A duality between $q$-multiplicities in tensor products and $q$-multiplicities of weights for the root systems $B,C$ or $D$

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A duality between q-multiplicities in tensor products and q-multiplicities of weights for the root systems B, C or D

Cédric Lecouvey

To cite this version:

Cédric Lecouvey. A duality betweenq-multiplicities in tensor products andq-multiplicities of weights for the root systemsB, C orD. 2004. �hal-00003061�

ccsd-00003061, version 1 - 12 Oct 2004

A duality between q-multiplicities in tensor products and q-multiplicities of weights for the root systems B, C or D

C´edric Lecouvey lecouvey@math.unicaen.fr

Abstract

Starting from Jacobi-Trudi’s type determinental expressions for the Schur functions of types B, C and D, we define a natural q-analogue of the multiplicity [V(λ) : M(µ)] when M(µ) is a tensor product of row or column shaped modules defined byµ. We prove that theseq-multiplicities are equal to certain Kostka-Foulkes polynomials related to the root systems C or D. Finally we express the corresponding multiplicities in terms of Kostka numbers

1 Introduction

Given two partitions λ and µ of length n, the Kostka number K_λ,µ^An is equal to the dimension of the weight space µ in the finite dimensional irreducible sl_n+1-module V(λ) of highest weightλ. The Schur duality is a classical result in representation theory establishing that K_λ,µ^An is also equal to the multiplicity of V(λ) in the tensor products

V(µ1Λ1)⊗ · · · ⊗V(µnΛ1) andV(Λ_µ^′₁)⊗ · · · ⊗V(Λ_µ^′_m)

whereµ^′ = (µ^′₁, ..., µ^′_m) is the conjugate partition ofµand the Λ_i’si= 1, ..., n−1 are the fundamental weights ofsl_n+1.Another way to defineK_λ,µ^An is to use the Jacobi-Trudi identity which gives a determi- nantal expression of the Schur functions_µ= char(V(µ)) in terms of the charactersh_k= char(V(kΛ₁)) of the k-th symmetric power representation. This formula can be rewritten

s_µ= Y

1≤i<j≤n

(1−R_i,j)h_µ (1)

where hµ =hµ1 · · · ·hµn and the Ri,j are the raising operators (see 3.2). Then one can prove that it makes sense to write

h_µ= Y

1≤i<j≤n

(1−R_i,j)⁻¹s_µ (2)

which gives the decomposition of h_µ on the basis of Schur functions. From this decomposition we derive the following expression for K_λ,µ^An:

K_λ,µ^An = X

σ∈Sn

(−1)^l(σ)P^An(σ(λ+ρ)−(µ+ρ)) (3) whereS_nis the symmetric group of ordernandP^An the ordinary Kostant’s partition function defined from the equality:

Y

αpositive root

1

(1−x^α) =X

β

P^An(β)x^β

with β running on the set of nonnegative integral combinations of positive roots of sl_n.

There exists a q-analogue K_λ,µ^An(q) of K_λ,µ^An obtained by replacing the ordinary Kostant’s partition function P^An by itsq-analogue P_q^An satisfying

Y

αpositive root

1

(1−qx^α) =X

β

P_q^An(β)x^β. So we have

K_λ,µ^An(q) = X

σ∈Sn

(−1)^l(σ)P_q^An(σ(λ+ρ)−(µ+ρ)) (4) which is a polynomial in q with nonnegative integer coefficients [8], [9]. In [11], Nakayashiki and Yamada have shown thatK_λ,µ^An(q) can also be computed from the combinatorialRmatrix corresponding to Kashiwara’s crystals associated to some U_q(slc_n)-modules.

For g= so2n+1, sp2n or so2n there also exist expressions similar to (3) for the multiplicities K_λ,µ^g of the weight µ in the finite dimensional irreducible module V(λ) but a so simple duality as for sl_n does not exist although it is possible to obtain certain duality results between multiplicities of weights and tensor product multiplicities of representations by using duals pairs of algebraic groups ([5]).

