Combinatorics of crystal graphs and Kostka–Foulkes polynomials for the root systems and

We use Kashiwara–Nakashima combinatorics of crystal graphs associated with the roots systems B n and D n to extend the results of Lecouvey [C. Lecouvey, Kostka–Foulkes polynomials, cyclage graphs and charge statistics for the root system C n , J. Algebraic Combin. (in press)] and Morris [A.-O. Morris, The characters of the group G L ( 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 B n , C n and D n generalizing the Lascoux–Schützenberger charge and from which it is possible to compute the Kostka–Foulkes polynomials K λ , 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 C n , J. Algebraic Combin. (in press)] 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 cannot be related in a simple way to the Kostka–Foulkes polynomials. Finally we give explicit formulas for K λ , μ ( q ) when | λ | ≤ 3 , or n = 2 and μ = 0 .