Super Krawtchouk Polynomials via Lie Superalgebras

TL;DR

This paper introduces a new class of multivariate super Krawtchouk polynomials derived from Lie superalgebra representation theory, extending classical results and exploring their orthogonality and recurrence properties.

Contribution

It develops a theory of super Krawtchouk polynomials using Lie superalgebras, generalizing classical polynomials and establishing their orthogonality and recurrence relations.

Findings

Proved orthogonality of super Krawtchouk polynomials

Constructed recurrence relations for these polynomials

Connected the polynomials to zonal spherical functions in quantum mechanics

Abstract

Multivariate extensions of the Krawtchouk polynomials have been studied by numerous authors in recent decades by exploring new connections to probability, representation theory and quantum integrability. We develop a theory of multivariate super Krawtchouk polynomials using the representation theory of the general linear Lie superalgebra, extending results of the first author in the classical setting. Specifically, in the present work we generalize the classical Krawtchouk polynomials, prove their orthogonality, construct certain recurrence relations, and discuss their connections with zonal spherical functions arising from a fermionic Fock-space framework in quantum mechanics.