Staircase Macdonald polynomials and the $q$-Discriminant

Abstract

We prove that a q-deformation D k (X; q) of the powers of the discriminant is equal, up to a normalization, to a specialization of a Macdonald polynomial indexed by a staircase partition. We investigate the expansion of D k (X; q) on different basis of symmetric functions. In particular, we show that its expansion on the monomial basis can be explicitly described in terms of standard tableaux and we generalize a result of King-Toumazet-Wybourne about the expansion of the q-discriminant on the Schur basis.

Macdonald polynomials at $t=q^k$

“…[3,5,7,13]). Note that in [2], we gave an expression of an other q-deformation of the powers of the discriminant as staircase Macdonald polynomials. This deformation is also relevant in the study of the expansion of i<j (x i − x j ) 2k in the Schur basis, since we generalized [2] a result of [5].…”