Macdonald polynomials for super-partitions

TL;DR

This paper generalizes Macdonald polynomials to super-Young diagrams with half-boxes, introducing super-Macdonald polynomials that incorporate Grassmann variables and extend their algebraic properties.

Contribution

It introduces super-Macdonald polynomials for super-Young diagrams, expanding the algebraic framework to include Grassmann variables and orthogonality relations.

Findings

Super-Macdonald polynomials extend classical Macdonald polynomials to super-Young diagrams.

They respect two orderings in the set of super-Young diagrams.

The polynomials are fully determined using orthogonality and triangular decompositions.

Abstract

We introduce generalization of famous Macdonald polynomials for the case of super-Young diagrams that contain half-boxes on the equal footing with full boxes. These super-Macdonald polynomials are polynomials of extended set of variables: usual $p_k$ variables are accompanied by anti-commuting Grassmann variables $\theta_k$. Starting from recently defined super-Schur polynomials and exploiting orthogonality relations with triangular decompositions we are able to fully determine super-Macdonald polynomials. These new polynomials have similar properties to canonical Macdonald polynomials -- they respect two different orderings in the set of (super)-Young diagrams simultaneously.