Polynomiality of the q,t-Kostka Revisited

Abstract

Let $K(q,t)= \|K_{\laμ}(q,t)\|_{\la,μ}$ be the Macdonald q,t-Kostka matrix and $K(t)=K(0,t)$ be the matrix of the Kostka-Foulkes polynomials K_{\laμ}(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 \ZZ[q,t] directly from the fact that the matrix $K(t{)}^{-1}$ has entries in \ZZ[t]. The proof uses only identities that can be found in the original paper [7] of Macdonald.

19 pages; to appear in a Volume dedicated to the memory of G. C. Rota edited by Domenico Senato U. of Basilicata