On the equivalence of two stability conditions of FB-modules

Transcription

1 arxiv: v2 [math.rt] 17 Jul 2018 On the equivalence of two stability conditions of FB-modules Alberto Arabia May 16, 2018 Abstract. We give a proof of the equivalence of two stability conditions in the work of Thomas Church and Benson Farb on FB-modules [1] : representation stability RS and character polynomiality PC. While the implication RS PC is a simple consequence of the Frobenius character formula, the converse seems not to have been documented, which motivated this work. An FB-module is a countable family W := {W m } m N of linear representations W m of the symmetric groups S m, assumed, in these notes, to be finite dimensional over the field of rational numbers Q, hereafter called an S m -module. The FB-module W is said to be representation stable RS in short, if there exists some N N, such that, for all m N, the decomposition in simple factors of W m has the stable form : W m λ V λ[m] n λ, where λ := λ 1... λ l is a partition verifying λ + λ 1 N; V λ[m] is the simple S m -module associated with λ[m] := m λ λ 1... λ l ; for m N, the family of natural numbers {n λ } λ is constant. The smallest such N is the rank of RS of W, which we will denote by rank rs W. The FB-module W is said to have a polynomial character PC in short, if there exists N N and a polynomial P W Q[X 1,..., X N ], such that, for all m N and all g S m, one has : χ Wm g = P W X 1 g,..., X N g, where χ Wm is the character of W m, and X i g is the number of cycles of length i in the decomposition of g in S m as a product of disjoint cycles. The polynomial P W, which is unique, is the polynomial character of W. The smallest such N is the rank of character polynomiality of W, it will be denoted by rank pc W. Regarding the equivalence of these conditions, we prove the following theorem. Theorem Let W be an FB-module. a rank pc W rank rs W. b If W is PC and has polynomial character P W, then rank rs W max{rank pc W, 2 deg w P W }, where deg w P W denotes the polynomial degree of P W, when stipulating that deg w X i := i. Université Paris-Diderot, IMJ-PRG, CNRS, Bâtiment Sophie Germain, bureau 608, Case 7012, Paris Cedex 13, France. Contact : alberto.arabia@imj-prg.fr. 1

2 Comments. i Theorem exhibit FB-modules W with rank pc W = 0 and arbitrarily big rank rs W. It also shows that the bounds in are optimal. ii Beyond FB-modules, we also look at FI-modules W, which are FB-modules with additional structure 1.2. There is a corresponding representation stability condition for FI-modules, its rank is denoted by rank fi rsw One has rank rs W rank fi rsw, but there is no equivalence with PC stability, in particular thm b for FI-modules and rank fi rs is false. An interesting by-product in this work is the fact that the weight w W m of an S m -module W m see 1.5 coincides with the lowest possible degree deg w P of a polynomial P expressing the character χ Wm Moreover, if W := {W m } is a PC FB-module, then deg w P W = w W m, for all m rank rs W. The next theorem follows from these observations. Proposition b. If W 1 and W 2 are FB-modules which are PC m N, then w W 1,m W 2,m = w W 1,m + w W 2,m, for all m 2w W 1,m + w W 2,m. In particular, w W 1 W 2 N = w W 1 N + w W 2 N, where W N denotes the truncation of W that replaces W n by 0 for all n < N. Contents 1 Preliminaries General notations The categories of FB and FI modules Projective FI-modules Young diagrams and Pieri s rule The weight of a representation and of an FB-module The FI-module V λ Representation stability of FI and of FB modules Character polynomiality of FI-modules Character polynomiality Frobenius character formula Basic examples of character polynomiality of FI-modules Polynomial weight vs. Representation weight On the minimal polynomial weight of a character PC versus RS Equivalence of the stability conditions Addendum on the weight of a tensor product Reinterpretation of the weight of a representation On the weight of the tensor product of representations

3 The first two sections are elementary. They introduce notations and wellknown properties of FI and FB-modules, which advanced readers can skip. 1. Preliminaries 1.1. General notations Given a group G, we denote by G-mod the category of G-modules, i.e. of finite dimensional linear representations of G over the field of rational numbers Q. We denote by S m the symmetric group defined over [[1, m]]. Its elements are the bijections of the interval of natural numbers [[1, m]]. For n m, we write S n S m for the inclusion which identifies a permutation g of [[1, n]] with its obvious extension to [[1, m]] that fixes all i > n. Q m and ɛq m denote respectively the trivial and the alternating or signature Q-linear representation of the group S m. S a S b denotes the subgroup of S a+b stabilizing the subset [[1, a]], or, equivalently, the subset [[a+1, a+b]]. The subgroup 1 a S b S a S b is the fixator of [[1, a]], i.e. the set of permutations fixing every i a. If W a is an S a -module and W b is an S b -module, we denote by W a W b the tensor product W a Q W b endowed with the S a S b -module structure defined by the componentwise action g a, g b w a w b := g a w a g b w b. A partition of m N\{0} is any decreasing sequence of natural numbers λ := λ 1... λ l > 0 such that m = i λ i. It is also denoted as the m- tuple 1 n 1, 2 n 2,..., m nm where n k := #{i λ i = k}, so that m = i i n i. The notation λ m says that λ is a partition of m, and λ is used for the number partitioned by λ. The partition λ is empty if λ = 0. Q cl S m denotes the Q-algebra of class functions of S m. These are the functions f : S m Q constant along the conjugacy classes of S m, i.e. such that fgxg 1 = fx, g, x S m. The scalar product is defined by _ _ Sm : Q cl S m Q cl S m Q f 1 f 2 Sm := 1 S m g S m f 1 g f 2 g 1, f 1, f 2 Q cl S m. If W m is an S m -module, χ Wm : S m Q denotes its character. The Schur s orthogonality relations state that if V 1 and V 2 are simple S m -modules, then χ V1 χ V2 Sm is equal to 1 if V 1 is isomorphic to V 2, and to 0 otherwise. Between here and the end of this preliminary section we recall concepts and terminology from the works of Church and Farb cf. [1]. 3

