On the tensor square of irreducible representations of reductive Lie superalgebras

TL;DR

This paper classifies certain irreducible super representations of semisimple Lie superalgebras over algebraically closed fields of characteristic zero, focusing on when their symmetric or alternating squares are irreducible or decompose simply.

Contribution

It provides a complete classification of irreducible super representations with specific properties of their symmetric or alternating squares for a class of Lie superalgebras.

Findings

Classification of irreducible super representations with irreducible or decomposable symmetric/alternating squares

Identification of cases where squares decompose into an irreducible plus trivial representation

Results applicable to semisimple Lie superalgebras over algebraically closed fields of characteristic zero

Abstract

For semisimple Lie superalgebras over an algebraically closed field of characteristic zero, whose category of finite dimensional super representations is semisismple, we classify all irreducible super representations for which the alternating or symmetric square representation is irreducible or decomposes into an irreducible representation and a trivial representation.