Orthogonality of Jack polynomials in superspace

TL;DR

This paper constructs and compares two types of orthogonal Jack polynomials in superspace, demonstrating their equivalence and extending classical symmetric function theory with an additional parameter.

Contribution

It introduces a new construction of Jack polynomials in superspace and proves their equivalence under different scalar products, extending classical symmetric function theory.

Findings

The constructed Jack polynomials are orthogonal with respect to a combinatorial scalar product.

They coincide with those orthogonal under an analytical scalar product from quantum mechanics.

The work generalizes classical symmetric functions to include an extra parameter.

Abstract

Jack polynomials in superspace, orthogonal with respect to a ``combinatorial'' scalar product, are constructed. They are shown to coincide with the Jack polynomials in superspace, orthogonal with respect to an ``analytical'' scalar product, introduced in hep-th/0209074 as eigenfunctions of a supersymmetric quantum mechanical many-body problem. The results of this article rely on generalizing (to include an extra parameter) the theory of classical symmetric functions in superspace developed recently in math.CO/0509408