Mathematics – Combinatorics
Scientific paper
2004-06-28
Mathematics
Combinatorics
Scientific paper
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 $| \lambda | \leq 3,$ or $n=2$ and $\mu =0$.
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