Orbits and invariants of super Weyl groupoid

TL;DR

This paper investigates the orbits and polynomial invariants of an affine super Weyl groupoid related to Lie superalgebras, revealing conditions for finite orbits and explicit invariants, and characterizing special parameters with infinite orbits.

Contribution

It provides a detailed analysis of the orbit structure and invariants of the super Weyl groupoid for Lie superalgebras, including explicit descriptions and parameter conditions.

Findings

For generic parameters, all orbits are finite and distinguishable by explicit invariants.

Identifies a special set of parameters where invariants are not finitely generated and orbits can be infinite.

Provides explicit descriptions of the invariants and orbit structures for the super Weyl groupoid.

Abstract

We study the orbits and polynomial invariants of certain affine action of the super Weyl groupoid of Lie superalgebra $\mathfrak {gl}(n,m)$, depending on a parameter. We show that for generic values of the parameter all the orbits are finite and separated by certain explicitly given invariants. We also describe explicitly the special set of parameters, for which the algebra of invariants is not finitely generated and does not separate the orbits, some of which are infinite.