Pieri rules for the Jack polynomials in superspace and the 6-vertex model

TL;DR

This paper derives Pieri rules for Jack polynomials in superspace, revealing a surprising connection to the 6-vertex model's partition function, and conjectures similar rules for Macdonald polynomials.

Contribution

It introduces Pieri rules for Jack polynomials in superspace with a novel determinant factor linked to the 6-vertex model, and proposes conjectures for Macdonald polynomials.

Findings

Pieri rules involve products of quotients of linear factors in α

An extra determinant in the rules relates to the 6-vertex model partition function

Conjectured Pieri rules for Macdonald polynomials in superspace

Abstract

We present Pieri rules for the Jack polynomials in superspace. The coefficients in the Pieri rules are, except for an extra determinant, products of quotients of linear factors in $\alpha$ (expressed, as in the usual Jack polynomial case, in terms of certain hook-lengths in a Ferrers' diagram). We show that, surprisingly, the extra determinant is related to the partition function of the 6-vertex model. We give, as a conjecture, the Pieri rules for the Macdonald polynomials in superspace.