Character Polynomials and the Restriction Problem
Sridhar Narayanan, Digjoy Paul, Amritanshu Prasad, Shraddha Srivastava
aa r X i v : . [ m a t h . R T ] J a n Character Polynomials and theRestriction Problem
Sridhar P. Narayanan, Digjoy Paul, Amritanshu Prasad,and Shraddha Srivastava
Abstract.
Character polynomials are used to study the restric-tion of a polynomial representation of a general linear group to itssubgroup of permutation matrices. A simple formula is obtainedfor computing inner products of class functions given by characterpolynomials. Character polynomials for symmetric and alternat-ing tensors are computed using generating functions with Eulerianfactorizations. These are used to compute character polynomi-als for Weyl modules, which exhibit a duality. By taking innerproducts of character polynomials for Weyl modules and characterpolynomials for Specht modules, stable restriction coefficients areeasily computed. Generating functions of dimensions of symmetricgroup invariants in Weyl modules are obtained. Partitions with tworows, two columns, and hook partitions whose Weyl modules havenon-zero vectors invariant under the symmetric group are charac-terized. A reformulation of the restriction problem in terms of arestriction functor from the category of strict polynomial functorsto the category of finitely generated FI-modules is obtained.
Contents
1. Introduction 22. Character Polynomials and their Moments 42.1. Moments and Stability 42.2. Symmetric and Alternating Tensors 62.3. Character Polynomials of Weyl Modules 82.4. Duality 113. The Restriction Problem 123.1. Character Polynomials of Specht Modules 12
Mathematics Subject Classification.
Key words and phrases. character polynomial, restriction problem. S n -invariant Vectors 165. Strict Polynomial Functors and FI-modules 205.1. Strict Polynomial Functors 205.2. FI-modules 215.3. The Restriction Functor 21Acknowledgements 23References 23
1. Introduction
Let K be a field of characteristic 0. Let P = K [ X , X , . . . ], aring of polynomials in infinitely many variables. Regard P as a gradedalgebra where the variable X i has degree i . Definition . For each n ≥
1, let V n be a representation of thesymmetric group S n . The collection { V n } ∞ n =1 is said to have eventuallypolynomial 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 ( w ) , X ( w ) , . . . ) , where X i ( w ) is the number of i -cycles in w . The collection { V n } is saidto 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 fam-ilies of representations of symmetric groups that occur naturally incombinatorics, topology and other areas. A survey of their historycan be found in the article of Garsia and Goupil [ ]. More recently,Church, Ellenberg and Farb [ ] developed the theory of FI-modules.They showed that each finitely generated FI-module gives rise to afamily of representations with eventually polynomial character.Any polynomial q ∈ P gives rise to a class function on S n for everypositive integer n . The value of this function at w ∈ S n is obtained bysubstituting for X i the number of i -cycles in w . For each n , we definethe moment of q as the average value of the associated class functionon S n . The ring P has a basis indexed by integer partitions, which wecall the binomial basis (Definition 2.2). We give an explicit formula forthe moment of a binomial basis element (Theorem 2.3). This formula HARACTER POLYNOMIALS AND THE RESTRICTION PROBLEM 3 can be used to compute inner products of class functions coming fromcharacter polynomials. It implies that such an inner product achieves aconstant value for large n (Corollary 2.4). This is a character-theoreticanalogue of [ , Theorem 1.13], which establishes representation stabil-ity for finitely generated FI-modules.For each partition λ , let W λ denote the Weyl functor (see [ , Defini-tion II.1.3]) associated to λ . Let P ( n, d ) denote the set of all partitionsof d with at most n parts. Then W λ ( K n ), as λ runs over P ( n, d ), arethe irreducible polynomial representations of the general linear group GL n ( K ) of degree d . For a partition µ = ( µ , . . . , µ m ) of size | µ | and an integer n ≥ µ + | µ | , let µ [ n ] denote the padded partition( n − | µ | , µ , µ , . . . , µ m ). Let V µ [ n ] denote the Specht module of S n cor-responding to µ [ n ]. Consider the decomposition of the restriction of W λ ( K n ) to S n into Specht modules:Res GL n ( K ) S n W λ ( K n ) = M µ V ⊕ r λµ ( n ) µ [ n ] , where the sum is over partitions µ such that n − | µ | ≥ µ . It is well-known that the coefficients r λµ ( n ) are eventually constant for large n (this result is attributed to D. E. Littlewood by Assaf and Speyer [ ]).Let r λµ be their eventually constant value, which is called the stablerestriction coefficient . Finding a combinatorial interpretation of r λµ isknown as the restriction problem .In this article we show that the family { Res GL n ( K ) S n W λ ( K n ) } haspolynomial character. We determine its character polynomial S λ (The-orem 2.8) by applying the Jacobi–Trudi identities to the characterpolynomials of symmetric and exterior powers of K n (Corollary 2.7).The character polynomials of symmetric and exterior tensor powersof K n have generating functions with Eulerian factorization (Theo-rem 2.6). Multiplying S λ by the character polynomial q µ of Specht mod-ules { V µ [ n ] } (which was computed by Macdonald [ , Example I.7.14(b)]and Garsia–Goupil [ ]), and then taking moments (Theorem 2.3) givesan algorithm to compute the stable restriction coefficients (Theorem 3.3).Assaf and Speyer [ ] and independently, Orellana and Zabrocki [ ]introduced Specht symmetric functions to study the restriction prob-lem. In Section 3.3 we explain the relationship between these twoapproaches.Notwithstanding several interesting recent developments [
2, 6, 13,14 ], a solution to the restriction problem remains elusive. Even r λ ∅ (here ∅ denotes the empty partition of 0, so r λ ∅ is the dimension ofthe space of S n -invariant vectors in W λ ( K n ) for large n ) is not well-understood. We provide generating functions in λ for the dimension NARAYANAN, PAUL, PRASAD, AND SRIVASTAVA of the space of S n -invariant vectors in W λ ( K n ) (Corollary 4.3). Usingour main generating function (Theorem 4.1) for the dimension of S n -invariants in mixed tensors, we are able to characterize partitions withtwo rows, two columns and hook partitions which have non-zero S n -invariant vectors.We conclude this paper by placing the restriction problem in thecontext of strict polynomial functors and FI-modules. Friedlander andSuslin [ ] introduced strict polynomial functors of degree d . The poly-nomial representations of degree d of GL n ( K ) are obtained by evaluat-ing strict polynomial functors of degree d at K n . Similarly, represen-tations of S n can be obtained by evaluating FI-modules at { , . . . , n } (see [ ]). We define a functor from the category of strict polynomialfunctors of degree d to the category of finitely generated FI-modulesfor every d (Section 5.3). This functor corresponds to restriction ofrepresentations from GL n ( K ) to S n under evaluation functors (Theo-rem 5.1).
