Representations of skew braces

TL;DR

This paper studies linear representations of skew left braces, revealing a correspondence with certain group representations and providing decompositions and examples related to their structure and irreducible representations.

Contribution

It establishes a correspondence between irreducible representations of skew left braces and their associated groups, and explores their decomposition and examples.

Findings

Isoclinic skew braces have isoclinic associated groups under certain conditions.

A one-to-one correspondence exists between irreducible representations of braces and their groups.

Explicit dimensions of irreducible representations are computed for prime power order braces.

Abstract

In this paper, we explore linear representations of skew left braces, which are known to provide bijective non-degenerate set-theoretical solutions to the Yang--Baxter equation that are not necessarily involutive. A skew left brace $(A, \cdot, \circ)$ induces an action $\lambda^{\op}: (A, \circ) \to \Aut (A, \cdot)$, which gives rise to the group $\Lambda_{A^{\op}} = (A, \cdot) \rtimes_{\lambda^{\op}} (A, \circ)$. We prove that if $A$ and $B$ are isoclinic skew left braces, then $\Lambda_{A^{\op}}$ and $\Lambda_{B^{\op}}$ are also isoclinic under some mild restrictions on the centers of the respective groups. Our key observation is that there is a one-to-one correspondence between the set of equivalence classes of irreducible representations of $(A, \cdot, \circ)$ and that of the group $\Lambda_{A^{\op}}$. We obtain a decomposition of the induced representation of the additive group $(A, \cdot)$ and of the multiplicative group $(A, \circ)$ corresponding to the regular representation of the group $\Lambda_{A^{\op}}$. As examples, we compute the dimensions of the irreducible representations for several skew left braces with prime power orders.