Character Polynomials and the Restriction Problem

Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.


Introduction
Let K be a field of characteristic 0. Let P = K[X 1 , X 2 , . . . ], a ring of polynomials in infinitely many variables. Regard P as a graded algebra where the variable X i has degree i. In this grading, the monomial X a 1 1 X a 2 2 · · · has degree i ia i . Definition 1.1. For each n ≥ 1, let V n be a representation of the symmetric group S n . The collection {V n } ∞ n=1 is said to have eventually polynomial character, if there exists q ∈ P and a positive integer N such that, for each n ≥ N and each w ∈ S n , trace(w; V n ) = q(X 1 (w), X 2 (w), . . . ), where X i (w) is the number of i-cycles in w. The collection {V n } is said to have polynomial character if N = 1. The polynomial q is called the character polynomial of {V n }.
Character polynomials have been used to study characters of families of representations of symmetric groups that occur naturally in combinatorics, topology and other areas. A survey of their history can be found in the article of Garsia and Goupil [6]. More recently, Church, Ellenberg and Farb [3] developed the theory of FI-modules. They showed that each finitely generated FI-module gives rise to a family of representations with eventually polynomial character.
Any polynomial q ∈ P gives rise to a class function on S n for every positive integer n. The value of this function at w ∈ S n is obtained by substituting for X i the number of i-cycles in w. For each n, we define the moment of q as the average value of the associated class function on S n . The ring P has a basis indexed by integer partitions, which we call the binomial basis (Definition 2.2). We give an explicit formula for the moment of a binomial basis element (Theorem 2.3). This formula can be used to compute inner products of class functions coming from character polynomials. It implies that such an inner product achieves a constant value for large n (Corollary 2.4). This is a character-theoretic analogue of [3,Theorem 1.13], which establishes representation stability for finitely generated FI-modules.
The homogeneous polynomial representations of degree d general linear groups GL n (K) and the representations of the symmetric group S d are linked via Schur-Weyl duality (see [7]). The restriction problem explores a different relationship between polynomial representation of GL n (K) and S n . It asks how an irreducible polynomial representation of GL n (K) decomposes when it is restricted to S n sitting inside GL n (K) as the subgroup of permutation matrices.
For each partition λ, let W λ denote the Weyl functor (see [1, Definition II.1.3]) associated to λ. Let P (n, d) denote the set of all partitions of d with at most n parts. Then W λ (K n ), as λ runs over P (n, d), are the irreducible polynomial representations of the general linear group GL n (K) of degree d.
Consider the decomposition of the restriction of W λ (K n ) to S n into Specht modules: , where the sum is over partitions µ such that n − |µ| ≥ µ 1 . It is wellknown that the coefficients r λµ (n) are eventually constant for large n (this result is attributed to D. E. Littlewood by Assaf and Speyer [2]). Let r λµ be their eventually constant value, which is called the stable restriction coefficient. Finding a combinatorial interpretation of r λµ is known as the restriction problem.
In this article we show that the family {Res GLn(K) Sn W λ (K n )} has polynomial character. We determine its character polynomial S λ (Theorem 2.7) by applying the Jacobi-Trudi identities to the character polynomials of symmetric and exterior powers of K n (Theorem 2.6). The character polynomials of symmetric and exterior tensor powers of K n have generating functions with Eulerian factorization (Theorem 2.6). Multiplying S λ by the character polynomial q µ of Specht modules {V µ[n] } (which was computed by Macdonald [16,Example I.7.14(b)] and Garsia-Goupil [6]), and then taking moments (Theorem 2.3) gives an algorithm to compute stable restriction coefficients (Theorem 3.3). Assaf and Speyer [2] and independently, Orellana and Zabrocki [18] introduced Specht symmetric functions to study the restriction problem. In Section 3.3 we explain the relationship between these two approaches.
Notwithstanding several interesting recent developments [2,8,18,19], a solution to the restriction problem remains elusive. Even r λ∅ (here ∅ denotes the empty partition of 0, so r λ∅ is the dimension of the space of S n -invariant vectors in W λ (K n ) for large n) appears to be a non-trivial and interesting problem. We provide generating functions in λ for the dimension of the space of S n -invariant vectors in W λ (K n ) (Corollary 4.3). Using our main generating function (Theorem 4.1) for the dimension of S n -invariants in mixed tensors, we are able to characterize partitions with two rows, two columns and hook partitions which have non-zero S n -invariant vectors. To the best of our knowledge, the specific problem of determining the S n -invariant vectors of a Weylmodule has not been studied before.
We conclude this paper by placing the restriction problem in the context of strict polynomial functors and FI-modules. Friedlander and Suslin [5] introduced strict polynomial functors of degree d. The polynomial representations of degree d of GL n (K) are obtained by evaluating strict polynomial functors of degree d at K n . Similarly, families of representations of S n with stability properties can be obtained by evaluating finitely generated FI-modules at {1, . . . , n} (see [3]). We define a functor from the category of strict polynomial functors of degree d to the category of finitely generated FI-modules for every d (Section 5.3). This functor corresponds to restriction of representations from GL n (K) to S n under evaluation functors (Theorem 5.1).

