Evaluation and normalization of Jack superpolynomials

TL;DR

This paper derives new evaluation formulas for Jack superpolynomials, simplifying their computation and providing insights into their properties, including a novel derivation of their combinatorial norm.

Contribution

It introduces simplified evaluation formulas for Jack superpolynomials and establishes conditions for non-vanishing coefficients in Pieri-type rules.

Findings

New evaluation formulas expressed via skew diagram fillings

Simplified dominance ordering on superpartitions

Derivation of the combinatorial norm of Jack superpolynomials

Abstract

Two evaluation formulas are derived for the Jack superpolynomials. The evaluation formulas are expressed in terms of products of fillings of skew diagrams. One of these formulas is nothing but the evaluation formula of the Jack polynomials with prescribed symmetry, which thereby receives here a remarkably simple formulation. Among the auxiliary results required to establish the evaluation formulas, the determination of the conditions ensuring the non-vanishing coefficients in a Pieri-type rule for Jack superpolynomials is worth pointing out. An important application of the evaluation formulas is a new derivation of the combinatorial norm of the Jack superpolynomials. We finally mention that the introduction of a simpler version of the dominance ordering on superpartitions is fundamental to establish our results.