A superpolynomial version of nonsymmetric Jack polynomials

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 et al. (Nucl Phys B606:547–582, 2001). The vector-valued Jack polynomials arise in standard modules of the rational Cherednik algebra and were originated by Griffeth (Trans Am Math Soc 362:6131–6157, 2010) for the family $G\left( n,\ell ,N\right)$ 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 3:159–170, 1999). The Poincaré series for supersymmetric polynomials graded by degree is obtained and is interpreted in terms of certain minimal polynomials. The squared norms of a special subset of these minimal polynomials are polynomials in the parameter. There is a brief discussion of antisymmetric polynomials and an application to wavefunctions of the Calogero–Moser quantum model on the circle.