2. Character Polynomials and their Moments2.1. Moments and Stability.
Definition . The moment of q ∈ P at n is definedas: h q i n = 1 n ! X w ∈ S n q ( X ( w ) , X ( w ) , . . . ) . We shall express integer partitions in exponential notation : given apartition α with largest part r , we write: α = 1 a a · · · r a r , 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 + 2 a + · · · + ra r . For everypartition α = (1 a · · · r a r ) define (cid:0) Xα (cid:1) ∈ P by: (cid:18) Xα (cid:19) = (cid:18) X a (cid:19)(cid:18) X a (cid:19) · · · (cid:18) X r a r (cid:19) . Definition . The basis of P consisting of el-ements (cid:26) (cid:18) Xα (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) α is an integer partition (cid:27) is called the binomial basis of P .For α ∈ Z r ≥ , define z α = Q ri =1 i a i a i !. This is the order of thecentralizer in S n of a permutation with cycle-type α . HARACTER POLYNOMIALS AND THE RESTRICTION PROBLEM 5
Theorem . For every α = ( a , . . . , a r ) ∈ Z r ≥ , we have: (cid:28)(cid:18) Xα (cid:19)(cid:29) n = ( if n < | α | , /z α otherwise. Proof.
We have: X n ≥ (cid:28)(cid:18) Xα (cid:19)(cid:29) n v n = X n ≥ n ! X w ∈ S n Y i ≥ (cid:18) X i ( w ) a i (cid:19) v iX i ( w ) . Replace the sum w ∈ S n by a sum over conjugacy classes in S n . If β = 1 b b · · · is a partition of n , then the number of elements in S n with cycle type β is n ! Q i i bi b i ! . We get: X n ≥ (cid:28)(cid:18) Xα (cid:19)(cid:29) n v n = X n ≥ X β ⊢ n Y i ≥ v ib i i b i b i ! (cid:18) b i a i (cid:19) = X b i ≥ a i Y i ≥ v ib i i b i b i ! (cid:18) b i a i (cid:19) = X b i ≥ a i Y i ≥ v ib i i b i a i !( b i − a i )!= X b i ≥ a i Y i ≥ v ia i i a i a i ! v i ( b i − a i ) i b i − a i ( b i − a i )! . Setting c i = b i − a i gives: X n ≥ (cid:28)(cid:18) Xα (cid:19)(cid:29) n v n = v | α | z α Y i ≥ X c i ≥ v ic i i c i c i != v | α | z α X n ≥ v n X γ ⊢ n z γ . Since P γ ⊢ n /z γ = 1 for every n , we get:(1) X n ≥ (cid:28)(cid:18) Xα (cid:19)(cid:29) n v n = v | α | z α − v , from which Theorem 2.3 follows. (cid:3) For two representations V and W of S n , let: h V, W i n = dim Hom S n ( V, W ) , NARAYANAN, PAUL, PRASAD, AND SRIVASTAVA which is the same as the Schur inner product of their characters: h V, W i n = 1 n ! X w ∈ S n trace( w ; V )trace( w, W ) . Corollary . For any q ∈ P of degree d , h q i n = h q i d for all n ≥ d . In particular, if { V n } and { W n } are families of representationswith polynomial characters of degree d and d , then h V n , W n i n stabilizesfor n ≥ d + d . Proof.
This follows from the fact that the polynomials (cid:0) Xα (cid:1) , as α runs over the set of integer partitions, form a basis of P . (cid:3) Definition . For a polynomial q ∈ P wedefine the stable moment h q i of q to be the eventually constant valueof h q i n : h q i = lim n →∞ h q i n . Let V n = V λ [ n ] , the Specht module of S n corresponding to thepadded partition λ [ n ]. It is well-known that { V n } is a family of repre-sentations with eventually polynomial character [ , Prop. I.1]. In otherwords, for every partition λ , there exists a polynomial q λ ∈ P such that(2) χ λ [ n ] ( w ) = q λ ( X ( w ) , X ( w ) , . . . ) for n ≥ | λ | + λ , where χ λ [ n ] denotes the character of the Specht module V λ [ n ] . Giventhree partitions λ , µ , and ν of the same integer k , let g λµν ( n ) denotethe multiplicity of V λ [ n ] in V µ [ n ] ⊗ V ν ( n ) . Then g λµν ( n ) = h q λ q µ q ν i n . By Corollary 2.4, g λµν ( n ) is eventually constant, recovering a well-known theorem of Murnaghan (see [ ]). Church, Ellenberg, and Farb[ , Section 3.4] point out that this result can also be obtained by show-ing that the families V µ [ n ] ⊗ V ν [ n ] and V λ [ n ] come from finitely generatedFI-modules. Let Sym d and ∧ d denote the symmetric and alternating tensor functors respectively. Thenfor every n ≥
0, Sym d ( K n ) and ∧ d ( K n ) can be regarded as represen-tations of S n . In this section, we will prove that they have polynomialcharacter by direct computation.Given w ∈ S n for any n and q ∈ P , let q ( w ) = q ( X ( w ) , X ( w ) , . . . ). HARACTER POLYNOMIALS AND THE RESTRICTION PROBLEM 7
Theorem . Let { H d } ∞ d =0 be the sequence of polynomials in P defined by: (3) ∞ X d =0 H d t d = ∞ Y i =1 (1 − t i ) − X i , an identity in the formal power series ring P [[ t ]] . Then for every n ≥ and every w ∈ S n , H d ( w ) = trace( w ; Sym d ( K n )) . Let { E d } ∞ d =0 be the sequence of polynomials in P defined by: (4) ∞ X d =0 E d t d = ∞ Y i =1 (1 − ( − t ) i ) X i . Then for every n ≥ and every w ∈ S n , E d ( w ) = trace( w ; ∧ d ( K n )) . Proof.
