The Smallest Singular Values and Vector-Valued Jack Polynomials

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.