Double Kostka Polynomials and Hall Bimodule

Double Kostka polynomials K-lambda,K-mu(t) are polynomials in t, indexed by double partitions lambda,mu. As in the ordinary case, K-lambda,K-mu(t) is defined in terms of Schur functions s(lambda)(x) and Hall Littlewood functions P-mu(x; t). In this paper, we study combinatorial properties of K-lambda,K-mu(t) and P-mu(x; t). In particular, we show that the Lascoux Schtitzenberger type formula holds for K-lambda,K-mu(t) in the case where mu = (-,mu"). 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(lambda) of M is sent to P-lambda (x; t) (up to scalar) under this isomorphism. This gives an alternate approach for their result.