4 1.2. The categories of FB and FI modules FB denotes the category of Finite sets and Bijections. An FB-module is, by definition, a covariant functor from the category FB to the category Vec f Q of finite dimensional Q-vector spaces and Q-linear maps : W : FB Vec f Q. To give an FB-module W is equivalent to giving the countable collection {W m := W[[1, m]]} m N, where W m is an S m -module. A morphism of FB-modules f : W Z corresponds to a family {f m : W m Z m } m N of morphisms of S m -modules. We have a canonical identification : Mor FB W, Z = m N Hom S m W m, Z m Notation. The category of FB-modules will be denoted by FB-mod. It is a semi-simple abelian category. FI denotes the category of Finite sets and Injections. An FI-module is a covariant functor from FI to Vec f Q : W : FI Vec f Q. To give an FI-module is equivalent to giving FI-1 an FB-module W := {W m } m N ; FI-2 for all m N, an interior map φw m : W m W m+1 in short φ m, which is Q-linear and such that, for all g S m S m+1, one has φ m g w = g φ m w. FI-3 for all n m, the image of φ n,m := φ n 1 φ m satisfies : φ n,m W m W n 1m S n m. Under this equivalence, a morphism of FI-modules f : W Z is simply a morphism of FB-modules which is compatible with the interior maps φ m, i.e. such that the diagrams W m f m φw m W m+1 f m+1 Z m φz m Z m+1 are commutative. Notation. The category of FI-modules will be denoted by FI-mod. It is an abelian category, which is not semi-simple. 4

5 Comments a The category FB-mod is equivalent to the full subcategory of FI-mod of FI-modules whose interior maps are null. The forgetful functor associates to an FI-module W := {φ m : W m W m+1 } the FB-module W := {W m }. It is an additive and exact functor which will be tacitly used. b More interesting, the subcategory FI-mod of FI-modules whose interior maps are injective and eventually exhaustive, i.e. such that the image φ m W m generates W m+1 as S m+1 -module for large m. Among these, the FI-modules V λ s which bind together the simple modules V λ[m] We will see that FB-modules W which are PC or RS are asymptotically isomorphic to direct sums of V λ s The truncations 1 The functor _ l : FI-mod FI-mod that truncates an FI-module {W m } m N by replacing by 0 its terms W m for m < l, is an additive exact functor. The cokernel of the natural inclusion _ l id FI is the truncation _ <l, it replaces by 0 the terms W m for m l. We thus have short exact sequences 0 W l W W <l 0, which are natural with respect to W. The full subcategory FI-mod l of FImodules W such that the inclusion W l W is an isomorphism is an abelian subcategory, and similarly for the full subcategory FI-mod <l of FI-modules W such that the quotient W W <l is an isomorphism. One has, Ext i FI FI-mod l, FI-mod l = 0, i > 0. The intersection FI-mod l FI-mod l is the semi-simple category S l -mod Projective FI-modules An obvious way to construct FI-modules of the type described in b is to start with à representation W n of some S n, and then set, for all m N : { 0, if m < n, PW n m := ind Sm S n Sm n Wn Q m n, otherwise. For each m n, the composition of the following natural maps ι m and κ m+1 : W n Q m n ι m W n Q m+1 n ψ m κ m+1 ind S m+1 S Wn n Sm+1 n Q m+1 n 1 This truncations are frequently called stupid or brutal. 5

6 gives the map ψ m whose image is invariant under 1 n S m+1 n. In particular, ψ m is S n S m n -linear, inducing thus the map φ m := indψ m : ind Sm S n Sm n Wn Q m n ind S m+1 S n Sm+1 n Wn Q m+1 n, which satisfies the requirements making the family an FI-module. PW m := {φ m : PW n m PW n m+1 } m N, All the previous constructions are natural on the category of S n -modules. As a consequence, we have defined a functor which is clearly additive and exact. P : S n -mod FI-mod, Proposition. Let W n be a representation of S n. a The interior maps of PW n are injective. They are exhaustive m n. b There is a natural identification of functors Mor FI PW n, _ = Hom Sn W n, _ n. c The FI-module PW n is a projective FI-module, which moreover is simple projective if and only if W n is a simple S n -module. All simple projective FI-modules are of this form. d The category FI-mod has enough projective objects. Proof. Left to the reader Young diagrams and Pieri s rule We recall the re-parametrization of irreducible representations of the symmetric groups due to Church and Farb The socle and the weight of a partition Let λ = λ 1 λ 2 λ l > 0 be a non-empty partition of m := i λ i. The socle of λ is the sub-partition λ := λ 2... λ l. The weight of λ is the number w λ := λ = λ λ l. Notice that the map λ λ is injective from the set of partitions of m N, so that it amounts the same, giving λ m or giving m, λ with m λ + λ 1. 6

