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

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