The Smallest Singular Values and Vector-Valued Jack Polynomials

TL;DR

This paper investigates the smallest singular values of vector-valued nonsymmetric Jack polynomials, constructing singular polynomials and analyzing their role in the orthogonality and positivity properties within symmetric group representations.

Contribution

It introduces a construction of singular polynomials for the smallest singular values and explores their significance in the orthogonality and positivity of Jack polynomials.

Findings

Constructed singular polynomials for the smallest singular values.

Bound the positivity region of the bilinear form for orthogonality.

Connected results to finite reflection groups and rational Cherednik algebra.

Abstract

There is a space of vector-valued nonsymmetric Jack polynomials associated with any irreducible representation of a symmetric group. Singular polynomials for the smallest singular values are constructed in terms of the Jack polynomials. The smallest singular values bound the region of positivity of the bilinear symmetric form for which the Jack polynomials are mutually orthogonal. As background there are some results about general finite reflection groups and singular values in the context of standard modules of the rational Cherednik algebra.