Polynomiality of the q,t-Kostka Revisited

Details Polynomiality of the q,t-Kostka Revisited Polynomiality of the q,t-Kostka Revisited

2000-08-25

arxiv.org/abs/math/0008199v1

Mathematics

Quantum Algebra

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

Scientific paper

Let $K(q,t)= \|K_{\la\mu}(q,t)\|_{\la,\mu}$ be the Macdonald q,t-Kostka matrix and $K(t)=K(0,t)$ be the matrix of the Kostka-Foulkes polynomials K_{\la\mu}(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.

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