Branching rules, Kostka-Foulkes polynomials and $q$-multiplicities in tensor product for the root systems $B_{n},C_{n}$ and $D_{n}$ - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Algebras and Representation Theory Année : 2006

Branching rules, Kostka-Foulkes polynomials and $q$-multiplicities in tensor product for the root systems $B_{n},C_{n}$ and $D_{n}$

Résumé

The Kostka-Foulkes polynomials $K$ related to a root system $\phi $ can be defined as alternated sums running over the Weyl group associated to $\phi .$ By restricting these sums over the elements of the symmetric group when $% \phi $ is of type $B,C$ or $D$, we obtain again a class $\widetilde{K}$ of Kostka-Foulkes polynomials. When $\phi $ is of type $C$ or $D$ there exists a duality beetween these polynomials and some natural $q$-multiplicities $U$ in tensor product \cite{lec}. In this paper we first establish identities for the $\widetilde{K}$ which implies in particular that they can be decomposed as sums of Kostka-Foulkes polynomials related to the root system of type $A$ with nonnegative integer coefficients.\ Moreover these coefficients are branching rule coefficients. This allows us to clarify the connection beetween the $q$-multiplicities $U$ and the polynomials defined by Shimozono and Zabrocki in \cite{SZ}. Finally we establish that the $q$-multiplicities $U$ defined for the tensor powers of the vector representation coincide up to a power of $q$ with the one dimension sum $X$ introduced in \cite{Ok} This shows that in this case the one dimension sums $% X$ are affine Kazhdan-Lusztig polynomials.
Fichier principal
Vignette du fichier
article9.pdf (285.8 Ko) Télécharger le fichier
Loading...

Dates et versions

hal-00003733 , version 1 (31-12-2004)
hal-00003733 , version 2 (17-01-2005)

Identifiants

Citer

Cédric Lecouvey. Branching rules, Kostka-Foulkes polynomials and $q$-multiplicities in tensor product for the root systems $B_{n},C_{n}$ and $D_{n}$. Algebras and Representation Theory, 2006, 9 (4), pp.377-402. ⟨10.1007/s10468-006-9020-7⟩. ⟨hal-00003733v2⟩
93 Consultations
177 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More