Inhomogeneous lattice paths, generalized Kostka polynomials and A$_{n-1}$ supernomials

Abstract

Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works and by Kashiwara's theory of crystal bases we define a statistic on paths yielding two novel classes of polynomials. One of these provides a generalization of the Kostka polynomials while the other, which we name the A$_{n-1}$ supernomial, is a $q$-deformation of the expansion coefficients of products of Schur polynomials. Many well-known results for Kostka polynomials are extended leading to representations of our polynomials in terms of a charge statistic on Littlewood-Richardson tableaux and in terms of fermionic configuration sums. Several identities for the generalized Kostka polynomials and the A$_{n-1}$ supernomials are proven or conjectured. Finally, a connection between the supernomials and Bailey's lemma is made.