7 In terms of Young diagrams, in order to get the socle λ of λ, one simply erases the first row of λ. The picture is thus : λ := λ :=. Conversely, given λ = λ 1 λ 2... λ l and any m λ + λ 1, Church and Farb introduce the notation λ[m] := m λ λ 1... λ l which, in terms of Young diagrams, corresponds to adding a first row with as many boxes as is necessary to raise the total number of boxes from λ to m. λ := λ[m] :=. Notice the following obvious notational facts : λ[m] m, λ[m] = λ, λ = λ[ λ ] Irreducible representations of the symmetric groups The irreducible representations of S m are parametrized by the partitions ν n. The simple S m -module associated with ν is classically denoted by V ν. The following proposition recalls a fundamental fact about the representations of the symmetric groups see [2], chap. 4, thm. 4.3, pp Proposition. The irreducible representations of the symmetric groups over a field k of characteristic zero are defined over the field of rational numbers. If W m is a k[s n ]-module of finite dimension, the values of the character χ Wm : S m k, g trg : W m W m, belong to the ring of integers Z k. Hint. Because W m is defined over Q, the trace of g S m belongs a priori to Q, and because the trace is the sum of the eigenvalues, which are roots of unity, it is also an algebraic integer. It is therefore an integer. 7

8 Pieri s rule Pieri s rule 2 gives the irreducible factors of the terms PV ν m, for any given partition ν n m. The rule says that in the decomposition PV ν m := ind Sm S n Sm n Vν Q m n = µ m V µ nνµ, 1 the nonzero multiplicities n ν µ are all equal to 1, and the corresponding partitions µ are those obtained from ν by adding m n boxes in different columns. For example, if ν := 3, 2, 2 and m {8, 9, 10, 11}, we have ind S 8 S 7 S 1 V = V V V ind S 9 S 7 S 2 V = V V V V V ind S 10 S 7 S 3 V = V V V V V V ind S 11 S 7 S 4 V = V V V V V V where, after a key observation of Church and Farb, for m ν + ν 1 = 10 in this example, the set of socles of the Young diagrams appearing in the decomposition 1 is constant. The following lemma obviously follows Lemma. Let ν n > 0. a For all µ m n, such that n ν µ 0, the weight of µ verifies ν = w ν w µ ν, and { w µ = w ν µ = ν[m] w µ = ν m ν + ν 1 & µ = ν[m]. b Let m 0 = ν + ν 1 and let P ν be a set of partitions µ such that ind Sm 0 S Vν n Sm0 Q n m0 n = V µ[m0 ]. µ Pν Then, ν[m 0 ] P ν and ν[m 0 ] 1 = ν[m 0 ] 2. For all m > m 0, we have ind Sm S n Sm n Vν Q m n = µ P ν V µ[m]. and, for all µ P ν, µ[m 0 ] 1 > µ[m 0 ] 2. 2 For a thorough introduction to these rules, read 4.3, p , and also Appendix A on the Littlewood-Richardson rules, p. 451, both in Fulton-Harris book [2]. 8

9 1.5. The weight of a representation and of an FB-module The definition of weight, is extended from partitions to representations and, more generally, to FB-modules. The weight of an S m -module W m is the upper bound of the weights of the partitions associated with its irreducible factors, i.e. if then W m µ m V τ nµ, w W m := sup { w µ nµ 0 }. For example, as a consequence of lemma a, we have { ν = w ν w PVν m ν, m ν, and w PV ν m = ν, m ν + ν 1. The weight of an FB-module W := W m is the upper-bound of the weights of its terms, i.e. w W := sup { w W m } m N. The weight at infinity of an FB-module W := W m is w W := lim w W N. N + It is easy to see that w W = w W N, for some N 0. We have w PV λ = w PV λ = λ The weight truncations Let p N. Given an S m -module W m, denote by W m w >p the sum of the irreducible factors of W m of weight > p. By Pieri s rule a, if W := {φ m : W m W m+1 } m N is an FImodule, one has φ m Wm w >p Wm+1 w >p, in which case, the family W w >p := { φ m W m w >p W m w >p }m N is a sub-fi-module of W. Let W w p := W/W w >p. The short exact sequence is natural with respect to W. 0 W w >p W W w p 0 9

10 Remark. The following are easy consequences of the definitions. a The full subcategory FI-mod w >p of FI-modules W such that the inclusion W w >p W is an isomorphism is an abelian subcategory. b The full subcategory FI-mod w p of FI-modules W such that the quotient W W w p is an isomorphism is an abelian subcategory. c Ext i FI FI-modw >p, FI-mod w p = 0, for all i N The FI-module V λ Given a partition λ, consider the projective FI-module PV λ of 1.3. Lemma a says that for all m λ the smallest weight of the irreducible components of PV λ m is exactly λ = w λ, which is the weight of a unique factor : the simple S m -module V λ[m] with multiplicity 1. The following proposition results from this simple observation and the weight filtration Proposition. Let λ be a nonempty partition. a The terms of the quotient FI-module V λ := PV λ w w λ = { φ m : V λ,m V λ,m+1 }m N { 0, for all m < λ, are V λ,m = V λ[m], otherwise. The interior maps φ m are injective, and are exhaustive for m λ. b If V λ := {φ m : V λ,m V λ,m+1 } m N is an FI-module such that φ m is injective and is exhaustive for all m N, then V λ N and V λ N cf are isomorphic FI-modules for m N Representation stability of FI and of FB modules Representation stable FI-modules An FI-module W = {φ m : W m W m+1 } m N is said to be representation stable for m N, in short RS m N, if the following conditions are satisfied. RS-1 The interior maps φ m are injective, and are exhaustive for m N. RS-2 For all m N, we have a stable decomposition in simple modules W m λ N V λ[m] n λ, where the n λ are independent of m N. We denote by rank fi rsw, the smallest such N, and we call it the rank of representation stability of W. 10

