Separating invariants for the klein four group and cyclic groups

TL;DR

This paper explicitly constructs separating invariants for indecomposable representations of the Klein four group and cyclic groups over fields with specific characteristics, using a recursive approach with orbit sums and products.

Contribution

It provides explicit, recursive constructions of separating invariants for these groups' representations, enhancing understanding of invariant rings in modular settings.

Findings

Explicit separating sets are constructed for Klein four and cyclic groups.

The separating sets are composed mainly of orbit sums and products.

The method is recursive and applicable to indecomposable representations.

Abstract

We consider indecomposable representations of the Klein four group over a field of characteristic $2$ and of a cyclic group of order $pm$ with $p,m$ coprime over a field of characteristic $p$. For each representation we explicitly describe a separating set in the corresponding ring of invariants. Our construction is recursive and the separating sets we obtain consist of almost entirely orbit sums and products.