Representations of skew 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 $λ^{\op}: (A, \circ) \to \Aut (A, \cdot)$, which gives rise to the group $Λ_{A^{\op}} = (A, \cdot) \rtimes_{λ^{\op}} (A, \circ)$. We prove that if $A$ and $B$ are isoclinic skew left braces, then $Λ_{A^{\op}}$ and $Λ_{B^{\op}}$ 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 $Λ_{A^{\op}}$. 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 $Λ_{A^{\op}}$. As examples, we compute the dimensions of the irreducible representations for several skew left braces with prime power orders.

17 pages, added Proposition 3.3: Proof that if $A$ is any skew left brace, then $\hat{I}{›}_A$ and $Λ_{A^{\op}}$ are isomorphic groups