Vector-Valued Jack Polynomials from Scratch

TL;DR

This paper presents an explicit algorithm for computing vector-valued Jack polynomials associated with the symmetric group using a Yang-Baxter graph, and explores their properties and variants.

Contribution

It introduces a new algorithm based on Yang-Baxter graphs for explicit computation of vector-valued Jack polynomials and extends existing theoretical results with new properties.

Findings

Algorithm for explicit computation of Jack polynomials

Analysis of normalization, symmetrization, and antisymmetrization properties

Introduction of a shifted version and discussion of vanishing properties

Abstract

Vector-valued Jack polynomials associated to the symmetric group ${\mathfrak S}_N$ are polynomials with multiplicities in an irreducible module of ${\mathfrak S}_N$ and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups $G(r,p,N)$ and studied by one of the authors (C. Dunkl) in the specialization $r=p=1$ (i.e. for the symmetric group). By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.