### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 115, 20 pages      arXiv:1804.09158      https://doi.org/10.3842/SIGMA.2018.115
Contribution to the Special Issue on the Representation Theory of the Symmetric Groups and Related Topics

### The Smallest Singular Values and Vector-Valued Jack Polynomials

Charles F. Dunkl
Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 22904-4137, USA

Received June 15, 2018, in final form October 22, 2018; Published online October 25, 2018

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.

Key words: nonsymmetric Jack polynomials; standard modules; Young tableaux.

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