Sym d K n has an obvious basis indexed by multisets of size d with elements drawn from [ n ]. The character of Sym d K n at w ∈ S n isthe number of such multisets that are fixed by w . In a multiset fixed by w , all the elements in any cycle of w appear with the same multiplicity.Therefore, a multiset fixed by w may be regarded as a multiset of cyclesof w with total weight d , where the weight of each i -cycle is i . Hencethe number of multisets of size d with elements drawn from the i -cyclesof w is the coefficient of t d in (1 − t i ) − X i ( w ) . It follows that the totalnumber of multisets of size d with elements drawn from all cycles of w is the coefficient of t d in Q i ≥ (1 − t i ) − X i ( w ) as claimed. A similarargument works for ∧ d K n , taking d -element subsets of [ n ] instead of d -element multisets with elements drawn from [ n ]. Also, each i -cyclecontributes ( − i +1 (cid:3) Expanding out the products in (3) and (4) gives:
Corollary . For every positive integer d , we have: H d = X α ⊢ d d Y i =1 (cid:18)(cid:18) X i a i (cid:19)(cid:19) , (5) E d = X α ⊢ d ( − a + a + ··· d Y i =1 (cid:18) X i a i (cid:19) . (6) Here the partition α has exponential notation a a · · · , and (cid:16)(cid:16) X i a i (cid:17)(cid:17) denotes the multiset binomial coefficient (cid:0) X i + a i − a i (cid:1) . NARAYANAN, PAUL, PRASAD, AND SRIVASTAVA
Applying theJacobi–Trudi identities [ , Section I.3] to the character polynomialsof Sym d and ∧ d gives character polynomials for Weyl functors. For apartition λ , let λ ′ denote its conjugate partition. Theorem . For every partition λ , the element of P defined by (7) S λ = det( H λ i + j − i ) = det( E λ ′ i + j − i ) 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 inTable 1. Theorem . Let λ be a partition of a positive integer d . For everypartition α = 1 a a · · · of d , the coefficient of (cid:0) Xα (cid:1) in the expansion of S λ in the binomial basis (Definition 2.2) is χ λ ( w α ) , where w α is a permutationwith cycle type α . The theorem will be a consequence of the following lemma:
Lemma . Let λ be a partition of a positive integer d . For every par-tition α of d , the coefficient of (cid:0) Xα (cid:1) in the expansion of H λ in the binomialbasis is σ λ ( w α ) , where σ λ ( w α ) is the value of the character σ λ of the permu-tation representation induced from the Young subgroup S n − d × S λ × · · · × S λ l of S n (see [ , Section 2.2] or [ , Section 2.3] ) at a permutation with cycletype α . Proof.
For a partition λ = ( λ , . . . , λ l ) of d , let λ = n − d . Let X n,λ = { ( T , . . . , T l ) | [ n ] = T ∪ · · · ∪ T l is a partition, with | T i | = λ i } . Let K [ X n,λ ] be the permutation representation associated to the action of S n on X n,λ . Then σ λ is the character of K [ X n,λ ], and σ λ ( w ) = | X wn,λ | , thecardinality of the number of fixed points of a permutation w in X n,λ . Take( T , . . . , T l ) ∈ X wn,λ . Then each T i is formed by taking a union of cycles of w . Suppose that b ij is the number of j cycles of w in T i . Then the array( b ij ) satisfies the constraints: b i + 2 b i + · · · = λ i for each i, (8) b j + b j + b j + · · · = X j ( w ) for each j. (9)Let B ( λ ; w ) denote the set of such arrays. We have:(10) trace( w, K [ X n,λ ]) = X ( b ij ) ∈ B ( λ ; w ) Y j (cid:18) X j ( w ) b j b j · · · (cid:19) . The multinomial coefficient (cid:0) X j ( w ) b j b j ··· (cid:1) is a polynomial in X j = X j ( w ) withtop degree term Q i ≥ X b ij j /b ij !. Therefore, trace( w, K [ X n,λ ]) is given by a HARACTER POLYNOMIALS AND THE RESTRICTION PROBLEM 9 λ S λ n = 1(1) X n = 2(2) X + X + X (1 ) X − X − X n = 3(3) X + X + X X + X + X (2 , X − X − X (1 ) X − X − X X + X + X n = 4(4) X + X + X X + X + X X + X + X X + X + X + X (3 , X + X + X X − X − X X − X − X − X − X (2 , X − X + X X + X − X X (2 , ) X − X − X X − X − X X − X + X + X + X (1 ) X − X − X X + X + X X + X + X X − X − X − X n = 5(5) X + X + X X + X + X X + X X + X X + X + X X + X X + X X + X X + X + X (4 , X + X + X X + X + X X − X − X X − X X − X X − X − X (3 , X + X + X X − X + X X + X X − X X − X − X X + X X + X X − X X (3 , ) X − X − X X − X X + X + X (2 , X − X − X X − X + X X + X X − X X + X + X X − X X − X X + X X (2 , ) X − X − X X + X + X X + X + X X + X X + X X − X − X (1 ) X − X − X X + X + X X + X X + X X − X − X X − X X − X X − X X + X + X Table 1.
