Schur Superpolynomials: Combinatorial Definition and Pieri Rule

TL;DR

This paper introduces conjectural combinatorial definitions for Schur superpolynomials and their duals, extending semi-standard tableaux, and explores the Pieri rule linking these formulations, advancing understanding of superpolynomial structures.

Contribution

It provides the first conjectural combinatorial definitions for Schur superpolynomials and their duals, along with a novel Pieri rule connecting their tableau representations.

Findings

Proposes combinatorial definitions for Schur superpolynomials and duals.

Establishes a conjectural Pieri rule linking the two formulations.

Explores extensions of the Schur bilinear identity.

Abstract

Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit $q=t=0$ and $q=t\rightarrow\infty$, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schur superpolynomials. Indeed, and this is an interesting novelty of the super case, the successive insertions of rows governed by this Pieri rule do not generate the tableaux underlying the Schur superpolynomials combinatorial construction, but rather those pertaining to their dual versions. As an aside, we present various extensions of the Schur bilinear identity.