No abstract

Double Kostka polynomials K λ,µ (t) are polynomials in t, indexed by double partitions λ, µ. As in the ordinary case, K λ,µ (t) is defined in terms of Schur functions s λ (x) and Hall-Littlewood functions P µ (x; t). In this paper, we study combinatorial properties of K λ,µ (t) and P µ (x; t). In particular, we show that the Lascoux-Schützenberger type formula holds for K λ,µ (t) in the case where µ = (−; µ ′′ ). Moreover, we show that the Hall bimodule M introduced by Finkelberg-Ginzburg-Travkin is isomorphic to the ring of symmetric functions (with two types of variables) and the natural basis u λ of M is sent to P λ (x; t) (up to scalar) under this isomorphism. This gives an alternate approach for their result.