11 Examples a PV λ is RS m λ +λ1,lemma b. b PW n RS m 2n, consequence of lemma b. c V λ is RS m λ, by definition Representation stable FB-modules An FB-module W = {W m } m N is said to be representation stable for m N, in short RS m N, if the previous condition RS-2 is satisfied. In that case, we say that W and λ P Vn λ λ as FB-modules, and we may write are asymptotically isomorphic W N λ N V λ n λ N 2 We denote by rank rs W, the smallest such N, and we call it the rank of representation stability of W. Warning. For an FI-module the stability rank as FB-module is, a priori, smaller than the stability rank as FI-module. It is very easy to construct examples where it is strictly smaller, for example take any RS FI-module and replace all its interior maps by the zero map Proposition a rank rs PV λ = rank fi rspv λ = λ + λ 1. b If the trivial representation Q n is contained in W n, then rank rs PW n = rank fi rspw n = 2n. c rank rs V λ = rank fi rsv λ = λ. d If W is an FB-module such that Z W, where Z is any of the FI-modules in the previous claims, then rank rs W rank rs Z. Proof. In c there is nothing to be proved. Let us denote by Z any of the other two FI-modules in the claims. We already gave the values of rank fi rsz in Next, because rank rs Z rank fi rsz, the equality will follow if we show that for m = rank fi rsz the S m -module Z m contains an irreducible 11

12 factor corresponding to a Young diagram having its two first lines of equal lengths : V... Z m Indeed, if rank rs Z < m, we would have Z m λ <m V λ[m] n λ, and the length of the first line of the Young diagram λ[m] in, would be strictly greater than that of the others, i.e. λ[m] 1 > λ[m] 2, which is contrary to our assumption cf b. Let us check in our cases. In a this results by Pieri s rule because m = λ + λ 1 is the smallest m such that λ is the socle of a diagram µ m, which means that µ 1 = µ 2. In b the Young diagram corresponding to the trivial module Q n has only one line with n boxes, and PQ n 2n verifies for m = 2n. c W m contains some irreducible S m -module of the form. 2. Character polynomiality of FI-modules 2.1. Character polynomiality Let Q cl S m denote the algebra of rational class functions on S m, i.e. functions f : S m Q which are constant on each conjugacy class of S m. Denote by Q[X] the ring of polynomials with rational coefficients and countably many variables X 1, X 2,..., endowed with the grading deg w that stipulates that : deg w X i := i The weight of a polynomial. To avoid confusion in the sequel with the usual degree degp of a polynomial P Q[X] which stipulates that degx i = 1, we will call deg w P the weight of P Proposition. Denote by X m,i : S m N the class function which assigns to g S m, the number X m,i g of i-cycles in the decomposition of g as product of disjoint cycles in S m. a The map ρ m : Q[X] Q cl S m X i g X m,i g is an homomorphism of algebras whose kernel contains the polynomials X1 +2X 2 + +mx m m and Xi X i 1 X i m/i

13 The restriction ρ m : Q[X 1,..., X m 1 ] Q cl S m is surjective. In particular, the characters of S m are represented by polynomials with rational coefficients and variables X 1,..., X m 1. b For n m and g S n, we have i ρ m X 1 ιg = ρ n X 1 g + m n, ii ρ m X i ιg = ρ n X i g, i > 1. S n ρ n ι ρ m S m Hint. a That the polynomials 4 belong to kerρ m is clear. Next, to see that ρ m is surjective, we need only show that the characteristic function of a conjugacy class of S m can be realized as a polynomial in X 1,..., X m. For k [[1, m]], let R k Z := ZZ 1 Z k Z m and consider D k Z := R k Z/R k k Q[Z]. This polynomial has the property that { 1, if Xi g = k, ρ m D k X i g = 0, otherwise. So that, if i i n i = m, we get { 1, if g is of type 1 n 1,..., m nm ρ m D n1 X 1 D n2 X 2 D nm X m g = 0, otherwise. b is clear Convention. In order to alleviate notations, we will write X i g for X m,i g despite the possible ambiguity in X 1 g cf b-i Definition. An FB-module W := {W m } m N is said to have a polynomial character for m N, in short PC m N, if there exists a polynomial P W Q[X] such that χ Wm = ρ m P W, m N. We denote by rank pc W, the smallest such N, and we call it the rank of character polynomiality of W Proposition and definition. The polynomial P W that asymptotically represents the characters of the terms W m of an FB-module W := {W m } m N is unique, it is called the polynomial character of the FB-module W. Proof. If P W were another polynomial representing χ Wm for m 0, the difference Q := P W P W, that we may assume to belong to Q[X 1,..., X N ], would be a polynomial representing the zero class function for all m N. 13 A