Character polynomials of Weyl modules polynomial with top degree terms:(11) X b i +2 b i + ··· = λ i Y i ≥ Y j ≥ X b ij j b ij ! . On the other hand, by (5), H λ = X b i +2 b i + ··· = λ i Y i ≥ Y j ≥ (cid:18)(cid:18) X j b ij (cid:19)(cid:19) . The terms of homogeneous degree d in this product come from the top degreeterms in each factor. Hence the top degree terms in H λ coincide with thetop degree terms in the expression on the right hand side of (10) and thelemma easily follows. (cid:3) Proof of Theorem 2.9.
For partitions λ and µ of d , let K µλ denotethe number of semistandard Young tableaux of shape µ and weight λ . Then K = ( K µλ ) is a unitriangular integer matrix with rows and columns indexedby partitions of d . Let K − µλ be the entries of the inverse matrix K − . Wehave: H λ = X µ K µλ S µ , (12) σ λ = X µ K µλ χ µ . (13)Therefore, S λ = X µ K − µλ H µ by (12) ≡ X µ K − µλ X α σ µ ( α ) (cid:18) Xα (cid:19) ignoring lower deg. terms (Lemma 2.10)= X α ⊢ d X µ K − µλ σ µ ( α ) (cid:18) Xα (cid:19) = X α χ λ ( α ) (cid:18) Xα (cid:19) by (13) , thereby completing the proof of Theorem 2.9. (cid:3) Theorem . For every partition λ = ( λ , . . . , λ l ) , S λ is the coefficientof t λ · · · t λ l l in Y i For every vector λ = ( λ , . . . , λ l ) with non-negative integer co-efficients, define: S λ = det( H λ i − i + j ) . HARACTER POLYNOMIALS AND THE RESTRICTION PROBLEM 11 When λ is a partition this coincides with the character polynomial of theWeyl module W λ . Then, for every partition λ , S λ is the coefficient of t λ in P λ ≥ S λ t λ . Here λ ≥ Z l ≥ ,and t λ = t λ · · · t λ l l . Now X λ ≥ S λ t λ = X λ ≥ X w ∈ S l sgn( w ) l Y r =1 H λ r − r + w ( r ) t λ r r = X w ∈ S l sgn( w ) l Y r =1 t r − w ( r ) r X λ r ≥ H λ r − r + w ( r ) t λ r − r + w ( r ) r = X w ∈ S l sgn( w ) l Y r =1 t r − w ( r ) r X λ r ≥ H λ r t λ r r = Y i Let τ d : P → P denote the linear involution defined by X µ ( − d −| µ | X µ . Comparing equations (3) and (4) shows that: τ d ( H d ) = E d . It follows that, if | µ | = d , then τ d ( H µ · · · H µ m ) = E µ · · · E µ m . When λ is a partition of d , then every term in the expansion of the Jacobi–Trudi determinants det( H λ i + j − i ) is of the form H µ or E µ for integer vector µ with | µ | = d . Therefore τ d (det( H λ i + j − i )) = det( E λ i + j − i ). By the Jacobi–Trudi identities (7), τ d ( S λ ) = τ d (det( H λ i + j − i )) = det( E λ i + j − i ) = S λ ′ , as claimed. (cid:3) 3. The Restriction Problem3.1. Character Polynomials of Specht Modules. For any partition λ , Macdonald [ , Example I.7.14(b)] gave the character polynomials q λ ∈ P of (2) as follows:(14) q λ = X { µ | λ − µ is a vertical strip } ( − | λ |−| µ | X α ⊢| µ | χ µ ( α ) (cid:18) Xα (cid:19) . It immediately follows that the leading coefficients of q λ in the binomialbasis are the same as those of S λ (see Theorem 2.9): Theorem . Let λ be a partition of a positive integer d . For everypartition α of d , the coefficient of (cid:0) Xα (cid:1) in the expansion of q λ in the binomialbasis of P is χ λ ( α ) . Corollary . The sets: S = { S λ | λ is an integer partition } , q = { q λ | λ is an integer partition } are bases of P . Proof. Regard P as a graded algebra where the degree of X i is i foreach i ≥ 1. Let P d denote the homogeneous elements of degree d in P .The degree d homogeneous parts of (cid:0) Xα (cid:1) , as α runs over all partitions of d ,form a basis of P d . Theorem 2.9 and Corollary 3.2 imply that the degree d homogeneous parts of S λ and q λ also form such a basis as λ runs over allpartitions of d , since the character table of S d forms a non-singular matrix.Therefore S and q are bases of P . (cid:3) The coefficients in the expan-sion of the elements of the basis S in terms of the basis q : S λ = X µ r λµ q µ are called the stable restriction coefficients. They determine the decompo-sition of a Weyl module W λ ( K n ) into irreducible representations of S n :Res GL n ( K ) S n W λ ( K n ) = M µ V ⊕ r λµ µ [ n ] . The following result, which is now immediate, is an algorithm for computingthe stable restriction coefficients: Theorem . For any partitions λ and µ , r λµ = h S λ q µ i . HARACTER POLYNOMIALS AND THE RESTRICTION PROBLEM 13 The polynomial S λ can be computed using Theorem 2.8, q µ using (14).After expanding the product in the binomial basis, the moment can be com-puted using Theorem 2.3. The matrix of the stable restriction coefficients r λµ , as λ and µ run over partitions of 0 ≤ n ≤ The blocks demarcate the partitions of each integer n , and within each block,the partitions of n are enumerated in reverse lexicographic order. Let Λ denote the ring ofsymmetric functions (as in [ , Section I.2]). Macdonald [ , Example I.7.13]constructed an isomorphism φ : Λ → P taking the Schur function s λ to thecharacter polynomial q λ . In this section we