Combinatorics of crystal graphs and Kostka–Foulkes polynomials for the root systems Bn,Cn and Dn

摘要

We use Kashiwara–Nakashima combinatorics of crystal graphs associated with the roots systems Bn and Dn to extend the results of Lecouvey [C. Lecouvey, Kostka–Foulkes polynomials, cyclage graphs and charge statistics for the root system Cn, J. Algebraic Combin. (in press)] and Morris [A.-O. Morris, The characters of the group GL(n,q), Math. Z. 81 (1963) 112–123] by showing that Morris-type recurrence formulas also exist for the orthogonal root systems. We derive from these formulas a statistic on Kashiwara–Nakashima tableaux of types Bn,Cn and Dn generalizing the Lascoux–Schützenberger charge and from which it is possible to compute the Kostka–Foulkes polynomials Kλ,μ(q) under certain conditions on (λ,μ). This statistic is different from that obtained in Lecouvey [C. Lecouvey, Kostka–Foulkes polynomials, cyclage graphs and charge statistics for the root system Cn, J. Algebraic Combin. (in press)] from the cyclage graph structure on tableaux of type Cn. We show that such a structure also exists for the tableaux of types Bn and Dn but cannot be related in a simple way to the Kostka–Foulkes polynomials. Finally we give explicit formulas for Kλ,μ(q) when |λ|≤3, or n=2 and μ=0.