14 If Q is not the null polynomial, we can write it as a polynomial in X N with coefficients Q i in Q[X 1,..., X N 1 ] : Q = Q 0 + Q 1 X N + Q 2 X 2 N + + Q r X r N, and Q r 0. Now, for any family a := {a 1,..., a N 1 } N, and any i N, it is easy to find m i N and g i S mi such that X 1 g i = a 1,..., X N 1 g i = a N 1 and X N g i > i. In that case, the polynomial in one variable Qa, X N has infinitely many roots and is, therefore, the null polynomial. In particular, in the decomposition, we would have Q r a = 0 for all choices of a, but this implies that Q r is the null polynomial in Q[X 1,..., X N 1 ], contrary to its definition. The polynomial Q must therefore be the null polynomial Frobenius character formula Given a partition λ := λ 1... λ l m, the celebrated Frobenius formula computes the character χ Vλ of the simple S m -module V λ. The important point for us about this formula is that it gives an expression of χ Vλ as a polynomial depending only on the socle λ of λ. Consequently, the same polynomial expresses the characters of all the terms of the FI-module V λ = {V λ,m } m, for m λ, which self-explains the property of character polynomiality of V λ Frobenius polynomial for χ Vλ Following Macdonald in his book [3] ex. I.7.14, p. 122, let y := {y 1,..., y l } be a set of l variables, where l := lλ. The discriminant of y is the antisymmetric homogeneous polynomial y := i<j y i y j. For d N, the d-power sum of y is the symmetric homogeneous polynomial P d y := y d y d l. The value χ Vλ g for g S m, is the coefficient of the monomial y λ 1+l 1 1 y λ 2+l 2 2 y λ 3+l 3 3 y λ l l, in the development of the product y P dy X dg. 5 d 1 This coefficient, denoted by X λ, is a polynomial in Q[X], we call it the Frobenius polynomial for χ Vλ. 14

15 Proposition. The Frobenius polynomial X λ for χ Vλ depends only on the socle λ of λ, and belongs to the ring Q[X 1,..., X λ2 +l 2]. Its weight is : deg w X λ = w λ. The characters of S m can be represented by polynomials in Q[X 1,..., X m 1 ] of weights m 1. Proof. Because the polynomial 5 is homogeneous, we can make y 1 = 1 without loosing information. In that case, χ Vλ g is the coefficient in y λ 2+l 2 2 y λ 3+l 3 3 y λ l l, after the development of the product ỹ 1 y j 1 + P dỹ X dg j>1 d 1 where ỹ := {y 2,..., y l }. But, in this product the first factor ỹ is already homogeneous of total degree l 2 + l 3 +, so that we must seek, in the development of the remaining factors, terms whose total degree is bounded by λ = λ λ l. But then, since we have Xd 1 + Pd = 1+ Xd P 1 d + Xd P 2 2 d + Xd P 3 3 d + and because deg tot Pd a = ad, we conclude that If d > λ 2 + l 2, the factor Xd 1 + P d contributes only to χ Vλ with its term 1 X d, so can be ignored. The product symbol d 1 in can therefore be replaced by λ 2 +l 2 d=1. The coefficient X d j is a polynomial of degree j in Xd and appears attached to monomials in ỹ of total degree jd. We can thus infer that, after development, the expression of χ Vλ of weight λ = w λ. is a polynomial in X 1,..., X λ2 +l 2 The following corollary of proposition is now immediate from the definition of representation stable FB-modules Corollary If W is an FB-module, then rank pc W rank rs W. Proof. Left to the reader. 15

16 2.3. Basic examples of character polynomiality of FI-modules The l-cycles of [[1, m]] Given m, l N, we denote by Γl m the set of l-tuples i 1,..., i l of pairwise distinct elements of [[1, m]] modulo cyclic permutation, i.e. such that i 1,..., i l = i 2,..., i l, i 1 = i 3,..., i l, i 1, i 2 = The symmetric group S m acts on Γ m l The elements of Γ m l Comments by g i 1,..., i l = gi 1, gi 2,..., gi l. 6 are called the l-cycles of [[1, m]]. a Given g S m, the set [[1, m]] is decomposed in g -orbits, each of which can be endowed with a cyclic order defined by g. For example, if x [[1, m]], we may consider x gx g 2 x x, which gives the well-known decomposition of g S m as product of disjoint cycles. b For m l, there is a difference between the cases l = 1 and l > 1. For all l 1, define the support of an l-cycle γ := i 1,..., i l the set {γ } of its coordinates : {γ } := {i 1,..., i l } [[1, m]]. Then, define γ S m by : { γij := i j+1 mod l, for i j {γ }, γ := γx := x, for x {γ }. i For l = 1, we have Γ1 m = [[1, m]] and the action 6 of S m on Γ1 m is the standard action of S m on [[1, m]]. The map _ : Γ1 m S m is uninteresting, as it is the constant map γ id [[1,m]]. ii For l > 1, the map _ : Γl m S m is injective, and the action 6 of S m on Γl m is the conjugation action of S m on itself, i.e. : g γ = g γ g 1. We will identify γ and γ if no confusion is likely to arise. In this sense, for g S m, the set of fixed points Γl mg := { } γ Γl m g γ = γ is simply : iii For all m, l > 0, we have : Γ m l g = { l-cycles γ S m gγ = γg }. 7 Γ m l = mm 1 m l 1 l 16

