Supersymmetric polynomials and algebro-combinatorial duality

TL;DR

This paper introduces a new combinatorial framework for supersymmetric polynomials extending classical families like Schur, Jack, and Macdonald, with applications to super-algebras and D-brane physics.

Contribution

It develops a systematic combinatorial definition for supersymmetric polynomial families and explores their algebraic and geometric properties, including super-Macdonald polynomials and their algebraic representations.

Findings

Super-Macdonald polynomials form a representation of a super-algebra.

Polynomials depend on odd Grassmann variables and are labeled by modified Young diagrams.

Connections to D-branes and BPS algebras are established.

Abstract

In this note we develop a systematic combinatorial definition for constructed earlier supersymmetric polynomial families. These polynomial families generalize canonical Schur, Jack and Macdonald families so that the new polynomials depend on odd Grassmann variables as well. Members of these families are labeled by respective modifications of Young diagrams. We show that the super-Macdonald polynomials form a representation of a super-algebra analog $\mathsf{T}(\widehat{\mathfrak{gl}}_{1|1})$ of Ding-Ioahara-Miki (quantum toroidal) algebra, emerging as a BPS algebra of D-branes on a conifold. A supersymmetric modification for Young tableaux and Kostka numbers are also discussed.