Moments and Stability.
Definition 2.1 (Moment). The moment of q ∈ P at n is defined as: We shall express integer partitions in exponential notation: given a partition α with largest part r, we write: where a i is the number of parts of α of size i for each 1 ≤ i ≤ r. Thus α is a partition of the integer |α| := a 1 + 2a 2 + · · · + ra r . For every integer partition α = 1 a 1 · · · r ar define X α ∈ P by: Definition 2.2 (Binomial basis). The basis of P consisting of elements X α α is an integer partition is called the binomial basis of P .
For an integer partition α = 1 a 1 2 a 2 · · · r ar , define z α = r i=1 i a i a i !. This is the order of the centralizer in S n of a permutation with cycletype α. Theorem 2.3. For every integer partition α = 1 a 1 2 a 2 · · · , we have: Proof. We have: Replace the sum w ∈ S n by a sum over conjugacy classes in S n . If β = 1 b 1 2 b 2 · · · is a partition of n, then the number of elements in S n with cycle type β is n!
Since γ⊢n 1/z γ = 1 for every n, we get: from which Theorem 2.3 follows.
For two representations V and W of S n , let: which is the same as the Schur inner product of their characters: Corollary 2.4. For any q ∈ P of degree d, q n = q d for all n ≥ d. In particular, if {V n } and {W n } are families of representations with polynomial characters of degree d 1 and d 2 , then V n , W n n stabilizes for n ≥ d 1 + d 2 .
Proof. This follows from the fact that the polynomials X α , as α runs over the set of integer partitions, form a basis of P . Definition 2.5 (Stable moment). For a polynomial q ∈ P we define the stable moment q of q to be the eventually constant value of q n : q = lim n→∞ q n .
, the Specht module of S n corresponding to the padded partition λ[n]. It is well-known that {V n } is a family of representations with eventually polynomial character [6, Proposition I.1]. In other words, for every partition λ, there exists a polynomial q λ ∈ P such that where χ λ[n] denotes the character of the Specht module V λ[n] . Given three partitions λ, µ, and ν of the same integer k, let g λµν (n) denote the multiplicity of V λ[n] in V µ[n] ⊗ V ν(n) . Then g λµν (n) = q λ q µ q ν n .