17 The FI-modules IE ν For m, l > 0, let IE m l be the vector space spanned by the set Γ m l, i.e. IE m l := γ Γ m l Q γ. Endow it with the linear action of S m induced by its action on the basis Γ m l. Notice that : IE m l = 0, for all m < l. For all m n, the set Γl n is a subset of Γl m, which is invariant under the action of 1 n S m n, so that the natural inclusions Γl m Γ m+1 l induce the interior maps clearly injective of an FI-module IE l := { } φie l m : IE l m IE m+1 l where φie l m is exhaustive for m l, since the natural map between orbit spaces: S m \Γ m l S m+1 \Γ m+1 l is bijective for m l. More generally, if ν := 1 n 1, 2 n 2,..., N n N = ν 1 ν 2 ν l is a nonempty partition, let IE m ν := IE m 1 n 1 IE m 2 n 2 IE m N n N, := IE ν1 IE ν2 IE νl, and define φie ν m : IE m ν IE m+1 ν The family as the tensor product of interior maps φie ν m := φie 1 n 1 φie N n N, := φie ν1 φie νl. IE ν := { } φie ν m : IE ν m IE ν m+1 m is an FI-module with all interior maps injective, and exhaustive for m ν Proposition a For m, l N, the character χ IE m l is expressed by the following polynomial of Q[X 1,..., X l ] of weight l and independent of m N : E l = φl X l + de φd ed=l, e 1 l X dx d 1 X d e 1, l where φ is the Euler s totient function. The FI-module IE l := {IE l m IE m+1 l } m has the following ranks rank pc IE l = 0 and rank rs IE l = rank fi rsie l = 2l. 17

18 b Given an ordered sequence Z := Z 1,..., Z N of polynomials of Q[X], and d N, denote by Q d [Z 1,..., Z N ] the subspace of polynomials of weight d relative to Z, i.e. the subspace spanned by the elements Z a 1 1 Z a N N where i i a i d. Then, for all N N, the inclusion : is an equality. Q d [E 1,..., E N ] Q d [X 1,..., X N ] c For every nonempty partition ν := 1 n 1, 2 n 2,..., N n N, we have i w IE m ν min{m, ν }. ii The FI-module IE ν := {IE ν m IE ν m+1 } m has the following ranks rank pc IE ν = 0 and rank rs IE ν = rank fi rsie ν = 2 ν. Proof. a Since the linear action of g S m is induced by its action on the basis Γ m l IE m l, the trace χ IE m l g is the cardinality of Γ m l g b-ii. When m < l, the set Γ m l is empty and IE m l = 0. And, since in S m there is no permutation g such that ρ m X d g l/d, we have ρ m E l = 0, which states a when m < l. When m l, following b, we consider two cases : l = 1. Then Γ m 1 = [[1, m]], χ IE m 1 g = [[1, m]] g = X 1 g, and a is obvious. l > 1. The set Γ m l g identifies, after b-ii-7, with the set of l-cycles γ S m such that gγ = γg. As a consequence, g {γ } = {γ } and the set {γ } appears endowed with two actions commuting to each other. But then, since the action of γ on {γ } is transitive, the g -orbits share the same cardinality. Denote it by d and set e := l/d. Then, g κ = γ e, for some κ [[1, d]], relatively prime to d. These observations are gathered in the following image: g γ e =g κ z 1 z 2 z 3 z 4 z e... γ γ γ γ γ γ γ γ γ... γ... e := l/d γ d If we choose the point to stand for the smallest coordinate in γ, this graphic representation becomes one-to-one, and thus helpful for the computation of the cardinality of the set Γl mg. Indeed, it can be used to reconstruct Γl mg by following the next steps. 18

19 1 Take an ordered sequence z 1,..., z e of d-cycles in g, among the X d g e := X d g X d g 1 X d g e 1. possible such sequences. 2 Shift to the first position the d-cycle containing the smallest coordinate in [[1, m]]. This means that X d g e has to be divided by e. 3 Fix the vertical orders of z 2,..., z e. There are d e 1 such orders. 4 Fix the number κ. There are φd possibilities. Therefore, Γ m l g = de=l φd de l X dg e. In both cases, l = 1 and l > 1, the previous expression is independent of m N, which means that rank pc IE l = 0. To estimate rank rs IE l, observe that, since the action of S m on Γl m is transitive, Γl m = S m 1,..., l, so that, denoting γ := 1,..., l, we have an isomorphism of S m -spaces : Γ m l = S m γ S m γ Sm l {pt}, because Stab Sm γ = γ S m l. Therefore, IE m l = ind Sm S l S m l ind S l γ Q Q m l. and IE l is the projective FI-module P ind S l γ Q for which we have already shown that it is RS m 2l cf b. We can be a more precise observing that the simple S l -module Q l appears with multiplicity 1 in ind S l γ Q, which is true since : Hom Sl ind S l γ Q, Q l = Hom γ Q, Q = Q. Thus, IE l contains PQ l = PV 0[l] as the sub-fi-module, and, thanks to d, rank rs IE l = rank fi rsie l = 2l. b In a, the equation l shows that E l = φl X l modulo a polynomial of weight l in the variables X d for d l and d l. In particular, E 1 = X 1. Now, given N 1, the system of equations l, for l N, can be inverted by recursively introducing some polynomials Q m l Z 1,..., Z l Q[Z] of weight l, such that X l = Q m l E 1,..., E l, l N. The equality, Q[X 1,..., X N ] = Q[E 1,..., E N ], 19

