Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems $B_{n},C_{n}$ and $D_{n}.$

TL;DR

This paper extends combinatorial formulas for Kostka-Foulkes polynomials to root systems of types B, C, and D using crystal graph combinatorics, introducing a new statistic for tableaux and explicit formulas in special cases.

Contribution

It introduces a new statistic on Kashiwara-Nakashima tableaux for types B, C, D and extends recurrence formulas for Kostka-Foulkes polynomials to these root systems.

Findings

Derived a generalized charge statistic for types B, C, D tableaux.

Established recurrence formulas for Kostka-Foulkes polynomials in orthogonal root systems.

Provided explicit formulas for polynomials when |λ| ≤ 3 or n=2 with μ=0.

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 $| \lambda | \leq 3,$ or $n=2$ and $\mu =0$.