N>=2 symmetric superpolynomials

TL;DR

This paper extends symmetric function theory to N=2 superpolynomials, systematically constructing bases and scalar products, and exploring their properties and deformations, laying groundwork for future N=2 Jack superpolynomials.

Contribution

It introduces the N=2 super-version of classical symmetric functions, constructs their bases and scalar products, and generalizes these to arbitrary N with a deformation for N=2.

Findings

Constructed N=2 supermonomial, elementary, homogeneous functions, and power sums.

Defined a natural N=2 scalar product preserving duality.

Generalized results to arbitrary N and introduced a one-parameter deformation for N=2.

Abstract

The theory of symmetric functions has been extended to the case where each variable is paired with an anticommuting one. The resulting expressions, dubbed superpolynomials, provide the natural N=1 supersymmetric version of the classical bases of symmetric functions. Here we consider the case where two independent anticommuting variables are attached to each ordinary variable. The N=2 super-version of the monomial, elementary, homogeneous symmetric functions, as well as the power sums, are then constructed systematically (using an exterior-differential formalism for the multiplicative bases), these functions being now indexed by a novel type of superpartitions. Moreover, the scalar product of power sums turns out to have a natural N=2 generalization which preserves the duality between the monomial and homogeneous bases. All these results are then generalized to an arbitrary value of N. Finally, for N=2, the scalar product and the homogenous functions are shown to have a one-parameter deformation, a result that prepares the ground for the yet-to-be-defined N=2 Jack superpolynomials.