study a different isomorphismΦ : Λ → P , due to Orellana and Zabrocki [ ], which takes s λ to S λ . Underthis isomorphism q λ is the image of ˜ s λ the Specht symmetric functions of[ 2, 13 ].Following [ , Proposition 12], define an algebra homomorphismΦ : Λ → P by:(15) Φ : p k X d | k dX d . For each k > 0, defineΞ k = 1 , e πi/k , e πi/k , . . . e k − πi/k , and for an integer partition µ = ( µ , . . . , µ m ),Ξ µ = Ξ µ , . . . , Ξ µ m . Let R n denote the space of K -valued class functions on S n . For every n ≥ n Λ : Λ → R n defined by:ev n Λ f ( w ) = f (Ξ µ ) , where µ is the cycle type of w . In other words, the symmetric function isevaluated on | µ | variables, whose values are given by the list Ξ µ . With thisdefinition, ev n Λ ( s λ ) is the character of Res GL n ( K ) S n W λ ( K n ).For each q ∈ P consider the function ev nP ( q ) ∈ R n given by:ev nP ( q )( w ) = q ( X ( w ) , X ( w ) , . . . ) . This defines a ring homomorphism ev nP : P → R n .Observe that ⊕ ∞ n =1 ev nP : P → ⊕ ∞ n =1 R n is injective, for if ev nP ( q ) ≡ n , then q vanishes whenever X , X , . . . take non-negative integer values,and hence q must be identically 0. Theorem . The algebra homomorphism Φ is the unique K -linearmap Λ → P such that the diagram (16) Λ Φ / / ev n Λ ❆❆❆❆❆❆❆❆ P ev nP ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ R n commutes for every n ≥ . Proof. From the definition of ev n Λ ,ev n Λ ( p k )( w ) = X d X d ( w ) d − X j =0 ( e πi/d ) jk . Now observe that d − X j =0 ( e πi/d ) jk = ( d if d | k, . It follows that ev n Λ ( p k )( w ) = X d | k dX d ( w ) = ev nP (Φ( p k )) . Since ⊕ n ev nP : P → ⊕ R n is injective, Φ( p k ) is completely determined bythe commutativity of (16). Since the polynomials { p k } k ≥ generate Λ, Φ iscompletely determined by its values on p k . (cid:3) Lemma . The homomorphism Φ : Λ → P is an isomorphism of rings. Proof. The inverse of Φ is obtained using the M¨obius inversion formula: X k d X d | k µ ( k/d ) p d . (cid:3) HARACTER POLYNOMIALS AND THE RESTRICTION PROBLEM 15 Theorem . For every partition λ , we have: Φ( s λ ) = S λ , (17) Φ(˜ s λ ) = q λ . (18) Proof. This follows immediately from Theorem 3.4. (cid:3) Remark . The second identity (18) is [ , Prop. 12]. 4. Moment Generating Functions4.1. The Main Generating Function. In this and the following sub-sections, we shall frequently use the following elementary identities:exp( t/i ) = X b ≥ i b b ! t b , (A) log 11 − t = ∞ X i =1 t i /i. (B)We shall use α = 1 a a · · · , β = 1 b b · · · , γ = 1 c c · · · . Let Par denotethe set of all integer partitions. We use the notation λ = ( λ , . . . , λ l ), µ = ( µ , . . . , µ m ). Also, t λ = t λ · · · t λ l l and u µ = u µ · · · u µ m m . We shallinterpret λ ≥ λ i ≥ i = 1 , . . . , l and µ ≥ µ i ≥ i = 1 , . . . , m .We use the notation R ⊏ [ l ] to signify that R is a multiset with elementsdrawn from [ l ]. We write t R for the monomial where t i is raised to themultiplicity of i in R . Similarly, for any S ⊂ [ m ], we write u S = Q i ∈ S u i . Theorem . We have: X n ≥ ,λ ≥ ,µ ≥ h H λ E µ i n t λ u µ v n = Y R ⊏ [ l ] Q S ⊂ [ m ] , | S | odd (1 + u S t R v ) Q S ⊂ [ m ] , | S | even (1 − u S t R v ) . Proof. Using (3) and (4), we have: X λ ≥ ,µ ≥ H λ E µ t λ u µ = l Y r =1 m Y s =1 Y i ≥ (cid:18) − ( − u s ) i − t ir (cid:19) X i . Now proceeding as in the proof of Theorem 2.3, X n ≥ ,λ ≥ ,µ ≥ h H λ E µ i n t λ u µ v n = Y i ≥ X b i ≥ v ib i i b i b i ! l Y r =1 m Y s =1 − ( − u s ) i − t ij ! b i (A) = Y i ≥ exp v i i l Y r =1 m Y s =1 (cid:20) − ( − u s ) i − t ir (cid:21)! = exp X i ≥ X R ⊏ [ l ] X S ⊂ [ m ] ( − | S | ( t R ( − | S | u S v ) i i (B) = Y R ⊏ [ l ] Y S ⊂ [ m ] (cid:16) − ( − | S | t R u S v (cid:17) ( − | S | +1 , which is equivalent to the desired expression. (cid:3) Corollary . We have: X λ ≥ ,µ ≥ h H λ E µ i t λ u µ = Y R,S (1 − ( − | S | u S t R ) ( − | S | +1 , where the product is over R ⊏ [ l ] , S ⊂ [ m ] , with at least one of R and S non-empty. Corollary . For every partition λ , h W λ i n is the coefficient of t λ v n in Y i From Theorem 4.1 we get:(19) X λ ≥ ,n ≥ h H λ i n t λ v n = Y R ⊏ [ l ] (1 − t R v ) − . Using this, the corollary can be deduced from Theorem 2.11 by taking mo-ments. (cid:3) S n -invariant Vectors. For a representation V n of S n , let V S n n de-note the subspace of S n -invariant vectors. If a family { V n } of representationshas polynomial character q ∈ P , then h q i n = dim( V S n n ) for all n ≥ . Therefore, for any partition λ , dim W λ ( K n ) S n = h S λ i n . In particular, W λ ( K n ) has a non-zero S n -invariant vector if and only if h S λ i n = 0. Definition . Let v ∈ Z l ≥ . A vector partition of v is an unordered collection v , . . . , v n of non-zero vectors in Z l ≥ such that v = v + · · · + v n . Let p n ( v ) (resp. p ≤ n ( v )) denote the number of vector partitions of