20 but also, then follows easily. Q d [X 1,..., X N ] = Q d [E 1,..., E N ], d N, c Since the action of S m on IE ν m := IE ν m 1 IE ν m l is induced by its component-wise action on the basis Γν m := Γν m 1 Γν m 2 Γν m l, the structure of Q[S m ]-module of IE ν m follows from the structure of S m -space of Γν m. Given γ := γ 1,..., γ l Γν m, we have S m γ = S m / Stab Sm γ = S m S {γ } S {γ } / Stab S γ {γ } where we set {γ } := {γ 1 } {γ l }. As a consequence the Q[S m ]-module IE ν m is a direct sum of induced modules of the form: ind Sm S {γ } S W n Q {γ } m n, where we can choose to have {γ } := [[1, n]], for n := {γ } min{m, ν }, in which case W n := Q[S n / Stab Sn γ] = ind Sn Stab Sn γ Q n. When m ν, the corollary a to Pieri s rule, gives the inequality and c-i follows. w ν w IE m ν ν, c-ii Since χ IE m ν = χ IE m 1 n1 χ IE m N n N and since rank pc IE l = 0, after a, it follows immediately that rank pc IE ν = 0. To finish, we notice that, on the one hand, the number of terms of the form is exactly de number of S m -orbits in Γν m, and, on the other hand, the sequence of inclusions S m \Γν m S m+1 \Γν m+1 is strict increasing until it stabilizes for m ν. As a consequence, the truncated FI-module IE ν ν is isomorphic to the sum of truncated projective FI-modules PW n ν, all of which being RS m 2 ν, after b. We therefore have proved that rank rs IE ν rank fi rsie ν 2 ν. To see that these are equalities, it suffices, by 1.7.4, to show that, for m = ν, one has : PQ m IE ν. But, this is clear if we choose γ Γν m such that {γ } = m as in, in which case follows, proceeding as in a, from the inclusion: Q m ind Sm Stab Sm {γ } Q m. 20

21 3. Polynomial weight vs. Representation weight 3.1. On the minimal polynomial weight of a character The Frobenius polynomial X λ Q[X] is of weight w λ, and there are infinitely many polynomials P λ Q[X 1,..., X m 1 ] such that ρ m P λ = ρ m X λ = χ Vλ. In this section, we advance the study of the relationship between weights of polynomials and weights of representations. We will see that w W m is the lowest possible weight for a polynomial P Wm Q[X 1,..., X m 1 ] verifying ρ m P Wm = χ Wm. For example, the polynomial X Wm := λ n λx λ, where X λ is the Frobenius polynomial of an irreducible factor V λ of multiplicity n λ in W m. Moreover, we will see that this polynomial X Wm is the only one satisfying these weights conditions, if and only if w W m m/ Theorem a For d, m N, denote by ρ m d the restriction of ρ m : Q[X] Q cl S m to the subspace Q d [X] of polynomials of weight d. The image of the map is the subspace ρ m d : Q d [X] Q cl S m Q d cl S m = χ Vλ λ = m & w λ d Q, 8 and kerρ m d = 0 if and only if d m/2. b Let P Q[X]. i The scalar product 1 P Sm is constant for m deg w P. ii The scalar product χ Vλ P Sm vanishes for λ m s.t. w λ > deg w P. iii Let W m be a representation of S m. If ρ m P = χ Wm, then: deg w P w W m. c Let W m = λ m V λ n λ be a representation of Sm. The polynomial X Wm := λ m n λx λ Q[X 1,..., X m 1 ] where X λ is the Frobenius polynomial for χ Vλ 2.2.2, verifies ρ m X Wm = χ Wm and deg w X Wm = w W m, and it is the only polynomial verifying simultaneously these two equalities, if and only if w W m m/2. Proof. a In 8, the inclusion is For the converse, it suffices to restrict oneself to the case of a monomial ρ m X n 1 1 X n d d for i i n i d, 21

22 or, thanks to b, what amounts to the same thing, to show that χ IE m 1 n1 χ IE m d n d χ Vλ λ = m & w λ d Q. Here, one recognizes, on the left, the character of IE ν m, for ν := 1 n 1,..., d n d. Now, since we already know after c, that w IE ν m ν, we conclude that w IE ν m d. The irreducible factors of IE ν m are therefore of the form V λ with w λ d, which settles the inclusion. For the last claim about kerρ m d, notice that the space Q d [X 1,..., X d ] admits as a basis the set of monomials Md := { X n 1 1 X n X n d d d i=1 i n i d }, while the space Q d cl S m has a basis indexed by the following set of partitions : Ld := { λ := 1 n 1, 2 n 2,..., λ n λ 2 2 λ 2 i=1 i n i d & λ 2 i=1 i n i + λ 2 m }, because {χ Vλ λ m} is linearly independent after Schur orthogonality. Since λ 2 d, the map ξ : Ld Md, which associates 1 n 1, 2 n 2,..., λ n λ 2 2 with the monomial X n 1 1 X n X n λ 2 λ 2 is well defined and injective. The condition for ξ to be bijective is that λ 2 be able to take the value d, in which case i=1,...,λ 2 i n i = d, and that condition appears to be simply that 2d m. b-i Again, thanks to b, we need only prove that the number: 1 χie m 1 gn 1 χie m 2 g n2 χ IE m r g nr S m, is constant for all m r i=1 i n i. But is simply the dimension of the subspace of invariant tensors in the S m -module IE m ν := IE m 1 n 1 IE m 2 n 2 IE m r nr, where ν := 1 n 1, 2 n 2,..., r nr. Let ν = ν 1 ν 2 ν l. The canonical basis B m ν of IE m ν is the set of tensors γ 1 γ 2 γ l, where γ i is a cycle of length ν i of S m. The basis Bν m is clearly in bijection with the set Tν m of Young tableaux τ of shape ν, with the i th row filled with elements of [[1, m]] so that they represent a cycle of length ν i of S m. The action of S m on Bν m induces in Tν m the natural action of S m on Young tableaux. Because of these identifications, the dimension of IE ν m Sm is the cardinality of the orbit space Tν m /S m, and the stability we are seeking to prove is 22

