A superpolynomial version of nonsymmetric Jack polynomials (Q6158142)

Superpolynomials involve both commuting and anti-commuting variables. Their interpretation as modules of the symmetric group \(\mathcal{S}_{N}\) allows to adapt the structure of the vector-valued nonsymmetric Jack polynomials to this framework. In [\textit{P. Desrosiers} et al., Nucl. Phys., B 606, No. 3, 547--582 (2001; Zbl 0969.81544)], the authors deal with the construction of such Jack superpolynomials by using differential operators which are motivated by equations of supersymmetric quantum mechanics. Later on, in a series of papers the above authors analyze their connections with superpartition ordering and they deduce determinantal formulas as well as the orthogonality of Jack polynomials in superspace. The approach in the paper under review is based on polynomials taking values in \(\mathcal{S}_{N}\)-modules. In such a way, the Young-type construction is applied to polynomials in anti-commuting variables of fixed degree. The basic definitions and the construction of tableau-like polynomials forming orthogonal bases of the two irreducible modules are introduced. A brief overview of the vector-valued nonsymmetric Jack polynomials as well as a description of the action of the group on such polynomials are presented. These polynomials constitute a particular case of the construction in [\textit{S. Griffeth}, Trans. Am. Math. Soc. 362, No. 11, 6131--6157 (2010; Zbl 1283.05264)], defined for the family \(G(n,p, N)\) of complex reflection groups. Taking into account the results in [\textit{C. F. Dunkl} and \textit{J.-G. Luque}, SIGMA, Symmetry Integrability Geom. Methods Appl. 7, Paper 026, 48 p. (2011; Zbl 1225.05239)], a natural inner product for the space of Jack polynomials such that they are mutually orthogonal is obtained. As a consequence, the squared norm of Jack polynomials is deduced in terms of a parameter \(\kappa.\) Following the techniques of \textit{T. H. Baker} and \textit{P. J. Forrester} [Ann. Comb. 3, No. 2--4, 159--170 (1999; Zbl 0937.33010)], in the paper under review the author generates supersymmetric Jack polynomials in an explicit way as well as their norms are deduced. On the other hand, the Poincaré-Hilbert series that yields the number of linearly independent supersymmetric polynomials of a given degree is obtained. Such a series provides information about the minimal supersymmetric polynomials which constitute a generating system of the space of polynomials on the ring of ordinary symmetric polynomials. The norms of certain minimal polynomials are given in terms of a denominator-free formula in terms of the parameter \(\kappa.\) The idea of polynomial-type norm formulas for minimal degree symmetric polynomials was studied for arbitrary irreducible \(\mathcal{S}_{N}\)-submodules in [\textit{C. F. Dunkl}, Sémin. Lothar. Comb. 64, B64a, 33 p. (2010; Zbl 1205.05241)]. Finally, the construction of antisymmetric polynomials as well as an interesting application of vector-valued Jack polynomials as wavefunctions of the Hamiltonian associated with the Calogero-Sutherland quantum mechanical model of identical particles on a circle with \(1/r^2\) interactions are given.