This implies that the quantifications of weight multiplicities and tensor product multiplicities can not coincide for the root systems B_n, C_n and D_n. The Kostka-Foulkes polynomials K_λ,µ^g (q) are the q-analogues of K_λ,µ^g defined as in (4) by quantifying the partition function corresponding to the root system associated to g (see 2.2). In [13], Hatayama, Kuniba, Okado and Takagi have introduced for type C_n a quantification X_λ,µ^Cn(q) of the multiplicity of V(λ) in the tensor product

W(µ₁Λ₁)⊗ · · · ⊗W(µ_nΛ₁) where for any i= 1, ..., n,

W(µ_iΛ₁) =V(µ_iΛ₁)⊕V((µ_i−2)Λ₁)⊕ · · · ⊕V((µ_imod2)Λ₁).

This quantification is based on the determination of the combinatorialRmatrix of someU_q^′(bg)-crystals in the spirit of [11]. Note that there also exist q-multiplicities for the sp₂-module V(λ) in a tensor product

V(Λ₁)^⊗k⊗V(Λ₂)^⊗l where k, l are positive integers obtained by Yamada [17].

In this paper we first use Jacobi-Trudi’s type determinantal expressions for the Schur functions associated to g to introduce q-analogues of the multiplicity ofV(λ) in the tensor products

(i) :h(µ) =V(µ₁Λ₁)⊗ · · · ⊗V(µ_nΛ₁),H(µ) =W(µ₁Λ₁)⊗ · · · ⊗W(µ_nΛ₁) (ii) :e(µ) =V(Λ_µ^′

1)⊗ · · · ⊗V(Λ_µ^′_m),E(µ) =W(Λ_µ^′

1)⊗ · · · ⊗W(Λ_µ^′_m) withn≥ |µ|

where

W(µiΛ1) =V(µiΛ1)⊕V((µi−2)Λ1)⊕ · · · ⊕V((µimod2)Λ1) W(Λ_k) =V(Λ_k)⊕V(Λ_k−2)⊕ · · · ⊕V(Λ_{kmod 2}) .

With the condition n≥ |µ|for (ii), these multiplicities are independent of the Lie algebra g of type Bn, Cn or Dn considered. When q = 1, we recover a remarkable property already used by Koike and Terada in [6]. Next we prove that these q-multiplicities are in fact equal to Kostka-Foulkes

polynomials associated to the root systems of types C and D.It is possible to extend the definition (4) of the Kostka-Foulkes polynomials associated to the root system A_n by replacing µ by γ ∈ Nⁿ whereγ is not a partition. In this caseK_λ,γ^An(q) may have nonnegative coefficients butK_λ,γ^An(1) is equal to the dimension of the weight space γ in V(λ). Now if we extend (4) by replacing λ by ξ ∈ Nⁿ, the polynomial K_ξ,µ^An(q) is equal up to a sign to a Kostka-Foulkes polynomial K_ν,µ^An(q) where ν is a partition. We obtained two expressions of theq-multiplicities defined above respectively in terms of the polynomialsK_λ,γ^An(q) andK_ξ,µ^An(q).By specializing atq= 1,this yields expressions of the corresponding multiplicities in terms of Kostka numbers.

In section 1 we recall the background on the root systems Bn, Cn and Dn and the corresponding Kostka-Foulkes polynomials. We review in section 2 the determinantal identities for Schur functions that we need in the sequel and we introduce the formalism suggested in [1] to prove the expressions of Schur functions in terms of raising and lowering operators implicitly contain in [15]. Thank to this formalism we are able to obtain expressions for multiplicities similar to (3). We quantify these multiplicities to obtain the desired q-analogues in section 3.We prove in Section 4 two duality theo- rems between our q-analogues and certain Kostka-Foulkes polynomials of types C and D.Finally we establish formulas expressing the associated multiplicities in terms of Kostka numbers.

Notation: In the sequel we frequently define similar objects for the root systems B_n C_n and D_n. When they are related to typeB_n(resp. C_n, D_n), we implicitly attach to them the labelB (resp. the labels C, D). To avoid cumbersome repetitions, we sometimes omit the labelsB, C and D when our definitions or statements are identical for the three root systems.

Note: While writing this work, I have been informed that Shimozono and Zabrocki [16] have introduced independently and by using creating operators essentially the same tensor power multiplicities. Thanks to this formalism they recover in particular Jacobi-Trudi’s type determinantal expressions of the Schur functions associated to the root systems B, C and Dwhich constitute the starting point of this article.

