Staircase Macdonald polynomials and the $q$-Discriminant

TL;DR

This paper establishes a connection between a $q$-deformed discriminant and Macdonald polynomials, providing explicit expansions in various symmetric function bases and generalizing previous results.

Contribution

It demonstrates that a $q$-disformed discriminant equals a specialization of a staircase-indexed Macdonald polynomial and extends known expansions to new bases.

Findings

$q$-discriminant equals a Macdonald polynomial specialization

Explicit monomial basis expansion in terms of standard tableaux

Generalization of King-Toumazet-Wybourne's Schur basis result

Abstract

We prove that a $q$-deformation $\Disc k\X q$ 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 $\Disc k\X q$ 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.