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 $K_{\lambda ,\mu }^{\phi }(q)$ 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 $B_{n},C_{n}$ 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 beetween these polynomials and some natural $q$-multiplicities $u_{\lambda ,\mu }(q)$ and $U_{\lambda ,\mu }(q)$ in tensor product \QCITE{cite}{}{lec}. 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 rule coefficients$.$ This allows us to clarify the connection beetween the $q$-multiplicities $u_{\lambda ,\mu }(q),U_{\lambda ,\mu }(q)$ and the polynomials $K_{\lambda ,\mu }^{\diamondsuit }(q)$ defined in \QCITE{cite}{}{SZ}. 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 in \QCITE{cite}{}{Ok} when all the parts of $\mu $ are equal to $1$ which partially proves some conjectures of \QCITE{cite}{}{lec} and \QCITE{cite}{}{SZ}.