Pieri rules for Schur functions in superspace

TL;DR

This paper establishes Pieri rules for Schur functions in superspace, providing foundational properties and dualities, which are essential for understanding their combinatorial and algebraic structures.

Contribution

It introduces Pieri rules for superspace Schur functions and explores their dualities, monomial expansions, and tableaux generating functions.

Findings

Pieri rules for $s_\Lambda$ and $ar s_\Lambda$ are proven.

Duality and monomial expansion properties are derived.

Tableaux generating functions for these bases are established.

Abstract

The Schur functions in superspace $s_\Lambda$ and $\bar s_\Lambda$ are the limits $q=t=0$ and $q=t=\infty$ respectively of the Macdonald polynomials in superspace. We prove Pieri rules for the bases $s_\Lambda$ and $\bar s_{\Lambda}$ (which happen to be essentially dual). As a consequence, we derive the basic properties of these bases such as dualities, monomial expansions, and tableaux generating functions.