23 equivalent to the natural map T m ν /S m T m+1 ν /S m+1, [τ mod S m ] [τ mod S m+1 ], being a bijection. But the necessary and sufficient condition for this is precisely that the total number of boxes ν be smaller or equal to m, since, in that case, a single permutation g S m will allow renumbering of all boxes simultaneously with numbers in the interval [[1, ν ]]. Hence is constant for m ν = i i n i. b-ii After a, ρ m P belongs to the linear span of the characters χ Vν of S m, with w ν deg w P, and, thanks to Schur orthogonality, these characters are orthogonal to any χ Vλ with w λ > deg w P, hence the claim. b-iii Indeed, if we had deg w P < w W m then, for any irreducible factor V λ of W m of weight w V λ = w W m, we would get, after b-ii : 0 = χ Vλ P Sm = χ Vλ W m Sm 0, which is a contradiction. Hence deg w P w W m. c Immediate consequence of b-iii and the study of kerρ m d in a Comments The proposition a showed that, for fixed m N, the character of the tensor product IE ν m := IE 1 m n 1 IE 2 m n 2 IE N m n N, as an S m -module, is the polynomial : E ν := E n 1 1 E n 2 2 E n N N of Q[X 1,..., X N ], whose weight deg w E ν = i i n i can be arbitrarily big. On the other hand, theorem c showed that the same character is given by the polynomial X IE m ν of Q[X 1,..., X m 1 ], of weight w IE ν m < m. This disagreement is due to the fact that, while E ν expresses a priori all the characters in the family {χ IE m ν } m N simultaneously, the Frobenius polynomial X IE m ν expresses a priori only one of them, viz. χ IE m ν. Heuristically speaking, the elements in kerρ m allow the reduction of the weight of the polynomial E ν in order to reach the required bound m. This distinction will become meaningful in the next section. It is also worth noting that while the explicit writing of E ν is very simple, thanks to formula a, the writing of X IE m ν can be quite complex insofar as it entails knowing the set of multiplicities n λ of the irreducible factors V λ in the decomposition IE ν m = λ m V n λ λ, and also in the explicit description of each X λ, for n λ 0. The following is a useful corollary to theorem c 23

24 Corollary. Let W be an FB-module which is PC m N and has polynomial character P W. a P W = X Wm, for all m max{2 deg w P W, N}. b w W m = deg w P W, for all m max{2 deg w P W, N}. c w W N = deg w P W. Proof. a,b Immediate after c. c For all m N, we have ρ m P W = χ Wm so that w W m deg w P W, again after c. Therefore, sup m N {w W m } = deg w P W, after b. 4. PC versus RS 4.1. Equivalence of the stability conditions We show the equivalence, for an FB-module W, between the properties RS and PC. In this, two more elements will deserve special attention. First, the ranks of validity of the properties, and, second, the weight of the polynomial P W. The complete statement is the following Theorem. Let W be an FB-module. a If W is RS, then W is PC and rank pc W rank rs W. b If W is PC with polynomial character P W, then W is RS and rank rs W max{rank pc W, 2 deg w P W }. Proof. a is corollary b Set d := deg w P W. After a, we know that P W = X Wm for all m 2d, so that, if we decompose W 2d in its simple Q[S 2d ]-submodules : W 2d = V λ[2d] n λ, λ d and if we denote λ := λ 1 λ 1 λ 2... λ l for λ := λ 1 λ 2... λ l, then the FI-module W := λ d V λ n λ, which is clearly RS m 2d, will be such that P W = P W, by construction. As a consequence, there exists an isomorphism of FB-modules: W max{2d,n} W max{2d,n}, and W is RS for m max{2d, N}, as stated Remark. The last proof shows, in, that an FB-module that is PC is asymptotically isomorphic, as FB-module, to an RS FI-module. 24

25 5. Addendum on the weight of a tensor product 5.1. Reinterpretation of the weight of a representation The theorem c gives an alternative definition of the weight of a representation W m of S m as the lowest possible weight of the polynomials expressing its character χ Wm On the weight of the tensor product of representations Proposition a If W 1,m and W 2,m are representations of S m, then w W 1,m W 2,m w W 1,m + w W 2,m. b If W 1 and W 2 are FB-modules which are PC m N, then w W 1,m W 2,m = w W 1,m + w W 2,m, for all m 2w W 1,m + w W 2,m. In particular, w W 1 W 2 N = w W 1 N + w W 2 N c w V λ[m] V µ[m] = λ + µ, m 2 λ + µ. Proof. aas χ W1 W 2 = χ W1 χ W2, we get ww 1 W 2 deg w X W1 X W2 =deg w X W1 +deg w X W2 = ww 1+wW 2, 1 2 inequality 1 after b-iii, and equality = 2 after c. b Let d i := deg w P Wi. We know after corollary a, that P Wi = X Wi,m, for m 2 max{d 1, d 2 }. Hence, X W1,m X W2,m = X W1,m W 2,m, for m 2d 1 +d 2, after the same corollary. Therefore, w W 1,m W 2,m = w W 1,m + w W 2,m, for m 2d 1 + d 2, again after c. c Particular case of b. References [1] T. Church, B. Farb. Representation theory and homological stability. Adv. Math. 245, [2] W. Fulton, J. Harris. Representation theory. A first course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991 [3] I.G. Macdonald. Symmetric functions and Hall polynomials. Second edition. Oxford University Press, New York,