Symmetric and Alternating
Tensors. Let Sym d and ∧ d denote the symmetric and alternating tensor functors respectively. For every n ≥ 0, Sym d (K n ) and ∧ d (K n ) can be regarded as representations of S n . In this section, we will prove that they have polynomial character by direct computation.
We shall work with generating functions that live in the ring P [[t]] of Laurent series in the variable t and coefficients in P .
which will be needed later. Also, given w ∈ S n for any n and q ∈ P , we write q(w) = q(X 1 (w), X 2 (w), . . . ).
d=0 be the sequence of polynomials in P defined by: As a consequence, for every n ≥ 1 and every w ∈ S n , be the sequence of polynomials in P defined by: Then for every n ≥ 1 and every w ∈ S n , For every positive integer d, we have: Proof. The standard basis of K n is indexed by the set [n] = {1, . . . , n}. The space Sym d K n has an induced basis indexed by multisets of size d with elements drawn from [n]. The trace of w ∈ S n on Sym d K n is the number of such multisets that are fixed by w. In a multiset that is fixed by w, the elements in each cycle of w appear with the same multiplicity. Hence a multiset fixed by w may be regarded as a multiset of cycles of w.
where the sum runs over all multisets M of cycles of w. Each of the X i (w) i-cycles of w contributes a factor (1 − t i ) −1 to this generating function, so Similarly, ∧ d K n has a basis indexed by subsets of [n] with d elements. Only subsets fixed by w contribute to the trace, and these are unions of the cycles of w. A cycle of length i changes the sign of the corresponding basis vector by a factor of −(−1) i . Assign weight where the sum runs over all subsets N of the set of cycles of w. Each of the X i (w) i-cycles of w contributes a factor (1−(−t) i ) to this generating function, so Expansion of the products in (5) and (6) using (3) and (4) gives (7) and (8) respectively.

Character Polynomials of Weyl Modules.
Applying the Jacobi-Trudi identities [16,Section I.3] to the character polynomials of Sym d and ∧ d gives character polynomials for Weyl functors. For a partition λ, let λ ′ denote its conjugate partition.
Theorem 2.7. For every partition λ, the element of P defined by is such that for every positive integer n and every w ∈ S n , S λ (w) = trace(w; W λ (K n )).
The polynomials S λ for partitions λ of integers at most 5 are given in Table 1. The highest degree coefficients in these expansions are character values of symmetric groups. More precisely, we have: The theorem will be a consequence of the following lemma: Lemma 2.9. Let λ be a partition of a positive integer d. For every partition α of d, the coefficient of X α in the expansion of H λ in the binomial basis is σ λ (w α ), the value of the character σ λ of the permutation representation of S d induced from the trivial representation of the Young subgroup S λ 1 × · · · × S λ l (see [9,Section 2.2] or [20, Section 2.3]) at a permutation w α with cycle type α.
Let K[X λ ] be the permutation representation associated to the action of S d on X λ . This representation is isomorphic to the representation of S d induced from the trivial representation of its Young subgroup S λ 1 ×· · ·×S λ l . Therefore σ λ is the character of K[X λ ], and σ λ (w α ) = |X wα λ |, the number of fixed points of a permutation w α in X λ . Take (T 1 , . . . , T l ) ∈ X wα λ . Then each T i is formed by taking a union of cycles of w α . Suppose that b ij is the number of j cycles of w α in T i . Then the array (b ij ) satisfies the constraints: where a j is the number of j-cycles in α. Let B(λ; α) denote the set of such arrays. We have: On the other hand, by (7), The terms of homogeneous degree d in this product come from the top degree terms in each factor. When |α| = |λ|, such a term has leading coefficient j X a j j if and only if (b ij ) ∈ B(λ; α). Hence the coefficient of X α is the expression on the right hand side of (12) and the lemma follows.
Proof of Theorem 2.8. For partitions λ and µ of d, let K µλ denote the number of semistandard Young tableaux of shape µ and weight λ. Then K = (K µλ ) is a unitriangular integer matrix with rows and columns indexed by partitions of d. We have: Let K −1 µλ be the entries of the inverse matrix K −1 . Then (14), thereby completing the proof of Theorem 2.8.
Theorem 2.10. For every partition λ = (λ 1 , . . . , λ l ), S λ is the coefficient Proof. For every vector λ = (λ 1 , . . . , λ l ) with non-negative integer coefficients, define: . When λ is a partition this coincides with the character polynomial of the Weyl module W λ . Then, for every partition λ, S λ is the coefficient of t λ in λ≥0 S λ t λ . Here λ ≥ 0 indicates that the sum is over all vectors in Z l ≥0 , and t λ = t λ 1 1 · · · t λ l l . Now Here we have used the convention that H d = 0 when d < 0.

