Generators of supersymmetric polynomials in positive characteristic

TL;DR

This paper proves two conjectures related to generators of supersymmetric polynomials and polynomial invariants in positive characteristic, extending previous work from characteristic zero to positive characteristic fields.

Contribution

It provides the first proof of conjectures on generators of supersymmetric polynomials and invariants in positive characteristic, advancing the understanding of Lie superalgebra invariants.

Findings

Proved conjectures on generators of supersymmetric polynomials in positive characteristic.

Extended the theory of polynomial invariants from characteristic zero to positive characteristic.

Established foundational results for the algebra of invariants of Lie superalgebras in positive characteristic.

Abstract

Kantor and Trishin described the algebra of polynomial invariants of the adjoint representation of the Lie supergalgebra $gl(m|n)$ and a related algebra $A_s$ of what they called pseudosymmetric polynomials over an algebraically closed field $K$ of characteristic zero. The algebra $A_s$ was investigated earlier by Stembridge who called the elements of $A_s$ supersymmetric polynomials and determined generators of $A_s$. The case of positive characteristic $p$ has been recently investigated by La Scala and Zubkov. They formulated two conjectures describing generators of polynomial invariants of the adjoint action of the general linear supergroup $GL(m|n)$ and generators of $A_s$, respectively. In the present paper we prove both conjectures.