimension d+i−1 i , and the higher pushforwards vanish. We thus find For the second and fourth equalities, we used the identities a b = a a−b and a+b−1 b = (−1) b −a b , and the third identity is the Chu-Vandermonde identity. This proves the lemma for λ = (n).
To see this, we first decompose the tensor product Sym n 1 (Q) ⊗ · · · ⊗ Sym nr (Q) into a direct sum of Schur functors S λ (Q) with ℓ(λ) ≤ r = rank(Q) where the bound comes from the Pieri rule. Now Borel-Weil-Bott (see [SS5,Theorem A.1] for a convenient formulation) says that Rπ * S λ (Q) = S λ (C d ), and we get a direct sum which is the same as the tensor product Sym n 1 (C d ) ⊗ · · · ⊗ Sym nr (C d ) (it is important that there are at most r factors in the tensor product so that all Schur functors appearing in it have at most r rows, and therefore do not annihilate Q).
We now prove the lemma for general λ by induction on dominance order (i.e., λ dominates µ if |λ| = |µ| and λ 1 + · · · + λ i ≥ µ 1 + · · · + µ i for all i). The base case is when λ has a single part, which we have already done. In general, s λ = s λ 1 · · · s λr + C where C is a sum of s µ with µ less dominant than λ. By induction, S (r) By the previous paragraph, Hence the statement also holds for λ.
Let K λ,µ be the Kostka number, i.e., the number of semistandard Young tableaux of shape λ and content µ, so that The matrix (K λ,µ ) is invertible; let K −1 λ,µ denote the entries of the inverse matrix, so that The above identities also hold after specializing to any finite number of variables, as this operation is linear. We thus find (1, . . . , 1).
We finally reach our main result: Theorem 7.8. We have where c λ is the integer given by In the case r = 1, we can give a simpler expression for the formal character by computing directly: Proposition 7.9. We have Θ A/a 1 = d−1 n=0 d−1 n σ n . Proof. We have A/a 1 = n≥0 Sym n (E) ⊗ Sym n (C ∞ ), and so where the binomial coefficient is the dimension of Sym n (E). Appealing to the identity from which the stated formula follows. 7.3. Enhanced Hilbert series of determinantal rings. Keep the same notation as the previous section. The formula for Θ A/ar gives a formula for H A/ar by a change of variables. In this section, we compute H A/ar directly, without using any of the theory behind Theorem 3.3. First consider the case r = 1. Then A/a 1 = n≥0 Sym n (C ∞ ) ⊕ dim Sym n (E) , and so the enhanced Hilbert series is Proof. The coefficient of t n 1 · · · t nr in exp(st 1 + s 2 t 2 + · · · ) is s n 1 +2n 2 +···+rnr n 1 ! · · · n r ! .
Using the product rule for derivatives, this can be written as Note that T i = 1 i! d i ds i (st 1 + s 2 t 2 + · · · )| s=1 , so we see that this expression is a polynomial in T 1 , . . . , T m−1 times exp(T 0 ).
Using [Ges,Theorem 16], we can give a determinantal formula for the enhanced Hilbert series of A/a r , in general, in terms of certain modifications of (7.10). Define, for any i ∈ Z, Then a 0 is the same as (7.10), and the enhanced Hilbert series of A/a r is the determinant det(a j−i ) r i,j=1 . From this formula, we deduce that the enhanced Hilbert series is a polynomial in T 1 , . . . , T d−1 times exp(rT 0 ). This gives an independent verification of the rationality theorem in this case. 7.4. Hilbert series of invariant rings. Let E be a finite dimensional representation of a reductive group G and let A be the tca A(E) G . Using the symmetric group model for V, A is given by A n = (E ⊗n ) G . Since A(E) is bounded, so is A, and so Theorem 6.1 assures us that H A is D-finite.
For the purposes of this section, we let H * A (t) = n≥0 dim(A n )t n . This is the nonexponential form of H A (t). The D-finiteness of H A is equivalent to the D-finiteness of H * A : each of the series n t n /n! and n n!t n is D-finite, and D-finiteness is preserved under Hadamard product by [Lip,Remark 2] (see also [Sta,Theorem 6.4.12] for a simpler reason for univariate series).
Example 7.12. Take G = SL(2) and E = C 2 . Then the dimension of (E ⊗n ) G is 0 for n odd and the Catalan number C n/2 for n even. (Recall that C k = 1 k+1 2k k .) We thus obtain We note that any algebraic function is D-finite. We have This is (essentially) a modified Bessel function of the first kind, and satisfies the differential equation t 2 d 2 y dt 2 + 3t dy dt − 4t 2 y = 0.
Example 7.13. Take G = SL(2) × SL(2) and E = C 2 ⊗ C 2 . Then (E ⊗n ) G is the tensor square of the space ((C 2 ) ⊗n ) SL(2) from the previous example, and so we find This series is not algebraic: indeed, if it were then the Hadamard product of it and the rational series k≥0 (k + 1) 2 t 2k would also be algebraic [Sta,Proposition 6.1.11], but this is the series k≥0 2k k 2 t 2k , which is known to be transcendental [WS,Theorem 3.3]. We thus see that H * A (t) need not always be algebraic. Thus Theorem 6.1 is optimal, in a sense. Remark 7.14. We point out that the D-finiteness of H * A (t) can be seen directly in this case. Let T be a maximal torus of G, and let α 1 , . . . , α r be characters of T giving an isomorphism with G r m . Let χ be the character of E, regarded as a function on T (and thus a Laurent polynomial in the α i 's). Let F (t, α) = n≥0 t n χ n = (1 − tχ) −1 . This is a rational function of t and the α's. By the Weyl integration formula (see §2.2 for the GL(n) case and [FH,§26.2] for the general case), we have where W is the Weyl group of G and ∆ is a certain Laurent polynomial in the α i . The D-finiteness of H * A (t) now follows from the following general fact: taking the constant term (with respect to the α i ) of the the product of a D-finite function with a Laurent polynomial is again D-finite.
Remark 7.15. There are some similar situations where one does not have D-finiteness. For k ≥ n, the canonical map ((C k ) ⊗n ) S k → ((C k+1 ) ⊗n ) S k+1 is an isomorphism. Let A n be the stable value of this vector space (precisely, we could define it as the direct limit). The dimension of A n is the Bell number B n . These numbers do not have a D-finite generating series: one has n≥0 B n t n n! = e e t −1 . This example can be made to look more like the previous ones by using the category Rep(S) of algebraic representations of the infinite symmetric group S, as studied in [SS3]. Let E = C ∞ be the permutation representation of S. Then the sequence (A n ) n≥0 is identified with A(E) S . Thus the conclusion of Theorem 6.1 does not apply to A(E) S .
We thank a referee for suggesting this example.
7.5. Generalizing character polynomials. Let E be a vector space of dimension m, and let M be a finitely generated module over A(E). Theorem 4.5 tells us that the enhanced Hilbert series can be written in the form H M (t) = where p i (t, T ) is a polynomial in t 1 , t 2 , . . . and T 1 , T 2 , . . .. In fact, using Corollary 3.5, we can be more precise: if we set deg(T d ) = d and deg(t d ) = 0, then each polynomial p i has degree ≤ i(m − i).