Duality.
Going through the entries of Table 1, the reader may have noticed that the coefficients in the expansion of S λ agree up to sign with those of S λ ′ for every partition λ. For each vector µ = (µ 1 , . . . , µ m ) with non-negative integer entries, let X µ = X µ 1 1 · · · X µm m , and |µ| = µ 1 + · · · + µ m .

Character Polynomials of Specht Modules.
Recall that for partitions λ and µ, we say that λ − µ is a vertical strip if the Young diagram of µ is contained inside the Young diagram of λ, and each row of the Young diagram of λ contains at most one box that is not in the Young diagram of µ [16, Section I.1].
For any partition λ, Macdonald [16, Example I.7.14(b)] gave the character polynomials q λ ∈ P of (2) as follows: It immediately follows that the leading coefficients of q λ in the binomial basis are the same as those of S λ (see Theorem 2.8): Theorem 3.1. Let λ be a partition of a positive integer d. For every partition α of d, the coefficient of X α in the expansion of q λ in the binomial basis of P is χ λ (α). Proof. Regard P as a graded algebra where the degree of X i is i for each i ≥ 1. Let P d denote the homogeneous elements of degree d in P . The degree d homogeneous parts of X α , as α runs over all partitions of d, form a basis of P d . Theorem 2.8 and the identity (15) imply that the degree d homogeneous parts of S λ and q λ also form such a basis as λ runs over all partitions of d, since the character table of S d forms a non-singular matrix. Therefore S and q are bases of P .

Stable Restriction Coefficients.
The coefficients in the expansion of the elements of the basis S in terms of the basis q: are called the stable restriction coefficients. They determine the decomposition of a Weyl module W λ (K n ) into irreducible representations of S n : The following result, which is now immediate, is an algorithm for computing the stable restriction coefficients: The polynomial S λ can be computed using Theorem 2.7, q µ using (15). After expanding the product in the binomial basis, the moment can be computed using Theorem 2.3. The matrix of the stable restriction coefficients r λµ , as λ and µ run over partitions of 0 ≤ n ≤ 5 is given by: The blocks demarcate the partitions of each integer n, and within each block, the partitions of n are enumerated in reverse lexicographic order.

Relation to Symmetric Functions.
Let Λ denote the ring of symmetric functions (as in [16,Section I.2]). Macdonald [16,Example I.7.13] constructed an isomorphism φ : Λ → P taking the Schur function s λ to the character polynomial q λ . In this section we study a different isomorphism Φ : Λ → P , due to Orellana and Zabrocki [18], which takes s λ to S λ . Under this isomorphism q λ is the image ofs λ the Specht symmetric functions of [2,18].
Following [18,Proposition 12], define an algebra homomorphism Φ : Λ → P by: For each k > 0, define Ξ k = 1, e 2πi/k , e 4πi/k , . . . e 2(k−1)πi/k , and for an integer partition µ = (µ 1 , . . . , µ m ), Let R n denote the space of K-valued class functions on S n . For every n ≥ 0 there is a map ev n Λ : Λ → R n defined by: , where µ is the cycle type of w. In other words, the symmetric function is evaluated on |µ| variables, whose values are given by the list Ξ µ , the remaining variables being set to 0. With this definition, ev n Λ (s λ ) is the character of Res For each q ∈ P consider the function ev n P (q) ∈ R n given by: ev n P (q)(w) = q(X 1 (w), X 2 (w), . . . ). This defines a ring homomorphism ev n P : P → R n . Observe that ⊕ ∞ n=1 ev n P : P → ⊕ ∞ n=1 R n is injective, for if ev n P (q) ≡ 0 for all n, then q vanishes whenever X 1 , X 2 , . . . take non-negative integer values, and hence q must be identically 0.
Theorem 3.4. The algebra homomorphism Φ is the unique K-linear map Λ → P such that the diagram commutes for every n ≥ 1.
Proof. From the definition of ev n Λ , Since ⊕ n ev n P : P → ⊕R n is injective, Φ(p k ) is completely determined by the commutativity of (17). Since the polynomials {p k } k≥1 generate Λ, Φ is completely determined by its values on p k .
Lemma 3.5. The homomorphism Φ : Λ → P is an isomorphism of rings.
Theorem 3.6. For every partition λ, we have: Proof. This follows immediately from Theorem 3.4.
We use the notation R ⊏ [l] to signify that R is a multiset with elements drawn from [l]. We write t R for the monomial where t i is raised to the multiplicity of i in R. Similarly, for any S ⊂ [m], we write u S = i∈S u i .
Proof. Using (5) and (6), we have: Now proceeding as in the proof of Theorem 2.3, which is equivalent to the desired expression.
Corollary 4.2. We have: where the product is over R ⊏ [l], S ⊂ [m], with at least one of R and S non-empty.
Proof. From Theorem 4.1 we get: (20) λ≥0,n≥0 Using this, the corollary can be deduced from Theorem 2.10 by taking moments.

