Jack superpolynomials: physical and combinatorial definitions

TL;DR

This paper explores the dual definitions of Jack superpolynomials, demonstrating their equivalence through physical and combinatorial approaches, and highlighting their significance in symmetric superpolynomial theory.

Contribution

It establishes the equivalence between the physical and combinatorial definitions of Jack superpolynomials, strengthening their theoretical foundation.

Findings

Physical and combinatorial definitions are equivalent.

Supports the correctness of underlying constructions.

Highlights the importance of Jack superpolynomials in symmetric superpolynomials.

Abstract

Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland. They are orthogonal with respect to the scalar product, dubbed physical, that is naturally induced by this quantum-mechanical problem. But Jack superpolynomials can also be defined more combinatorially, starting from the multiplicative bases of symmetric superpolynomials, enforcing orthogonality with respect to a one-parameter deformation of the combinatorial scalar product. Both constructions turns out to be equivalent. This provides strong support for the correctness of the various underlying constructions and for the pivotal role of Jack superpolynomials in the theory of symmetric superpolynomials.