v withexactly (resp. at most) n parts. HARACTER POLYNOMIALS AND THE RESTRICTION PROBLEM 17 Theorem . For every partition λ = ( λ , . . . , λ l ) , dim W λ ( K n ) S n = X w ∈ S l sgn( w ) p ≤ n ( λ − w (1) , . . . , λ l − l + w ( l )) . Proof. The coefficient of t λ v n in the right hand side of (19) is p ≤ n ( λ ).Therefore,(20) h H λ i n = p ≤ n ( λ ) for every λ ∈ Z l ≥ . By the Jacobi–Trudi identity (7), S λ = X w ∈ S l H λ − w (1) ,...,λ l − l + w ( l ) , so by (20), h S λ i n = X w ∈ S l sgn( w ) p ≤ n ( λ − w (1) , . . . , λ l − l + w ( l )) , as claimed. (cid:3) Remark . In general, we do not know of a combinatorial proof of thenon-negativity of P w ∈ S l sgn( w ) p ≤ n ( λ − w (1) , . . . , λ l − l + w ( l )), whichfollows from Theorem 4.5. When l = 2, this is the main result of Kim andHahn [ ], who refer to it as a conjecture of Landman, Brown and Portier[ ].The problem of characterizing those partitions λ for which W λ ( K n ) hasa non-zero S n -invariant vector for large n appears to be quite hard. Thefollowing result solves this problem for partition with two rows, two columns,and for hook-partitions. Theorem . Let λ be a partition. (4.7.1) If λ = ( λ , λ ) , then h S λ i > unless λ = (1 , . (4.7.2) If λ ′ = ( λ , λ ) , then h S λ i > if and only if λ = λ (in whichcase h S λ i = 2 ) or λ = λ + 1 (in which case h S λ i = 1 ). (4.7.3) If λ = ( a + 1 , b ) , then h S λ i > if and only if a ≥ (cid:0) b +12 (cid:1) . Proof of (4.7.1). By Theorem 4.5 we need to show that, for every λ ≥ λ ≥ p ≤ n ( λ , λ ) > p ≤ n ( λ + 1 , λ − n , unless λ = λ = 1. From the main result of Kimand Hahn [ ] (the result on the last line of the first page), it follows that p n ( λ , λ ) ≥ p n ( λ + 1 , λ − 1) for all n ≥ p n ( λ , λ ) > p n ( λ + 1 , λ − 1) for at leastone value of n . When k ≥ l ≥ k and l is even, p ( k, l ) > p ( k + 1 , l − k and l are odd and ( k, l ) = (1 , p ( k, l ) > p ( k + 1 , l − (cid:3) Lemma . For all k, l ≥ , p ( k, l ) = ( ( k +1)( l +1) − if both k and l are even, ( k +1)( l +1)2 − otherwise , (21) p ( k, l ) = 16 ( A + 3 B + 2 C ) , (22) where A = (cid:18) k + 22 (cid:19)(cid:18) l + 22 (cid:19) − k + 1)( l + 1) + 3 ,B = ( k/ l/ − if k and l are even, ( k + 1)( l + 2) / − if k is odd and l is even, ( k + 2)( l + 1) / − if k is even and l is odd, ( k + 1)( l + 1) / − otherwise ,C = ( if k and l are divisible by , otherwise. Proof. Consider the set of all ordered triples (( k , l ) , ( k , l ) , ( k , l ))such that P i ( k i , l i ) = ( k, l ) and no ( k i , l i ) = (0 , S acts bypermutation on the set of all such triples, and the number of orbits if p ( k, l ).The quantities A , B , and C in Lemma 4.8 are the number of such triplesthat are fixed by permutations in S of cycle types (1 , , , p ( k, l ) then follows from Burnside’s lemma.The formula for p ( k, l ) is obtained in a similar fashion. (cid:3) Lemma . For all integers k ≥ l ≥ such that at least one of k and l is even, p ( k, l ) > p ( k + 1 , l − . Proof. By Lemma 4.8, we also have: p ( k + 1 , l − 1) = ( ( k +2) l − if k and l are odd, ( k +2) l − . Thus, if k and l are both even and k ≥ l , then p ( k, l ) − p ( k + 1 , l − 1) = ( k + 1)( l + 1) − − (cid:18) ( k + 2) l − (cid:19) = k − l > . If one of k and l is even and the other is odd, then p ( k, l ) − p ( k + 1 , l − 1) = ( k + 1)( l + 1)2 − − (cid:18) ( k + 2) l − (cid:19) = k − l + 12 > , thereby completing the proof of Lemma 4.9. (cid:3) HARACTER POLYNOMIALS AND THE RESTRICTION PROBLEM 19 Lemma . If k ≥ l ≥ , both k and l are odd, and ( k, l ) = (1 , , then p ( k, l ) > p ( k + 1 , l − . Proof. When k and l are odd, Lemma 4.8 gives:12( p ( k, l ) − p ( k + 1 , l − ( ( kl + 2 l )( k − l ) + k ( k − 3) + 3 l + 4 if 3 | k and 3 | l, ( kl + 2 l )( k − l ) + k ( k − 3) + 3 l otherwise . This is clearly positive for all k ≥ l such that k ≥ l ≥ (cid:3) Proof of (4.7.2). By the second Jacobi–Trudi identity,(23) W λ ′ = E λ E λ − E λ +1 E λ − . Taking l = 0 and m = 2 Corollary 4.2 gives: X λ ,λ ≥ h E λ E λ i u λ u λ = (1 + u )(1 + u )(1 − u u ) . Therefore the coefficient of u λ u λ is the number p ′ ( λ , λ ) of ways of writing( λ , λ ) as a sum of vectors of the form (1 , , 0) and (0 , , 1) and (1 , 0) are used at most once. Clearly p ′ ( λ , λ ) = λ = λ ≥ , | λ − λ | = 1 , . By (23), for any partition λ = ( λ , λ ) with two parts, h W λ ′ i = p ′ ( λ , λ ) − p ′ ( λ + 1 , λ − 1) = λ ≥ λ ≥ , λ − λ = 1 , , as claimed. (cid:3) Proof of (4.7.3). Using Pieri’s rule, we have: h k e l = s ( k − | l ) + s ( k | l − , whence s ( a | b ) = h a +1 e b − h a +2 e b − + · · · + ( − b h a + b +1 e . It follows that(24) (cid:10) W ( a | b ) (cid:11) = b X j =0 ( − j h H a + j +1 E b − j i . Taking l = m = 1 in Corollary 4.2 gives:(25) X a,b ≥ h H i E j i t i u j = Q ∞ k =0 (1 + t k u ) Q ∞ k =1 (1 − t k ) . The coefficient of t i u j in the above expression is