2 Background on the root systems B_n, C_n and D_n

2.1 Convention for the positive roots

Consider an integer n ≥ 1. The weight lattice for the root system C_n (resp. B_n and D_n) can be identified with PCn = Zⁿ (resp. PBn = PDn

Z 2

n

) equipped with the orthonormal basis εi, i = 1, ..., n. We take for the simple roots





α^B_nⁿ =ε_n andα^B_iⁿ =ε_i−ε_i+1, i= 1, ..., n−1 for the root systemB_n α^C_nⁿ = 2ε_n and α^C_iⁿ =ε_i−ε_i+1,i= 1, ..., n−1 for the root systemC_n α^D_nⁿ =ε_n+ε_n−1 and α^D_i ⁿ =ε_i−ε_i+1, i= 1, ..., n−1 for the root systemD_n

. (5)

Then the set of positive roots are





R⁺_Bn ={ε_i−ε_j, ε_i+ε_j with 1≤i < j≤n} ∪ {ε_i with 1≤i≤n}for the root system B_n R⁺_Cn ={ε_i−ε_j, ε_i+ε_j with 1≤i < j ≤n} ∪ {2ε_i with 1≤i≤n} for the root system C_n R⁺_Dn ={ε_i−ε_j, ε_i+ε_j with 1≤i < j≤n} for the root systemD_n

.

Denote respectively by P_B⁺

n, P_C⁺

n and P_D⁺

nthe sets of dominant weights of so_2n+1, sp_2n and so_2n.

Letλ= (λ₁, ..., λ_n) be a partition withnparts. We will classically identifyλwith the dominant weight P_n

i=1λ_iε_i.Note that there exists dominant weights associated to the orthogonal root systems whose coordinates on the basisεi, i= 1, ..., nare not positive integers (hence which can not be regarded as a partition). For each root system of type B_n, C_n orD_n, the set of weights having nonnegative integer coordinates on the basis ε₁, ..., ε_n can be identify with the set π_n⁺ of partitions of length n. For any partition λ,the weights of the finite dimensional so2n+1, sp2n orso2n-module of highest weight λare all in π_n=Zⁿ.For any α∈π_n we write|α|=α₁+· · ·+α_n.

The conjugate partition of the partitionλis denotedλ^′ as usual. Considerλ, µtwo partitions of length n and set m = max(λ1, µ1). Then by adding to λ^′ and µ^′ the required numbers of parts 0 we will consider them as partitions of length m.

The Weyl group W_Bn = W_Cn of so_2n+1 and sp_2n is identified to the sub-group of the permutation group of the set{n, ...,2,1,1,2, ..., n}generated bys_i= (i, i+1)(i, i+ 1), i= 1, ..., n−1 ands_n= (n, n) where for a6=b (a, b) is the simple transposition which switches aand b.We denote byl_B the length function corresponding to the set of generators s_i, i= 1, ...n.

The Weyl groupW_Dn ofso_2nis identified to the sub group ofW_Bn generated bys_i= (i, i+ 1)(i, i+ 1), i= 1, ..., n−1 and s^′_n = (n, n−1)(n−1, n). We denote by l_D the length function corresponding to the set of generators s^′_n ands_i, i= 1, ...n−1.

Note that W_Dn ⊂ W_Bn and any w ∈W_Bn verifies w(i) =w(i) for i∈ {1, ..., n}. The action of w on β = (β1, ..., βn)∈Pn is given by

w·(β1, ..., βn) = (β₁^w, ..., β_n^w) where β_i^w =β_w(i) ifσ(i)∈ {1, ..., n} and β_i^w =−β_w(i) otherwise.

The half sums ρBn, ρCn and ρDn of the positive roots associated to each root system Bn, Cn and Dn

verify:

ρBn = (n− 1 2, n−3

2, ...,1

2), ρCn = (n, n−1, ...,1) andρBn = (n−1, n−2, ...,0).

In the sequel we identify the symmetric group S_n with the sub group of W_Bn or W_Dn generated by the s_i’s,i= 1, ..., n−1.

2.2 Schur functions and Kostka-Foulkes polynomials

We now briefly review the notions of Schur functions and Kostka-Foulkes polynomials associated to the roots systems B_n, C_n and D_n and refer the reader to [12] for more details. For any weight β = (β₁, ..., β_n)∈π_n we setx^β =x^β₁¹ · · ·x^β_nⁿ wherex₁, ..., x_n are fixed indeterminates. We set

a^B_βⁿ = X

w∈W_Bn

(−1)^l(σ)(w·x^β)

where w·x^µ=x^w(µ).The Schur function s^B_βⁿ is defined as in [12] by

s^B_βⁿ = a^B_β+ρⁿ

Bn

a^B_ρBn .

