Branching Rules, Kostka–Foulkes Polynomials and q-multiplicities in Tensor Product for the Root Systems B _n , C _n and D _n

Abstract

The Kostka–Foulkes polynomials $K_{\lambda ,\mu }^{\phi }(q)$ related to a root system $\phi$ can be defined as alternating 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 $B_{n},C_{n}$ or $D_{n}$, we obtain again a class $\widetilde{K}_{\lambda ,\mu }^{\phi }(q)$ of Kostka–Foulkes polynomials. When $\phi$ is of type $C_{n}$ or $D_{n}$ there exists a duality between these polynomials and some natural $q$-multiplicities $u_{\lambda ,\mu }(q)$ and $U_{\lambda ,\mu }(q)$ in tensor products [11]. In this paper we first establish identities for the $\widetilde{K}_{\lambda ,\mu }^{\phi }(q)$ which implies in particular that they can be decomposed as sums of Kostka–Foulkes polynomials $K_{\lambda ,\mu }^{A_{n-1}}(q)$ with nonnegative integer coefficients. Moreover these coefficients are branching coefficients$.$ This allows us to clarify the connection between the $q$-multiplicities $u_{\lambda ,\mu }(q),U_{\lambda ,\mu }(q)$ and the polynomials $K_{\lambda ,\mu }^{\diamondsuit }(q)$ defined by Shimozono and Zabrocki. Finally we show that $u_{\lambda ,\mu }(q)$ and $U_{\lambda ,\mu }(q)$ coincide up to a power of $q$ with the one dimension sum introduced by Hatayama and co-workers when all the parts of $\mu$ are equal to $1$, which partially proves some conjectures of Lecouvey and Shimozono and Zabrocki.