the number ˜ p ( i, j ) of waysof writing the vector ( i, j ) as a sum of vectors of the form ( a, 0) where a > a, 1) where a ≥ 0, and vectors of the form ( a, 1) are used at most once.If ˜ p ( i, j ) > 0, then j distinct vectors of the form ( a, 1) are used, so that i ≥ (cid:0) j (cid:1) . Therefore(26) ˜ p ( i, j ) = 0 for all i < (cid:18) j (cid:19) . If a < (cid:0) b +12 (cid:1) , then a − j < (cid:0) b + j +12 (cid:1) for all j ≥ 0. By (26) h H a − j E b + j +1 i =0 for all j ≥ 0. This allows us to extend the index of summation in the righthand side of (24) without changing the sum: (cid:10) W ( a | b ) (cid:11) = b X j = − a − ( − j h H a + j +1 E b − j i = X k + l = a + b +1 h H k E l i = (cid:10) W (0 | a + b +1) (cid:11) . Taking l = 0 and m = 1 in Corollary 4.2 can be used to show that h E k i = 0for all k > 1, so (cid:10) W (0 | a + b +1) (cid:11) = h E a + b +2 i = 0 for all a, b ≥ 0. Therefore, (cid:10) W ( a | b ) (cid:11) = 0 for a < (cid:0) b +12 (cid:1) .Conversely, suppose a ≥ (cid:0) b +12 (cid:1) . The hook partition λ = ( a + 1 , b ) is ofsize a + b + 1, and so h S λ i a + b +1 = h S λ i by Corollary 2.4. Therefore it sufficesto show that W λ ( K a + b +1 ) contains an S a + b +1 -invariant vector. Then thehook partition λ = ( a + 1 , b ) dominates the partition µ = ( a − (cid:0) b (cid:1) + 1 , b, b − , . . . , W ( a | b ) ( K a + b +1 ) contains anon-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 } arelinearly independent, and generate a representation that is isomorphic tothe regular representation of S n . In particular, the trivial representation iscontained in W ( a | b ) ( K a + b +1 ). (cid:3) 5. Strict Polynomial Functors and FI -modules In this section we may take K to be any field (not necessarily of charac-teristic zero). The category Rep Γ d of strict polynomial functors was intro-duced by Friedlander and Suslin [ ] as a tool to unify homogeneous poly-nomial representations of GL n ( K ) of degree d across all n . Later, Church,Ellenberg and Farb [ ] introduced the category of FI-modules to unify rep-resentations of S n across all n . In this section, we lift the restriction functorRes GL n ( K ) S n to a functor Res d : Rep Γ d → FI-Mod. The Schur category (also knownas the divided power category, see [ 9, 16 ]) Γ d is the category whose objects HARACTER POLYNOMIALS AND THE RESTRICTION PROBLEM 21 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 functor category Fun(Γ d , Vec),where Vec denotes the category of finite-dimensional vector spaces over K (see [ , Section 2]). When K has characteristic zero Rep Γ d is semisim-ple, and its simple objects are functors known as Weyl functors . For eachpartition 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 rep-resentations 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 → Vecdefine 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 ahomogeneous polynomial representation of degree d . For each n ≥ λ ∈ P ( n, d ), ev n ( W λ ) = W λ ( K n ) is the irreducible polynomial representa-tion of GL n ( K ) corresponding to λ . FI -modules. The category FI is the one that has finite sets asobjects, and injective functions as morphisms. The category of FI-modulesis the functor category FI-Mod = Fun(FI , Vec). Let Rep S n denote thecategory of representations of S n over K . The evaluation functor ev n :FI-Mod → Rep S n is defined as follows: For an FI-module V : FI → Vecdefine ev n ( V ) = V ([ n ]) where [ n ] is the set { , . . . , n } .For each partition λ there exists an FI-module V ( λ ) (see [ , Proposi-tion 3.4.1]) such that, for every n ≥ | λ | + λ , we have:ev n ( V ( λ )) = V λ [ n ] . For each object A of FI let F [ A ] bethe vector space of all functions A → K . Given an injective function i : A → B , define F ( i ) : F ( A ) → F ( B ) by: F ( i )( f )( b ) = ( f ( a ) if there exists a ∈ A such that i ( a ) = b, , for all f ∈ F ( A ). Then F : FI → Γ is a functor. For every positive integer d , define F d : FI → Γ d by F d ( A ) = F ( A ) , F d ( i ) = F ( i ) ⊗ d . The restriction functor Res d : Rep Γ d → FI-Modis defined by: Res d F = F ◦ F d for every object F of Rep Γ d . Theorem . The diagram of functors Rep Γ d Res d / / ev n (cid:15) (cid:15) FI -Mod ev n (cid:15) (cid:15) Rep d GL n ( K ) Res GLn ( K ) Sn / / Rep S n commutes, in the sense that Res GL n ( K ) S n ◦ ev n is naturally isomorphic to ev n ◦ Res d . Proof. Given F ∈ Rep Γ d , ev n ( K ) is F ( K n ). Given w ∈ S n , let T w ∈ GL n ( K ) denote the linear map the takes the i th coordinate vector e i of K n to e w ( i ) . An element w ∈ S n acts on F ( K n ) via F ( T ⊗ dw ). On the other hand,ev n ◦ Res d ( F ) = F ◦ F d ([ n ])= F ( F ([ n ]) . An element w ∈ S n acts on F ( F ([ n ])) by F ( F d ( w )) = F ( F ( w ) ⊗ d ). Thesetwo actions of S n coincide under the isomorphism K n → F ([ n ]) given by