Supersymmetric elements in divided powers algebras

Abstract

Description of adjoint invariants of general Linear Lie superalgebras $\mathfrak{gl}(m|n)$ by Kantor and Trishin is given in terms of supersymmetric polynomials. Later, generators of invariants of the adjoint action of the general linear supergroup $GL(m|n)$ and generators of supersymmetric polynomials were determined over fields of positive characteristic. In this paper, we introduce the concept of supersymmetric elements in the divided powers algebra $Div[x_1,\dots ,x_m,y_1,\dots ,y_n]$, and give a characterization of supersymmetric elements via a system of linear equations. Then we determine generators of supersymmetric elements for divided powers algebras in the cases when $n=0$, $n=1$, and $m\le 2,n=2$.