Whenν ∈π_n⁺, s^B_νⁿ is the Weyl character ofV(ν) the finite dimensional irreducible module with highest weight ν.For any w∈W_Bn,the dot action ofw on β∈π_n is defined by

w◦β=w·(β+ρ_Bn)−ρ_Bn.

We have the following straightening law for the Schur functions. For any β ∈π_n, s^B_βⁿ = 0 or there exists a unique ν ∈π_n⁺ such that s^B_βⁿ = (−1)^l(w)s^B_νⁿ withw∈W_Bn and ν =w◦β. SetK=Z[q, q⁻¹] and write K[π_n] for the K-module generated by the x^β,β ∈π_n. SetC_Bn = K[π_n]^WBn ={f ∈K[π_n], w·f =f for any w∈WBn}.Then {s^B_νⁿ}, ν ∈π_n⁺ is a basis of K[πn]^WBn.

We defines^C_βⁿ ands^D_βⁿ belonging toC_Cn =C_Bn and C_Dn in the same way and we obtain similarly that {s^C_νⁿ, ν ∈π_n⁺}and {s^D_νⁿ, ν ∈π_n⁺}are respectively bases ofC_Cn and C_Dn.

The q-analogue P_q^Bn of Kostant’s partition function corresponding to the root system B_n is defined by the equality

Y

α∈R⁺_Bn

1

1−qx^α = X

β∈πn

P_q^Bn(β)x^β.

Note that P_q^Bn(β) = 0 if β is not a linear combination of positive roots of R_B⁺_nwith nonnegative coefficients. We write similarly P_q^Cn and P_q^Dn for theq-partition functions associated respectively to the root systemsC_nandD_n. Givenλandµtwo partitions of lengthn,the Kostka-Foulkes polynomials of types Bn, Cn and Dn are then respectively defined by

K_λ,µ^Bn(q) = X

σ∈W_Bn

(−1)^l(σ)P_q^Bn(σ(λ+ρ_Bn)−(µ+ρ_Bn)), K_λ,µ^Cn(q) = X

σ∈W_Cn

(−1)^l(σ)P_q^Cn(σ(λ+ρ_Cn)−(µ+ρ_Cn)), K_λ,µ^Dn(q) = X

σ∈W_Dn

(−1)^l(σ)P_q^Dn(σ(λ+ρDn)−(µ+ρDn)).

Remarks:

(i) : We have K_λ,µ(q) = 0 when|λ|<|µ|.

(ii) : When |λ|=|µ|, K_λ,µ^Bn(q) = K_λ,µ^Cn(q) =K_λ,µ^Dn(q) = K_λ,µ^An−1(q) that is, the Kostka-Foulkes polyno- mials associated to the root systems B_n, C_nandD_nare Kostka-Foulkes polynomials associated to the root system A_n−1.

3 Determinantal identities and multiplicities of representations

3.1 Determinantal identities for Schur functions

Consider k∈Z. When k is a nonnegative integer, write (k)_n= (k,0, ...,0) for the partition of length n with a unique non-zero part equal tok. Then set

h^B_kⁿ =s^B_(k)ⁿ

n, h^C_kⁿ =s^C_(k)ⁿ

n, h^D_kⁿ =s^D_(k)ⁿ

n

and

H_k^Bn =h^B_kⁿ+h^B_k−2ⁿ +· · ·+h^B_kⁿ_{mod 2}, H_k^Cn =h^C_kⁿ+h^C_k−2ⁿ +· · ·+h^B_kⁿ_{mod 2}, H_k^Dn =h^D_kⁿ+h^D_k−2ⁿ +· · ·+h^D_{kmod 2}ⁿ .

When k is a negative integer we seth^B_kⁿ =h^C_kⁿ =h^D_kⁿ = 0 andH_k^Bn =H_k^Cn =H_k^Dn = 0.