S n -invariant Vectors.
For a representation V n of S n , let V Sn n denote the subspace of S n -invariant vectors. If a family {V n } of representations has polynomial character q ∈ P , then q n = dim(V Sn n ) for all n ≥ 0. Therefore, for any partition λ, dim W λ (K n ) Sn = S λ n . In particular, W λ (K n ) has a non-zero S n -invariant vector if and only if S λ n = 0.
Proof. We use Littlewood's plethystic formula [15, Theorem XI] (see also, [17, Theorem 2.6]) for restriction coefficients. This formula asserts that, for every partition λ with at most n parts, and every partition µ of n, the multiplicity of the Specht module V µ in Res The inequality above holds because the plethystic substitution of a Schurpositive symmetric function into another is Schur-positive.
Definition 4.5 (Vector Partitions). Let v ∈ Z l ≥0 . A vector partition of v is an unordered collection v 1 , . . . , v n of non-zero vectors in Z l ≥0 such that v = v 1 + · · · + v n .
Let p n (v) (resp. p ≤n (v)) denote the number of vector partitions of v with exactly (resp. at most) n parts.
Remark 4.7. In general, we do not know of a combinatorial proof of the non-negativity of w∈S l sgn(w)p ≤n (λ 1 − 1 + w(1), . . . , λ l − l + w(l)), which follows from Theorem 4.6. When l = 2, this is the main result of Kim and Hahn [10], who refer to it as a conjecture of Landman, Brown and Portier [14].
The problem of characterizing those partitions λ for which W λ (K n ) has a non-zero S n -invariant vector for large n appears to be quite hard. The following result solves this problem for partition with two rows, two columns, and for hook-partitions.  Proof of (4.8.1). By Theorem 4.6 we need to show that, for every λ 1 ≥ λ 2 ≥ 1, p ≤n (λ 1 , λ 2 ) > p ≤n (λ 1 + 1, λ 2 − 1) for sufficiently large n, unless λ 1 = λ 2 = 1. From the main result of Kim and Hahn [10] (the result on the last line of the first page), it follows that p n (λ 1 , λ 2 ) ≥ p n (λ 1 + 1, λ 2 − 1) for all n ≥ 1.
Therefore, it suffices to prove that p n (λ 1 , λ 2 ) > p n (λ 1 + 1, λ 2 − 1) for at least one value of n. When k ≥ l ≥ 1 are such that at least one of k and l is even, p 2 (k, l) > p 2 (k + 1, l − 1). When both k and l are odd and (k, l) = (1, 1), p 3 (k, l) > p 3 (k + 1, l − 1). These inequalities will be proved in Lemmas 4.10 and 4.11 below. if both k and l are even,  Thus, if k and l are both even and k ≥ l, then If one of k and l is even and the other is odd, then thereby completing the proof of Lemma 4.10.
Proof. When k and l are odd, Lemma 4.9 gives: This is clearly positive for all k ≥ l such that k ≥ 3 and l ≥ 1.
The coefficient of t i u j in the above expression is the numberp(i, j) of ways of writing the vector (i, j) as a sum of vectors of the form (a, 0) where a > 0, and (a, 1) where a ≥ 0, and vectors of the form (a, 1) are used at most once.
Ifp(i, j) > 0, then j distinct vectors of the form (a, 1) are used, so that i ≥ j 2 . Therefore If a < b+1 2 , then a − j < b+j+1 2 for all j ≥ 0. By (27) H a−j E b+j+1 = 0 for all j ≥ 0. This allows us to extend the index of summation in the right hand side of (25) without changing the sum: Taking l = 0 and m = 1 in Corollary 4.2 can be used to show that E k = 0 for all k > 1, so W (0|a+b+1) = E a+b+2 = 0 for all a, b ≥ 0. Therefore, W (a|b) = 0 for a < b+1 2 . Conversely, suppose a ≥ b+1 2 . By Theorem 4.4, it suffices to show that W λ (K n ) contains a non-zero S n -invariant vector for some positive integer n. We shall show that W λ (K b+1 ) contains a non-zero S b+1 -invariant vector. The hook partition λ = (a + 1, 1 b ) dominates the partition µ = (a − b 2 + 1, b, b − 1, . . . , 2, 1), which has b + 1 distinct parts. Therefore W (a|b) (K a+b+1 ) contains a non-zero vector v with weight µ. For each w ∈ S n let v w = ρ (a|b) (w)v. Then v w lies in the weight space of w · µ. Hence the vectors {v w | w ∈ S n } are linearly independent, and generate a representation that is isomorphic to the regular representation of S n . In particular, the trivial representation is contained in W (a|b) (K a+b+1 ).