e i δ i , where δ i is the Kronecker delta function on [ n ] supported at i .Thus we get an isomorphism Res GL n ( K ) S n ◦ ev n → ev n ◦ Res d ( F ) of representa-tions of S n . The naturality of this isomorphism follows tautologically fromunwinding the definitions. (cid:3) Theorem . For every strict polynomial functor F of degree d , Res d F is a finitely generated FI -module. Proof. It suffices to prove the theorem for F = Γ λ since it generatesRep Γ d [ , Proposition 2.9]. Now Γ λ is a tensor product of strict polyno-mial functors of the form Γ d . Also, Res d takes the external tensor product ofstrict polynomial functors to the tensor product of FI-modules, and a tensorproduct of finitely generated FI-modules is finitely generated [ , Proposi-tion 2.3.6]. Therefore it suffices to show that Res d Γ d is finitely generatedfor each d .By definition Res d Γ d ( A ) = Γ d ( F ( A )). Let S = { δ λ · · · δ λ l l ∈ Γ d F ([ l ]) | λ ⊢ d } . For any finite set A , { δ λ a · · · δ λ l a l | a , . . . , a l ∈ A, λ ⊢ d } spans Γ d ( F ( A )). But δ λ a · · · δ λ l a l is the image of Γ d ◦ F ( ι ) for the injectivemap ι : [ l ] → A defined by ι ( i ) = a i . Therefore Res d Γ d is a finitely generatedFI-module; it is generated in degree d . (cid:3) HARACTER POLYNOMIALS AND THE RESTRICTION PROBLEM 23 Acknowledgements. We thank Aprameyo Pal, K. N. Raghavan, AnneSchilling, and S. Viswanath for helpful discussions and encouragement. Wethank Brian Hopkins for his answer on mathoverflow https://mathoverflow.net/q/340231 which helped us prove part (1) of Theorem 4.7. We thank Dipendra Prasadfor an argument in the proof of the sufficiency of the condition in part (3)of Theorem 4.7. References [1] K. Akin, D. A. Buchsbaum, and J. Weyman. Schur functors and Schur com-plexes. Adv. in Math. , 44(3):207–278, 1982. doi:10.1016/0001-8708(82)90039-1.[2] S. Assaf and D. Speyer. Specht modules decompose as alternatingsums of restrictions of schur modules. Proc. Amer. Math. Soc. , 2019.doi:10.1090/proc/14815.[3] T. Church, J. S. Ellenberg, and B. Farb. FI-modules and stability for rep-resentations of symmetric groups. Duke Math. J. , 164(9):1833–1910, 2015.doi:10.1215/00127094-3120274.[4] E. M. Friedlander and A. Suslin. Cohomology of finite group schemes over afield. Invent. Math. , 127(2):209–270, 1997. doi:10.1007/s002220050119.[5] A. M. Garsia and A. Goupil. Character polynomials, their q -analogs and theKronecker product. Electron. J. Combin. , 16(2, Special volume in honor ofAnders Bj¨orner):Research Paper 19, 40, 2009. doi:10.1016/j.jcta.2019.02.019.[6] N. Harman. Representations of monomial matrices and restriction from GL n to S n . arXiv e-prints , arXiv:1804.04702, Apr 2018.[7] G. James and A. Kerber. The representation theory of the symmetric group ,volume 16 of Encyclopedia of Mathematics and its Applications . Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn,With an introduction by Gilbert de B. Robinson.[8] J. K. Kim and S. G. Hahn. Partitions of bipartite numbers. Graphs and Com-binatorics , 13(1):73–78, Mar 1997. doi:10.1007/BF01202238.[9] H. Krause. Koszul, Ringel and Serre duality for strict polynomial functors. Compos. Math. , 149(6):996–1018, 2013. doi:10.1112/S0010437X12000814.[10] B. M. Landman, E. A. Brown, and F. J. Portier. Partitions of bi-partitenumbers into at mostj parts. Graphs and Combinatorics , 8(1):65–73, 1992.doi:10.1007/BF01271709.[11] D. E. Littlewood. Products and plethysms of characters with orthogo-nal, symplectic and symmetric groups. Canadian J. Math. , 10:17–32, 1958.doi:10.4153/CJM-1958-002-7.[12] I. G. Macdonald. Symmetric functions and Hall polynomials . Oxford Clas-sic Texts in the Physical Sciences. The Clarendon Press, Oxford UniversityPress, New York, second edition, 2015. With contribution by A. V. Zelevinskyand a foreword by Richard Stanley, Reprint of the 2008 paperback edition [MR1354144].[13] R. Orellana and M. Zabrocki. Symmetric group characters as symmetric func-tions. arXiv e-prints , arXiv:1605.06672v4, 2018.[14] R. Orellana and M. Zabrocki. The Hopf structure of symmetric group charac-ters as symmetric functions. arXiv e-prints , arXiv:1901.00378, Dec 2018. [15] A. Prasad. Representation theory , volume 147 of Cambridge Studies in Ad-vanced Mathematics . Cambridge University Press, Delhi, 2015. A combinato-rial viewpoint.[16] W. van der Glen. Lectures on Bifunctors and Finite Generation of RationalCohomology Algebras , pages 41–65. Springer International Publishing, Cham,2015. doi:10.1007/978-3-319-21305-7 3. The Institute of Mathematical Sciences (HBNI), Chennai E-mail address : [email protected] The Institute of Mathematical Sciences (HBNI), Chennai E-mail address : [email protected] The Institute of Mathematical Sciences (HBNI), Chennai E-mail address : [email protected] Tata Institute of Fundamental Research, Mumbai E-mail address ::