A Superpolynomial Version of Nonsymmetric Jack Polynomials

TL;DR

This paper develops a superpolynomial extension of nonsymmetric Jack polynomials, providing explicit orthogonal bases, norm formulas, and applications to supersymmetric functions and quantum models.

Contribution

It introduces a superpolynomial framework for nonsymmetric Jack polynomials, including orthogonal bases, norm formulas, and connections to supersymmetric and quantum models.

Findings

Explicit orthogonal basis of anti-commuting polynomials

Closed-form squared norms of the polynomials

Application to Calogero-Moser quantum wavefunctions

Abstract

Superpolynomials consist of commuting and anti-commuting variables. By considering the anti-commuting variables as a module of the symmetric group the theory of vector-valued nonsymmetric Jack polynomials can be specialized to superpolynomials. The theory significantly differs from the supersymmetric Jack polynomials introduced and studied in several papers by Desrosiers, Mathieu and Lapointe (Nucl. Phys. B606, 2001). The vector-valued Jack polynomials arise in standard modules of the rational Cherednik algebra and were originated by Griffeth (T.A.M.S. 362, 2010) for the family G(n,p,N) of complex reflection groups. In the present situation there is an orthogonal basis of anti-commuting polynomials which corresponds to hook tableaux arising in Young's representations of the symmetric group. The basis is then used to construct nonsymmetric Jack polynomials by specializing the machinery set up in a paper by Luque and the author (SIGMA 7,2011). There is an inner product for which these polynomials form an orthogonal basis, and the squared norms are explicitly found. Supersymmetric polynomials are obtained as linear combinations of the nonsymmetric Jack polynomials contained in a submodule; this is based on an idea of Baker and Forrester (Ann. Comb. 3, 1999). The Poincar\'e series for supersymmetric polynomials graded by degree is obtained and is interpreted in terms of certain minimal polynomials. There is a brief discussion of antisymmetric polynomials and an application to wavefunctions of the Calogero-Moser quantum model on the circle.