Strict Polynomial Functors and FI-modules
In this section we may take K to be any field (not necessarily of characteristic zero). Friedlander and Suslin [5] introduced strict polynomial functors to unify homogeneous polynomial representations of GL n (K) of degree d across all n. Later, Church, Ellenberg and Farb [3] introduced FI-modules to unify representations of S n across all n. In this section, we lift the restriction functor Res GLn(K) Sn to a functor from the category of strict polynomial functors to the category of FI-modules.

Strict Polynomial Functors.
The Schur category (also known as the divided power category, see [22]) Γ d is the category whose objects are finite dimensional vector spaces over K. Given objects V and W , Hom Γ d (V, W ) = Hom S d (V ⊗d , W ⊗d ).
The category of strict polynomial functors is the category Rep Γ d whose objects are K-linear covariant functors from Γ d to the category of K-vector spaces, and morphisms are natural transformations between functors (see [5,Section 2]). When K has characteristic zero, Rep Γ d is semisimple, and its simple objects are functors known as Weyl functors (see [11]). For each partition λ of d let W λ denote the Weyl functor corresponding to λ.
Let Rep d GL n (K) denote the category of homogeneous polynomial representations of GL n (K) of degree d. Define a functor ev n : Rep Γ d → Rep d GL n (K) as follows: for each strict polynomial functor F : Γ d → Vec define ev n (F ) = F (K n ). Let T ∈ GL n (K) act on F (K n ) by F (T ⊗d ). This makes F (K n ) a representation of GL n (K) which turns out to be a homogeneous polynomial representation of degree d. For each n ≥ 1 and λ ∈ P (n, d), ev n (W λ ) = W λ (K n ) is the irreducible polynomial representation of GL n (K) corresponding to λ (consistent with the notation of Section 1).

5.2.
FI-modules. The category FI is the one that has finite sets as objects, and injective functions as morphisms. The category of FI-modules is the category FI−Mod whose objects are covariant functors from FI to the category of K-vector spaces, and morphisms are natural transformations of functors. Let Rep S n denote the category of representations of S n over K. The evaluation functor ev n : FI−Mod → Rep