Polynomiality of the Q, T-kostka Revisited

Let K(q, t) = K λµ (q, t) λ,µ be the Macdonald q, t-Kostka matrix and K(t) = K(0, t) be the matrix of the Kostka-Foulkes polynomials K λµ (t). In this paper we present a new proof of the polynomiality of the q, t-Kostka coefficients that is both short and elementary. More precisely, we derive that K(q, t) has entries in Z[q, t] directly from the fact that the matrix K(t) −1 has entries in Z[t]. The proof uses only identities that can be found in the original paper [7] of Macdonald. Introduction The polynomiality problem for the q, t-Kostka coefficients [11], was posed by Macdonald in the fall 1988 meeting of the Lotharingian seminar. It remained open for quite a few years, when suddenly in 1996, several proofs of varied difficulty appeared in a period of only a few months. At the present there are three basically different approaches to proving the polynomiality of the q, t-Kostka coefficients: