Representation theory of skew braces

TL;DR

This paper explores the representation theory of skew braces, extending classical group representation results like Maschke's and Clifford's theorems to skew braces, and discusses the complexity of classifying such representations.

Contribution

It generalizes key results from group representation theory to skew braces and highlights the increased complexity in classifying skew brace representations.

Findings

Maschke's theorem extends to skew braces

Clifford's theorem extends to skew braces

Skew brace representations are harder to classify than group representations

Abstract

According to Letourmy and Vendramin, a representation of a skew brace is a pair of representations on the same vector space, one for the additive group and the other for the multiplicative group, that satisfies a certain compatibility condition. Following their definition, we shall explain how some of the results from representation theory of groups, such as Maschke's theorem and Clifford's theorem, extend naturally to that of skew braces. We shall also give some concrete examples to illustrate that skew brace representations are more difficult to classify than group representations.