Grothendieck rings of basic classical Lie superalgebras

TL;DR

This paper explicitly describes the Grothendieck rings of finite-dimensional representations of basic classical Lie superalgebras using generalized root systems and their invariance under Weyl groupoid actions.

Contribution

It provides a new interpretation of these Grothendieck rings as invariant subrings under Weyl groupoid actions, linking representation theory with root system symmetries.

Findings

Explicit descriptions of Grothendieck rings for basic classical Lie superalgebras.

Identification of these rings as invariant subrings under Weyl groupoid actions.

Connection between representation rings and generalized root systems.

Abstract

The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corresponding generalised root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called Weyl groupoids.