For any α= (α₁, ..., α_n)∈Zⁿ define

u^B_αⁿ = det









h^B_α1ⁿ h^B_α1ⁿ₊₁+h^B_α1ⁿ₋₁ · · · ·· h^B_α1ⁿ_+n−1+h^B_α1ⁿ_−n+1 h^B_α2ⁿ₋₁ h^B_α2ⁿ+h^B_α2ⁿ₋₂ · · · ·· h^B_α2ⁿ_+n−2+h^B_α2ⁿ_−n

· · · · · ·· ·

· · · · · ·· ·

h^B_αnⁿ_−n+1 h^B_αnⁿ_−n+2+h^B_αnⁿ_−n · · · ·· h^B_αnⁿ+h^B_αnⁿ_−2n+2







. (6)

By using the equalities h^B_kⁿ =H_k^Bn−H_k−2^Bn and simple computations on determinants we have also

u^B_αⁿ = det









H_α^Bn

1 −H_α^B₁ⁿ₋₂ H_α^B₁ⁿ₊₁−H_α^B₁ⁿ₋₁ · · · ·· H_α^B₁ⁿ_+n−1−H_α^B₁ⁿ_−n−1 H_α^B₂ⁿ₋₁−H_α^B₂ⁿ₋₃ H_α^B₂ⁿ−H_α^B₂ⁿ₋₄ · · · ·· H_α^B₂ⁿ_+n−2−H_α^B₂ⁿ_−n−2

· · · · · ·· ·

· · · · · ·· ·

H_α^B_nⁿ_−n+1−H_α^B_nⁿ_−n−1 H_α^B_nⁿ_−n+2−H_α^B_nⁿ_−n−2 · · · ·· H_α^B_nⁿ−H_α^B_nⁿ_−2n−2







 .

(7) We define u^C_αⁿ and u^D_αⁿsimilarly by replacing h^B_kⁿ respectively by h^C_kⁿ and h^D_kⁿ.

Considerp and ntwo integers such that n≥1. When p is nonnegative and n≥p, write (1^p)_n = (1, ...,1,0, ...,0) for the partition of length nhaving p non zero parts equal to 1.We set







e^B_pⁿ =s^B₍₁ⁿp)n, e^C_pⁿ =s^C₍₁ⁿp)n, e^D_pⁿ =s^D₍₁ⁿp)n if 0≤p≤n e^B_pⁿ =e^B_2p−nⁿ , e^C_pⁿ =e^C_2p−nⁿ , e^D_kⁿ =e^D_2p−nⁿ ifn+ 1≤p≤2n e^B_pⁿ =e^C_pⁿ =e^D_pⁿ = 0 otherwise

and

E_k^Bn =e^B_kⁿ+e^B_k−2ⁿ +· · ·+e^B_kⁿ_{mod 2}, E_k^Cn =e^C_kⁿ+e^C_k−2ⁿ +· · ·+e^B_{kmod 2}ⁿ , E_k^Dn =e^D_kⁿ+e^D_k−2ⁿ +· · ·+e^D_{kmod 2}ⁿ .

For any β= (β₁, ..., β_n)∈Zⁿdefine

v^B_βⁿ = det







 e^B_βⁿ

1 e^B_βⁿ

1+1+e^B_βⁿ

1−1 · · · ·· e^B_βⁿ

1+n−1+e^B_βⁿ

1−n+1

e^B_β2ⁿ₋₁ e^B_β2ⁿ+e^B_β2ⁿ₋₂ · · · ·· e^B_β2ⁿ_+n−2+e^B_β2ⁿ_−n

· · · · · ·· ·

· · · · · ·· ·

e^B_βnⁿ_−n+1 e^B_βnⁿ_−n+2+e^B_βnⁿ_−n · · · ·· e^B_βnⁿ+e^B_βnⁿ_−2n+2







 .

By using the equalities e^B_kⁿ =E_k^Bn −E_k−2^Bn and simple computations on determinants we have also

v^B_βⁿ = det









E_β^Bn

1 −E_β^Bn

1−2 E_β^Bn

1+1−E_β^Bn

1−1 · · · ·· E_β^Bn

1+n−1−E_β^Bn

1−n−1

E_β^B₂ⁿ₋₁−E_β^B₂ⁿ₋₃ E_β^B₂ⁿ−E_β^B₂ⁿ₋₄ · · · ·· E_β^B₂ⁿ_+n−2−E_β^B₂ⁿ_−n−2

· · · · · ·· ·

· · · · · ·· ·

E_β^B_nⁿ_−n+1−E_β^B_nⁿ_−n−1 E_β^B_nⁿ_−n+2−E_β^B_nⁿ_−n−2 · · · ·· E_β^B_nⁿ−E_β^B_nⁿ_−2n−2









The determinants v^C_βⁿ, v^D_βⁿ are defined similarly.