Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems $B_n,C_n$ and $D_n.$

Abstract

We use Kashiwara-Nakashima's combinatorics of crystal graphs associated to the roots sytems $B_n$ and $D_n$ to extend the results of \QCITE{cite}{}{lec3} and \QCITE{cite}{}{Mor} by showing that Morris type recurrence formulas also exist for the orthogonal root systems. We derive from these formulas a statistic on Kashiwara-Nakashima's tableaux of types $B_n,C_n$ and $D_n$ generalizing Lascoux-Sch\UNICODE{0xfc}tzenberger's charge and from which it is possible to compute the Kostka-Foulkes polynomials $K_{\lambda ,\mu }(q)$ with restrictive conditions on $(\lambda ,\mu )$ . This statistic is different from that obtained in \QCITE{cite}{}{lec3} from the cyclage graph structure on tableaux of type $C_n$. We show that such a structure also exists for the tableaux of types $B_n$ and $D_n$ but can not be simply related to the Kostka-Foulkes polynomials. Finally we give explicit formulas for $K_{\lambda ,\mu }(q)$ when $|\lambda |\le 3,$ or $n=2$ and $\mu =0$.