Representations of Skew Left Braces of order $pq$

TL;DR

This work determines the irreducible representations of skew left braces of order $pq$ by translating the problem to the representation theory of their associated semidirect product groups $\Lambda_A$ of order $p^2q^2$ and exploiting the isomorphism $\Lambda_A \cong \Lambda_{A^{\operatorname{op}}}$. Building on the AB2020 classification of skew left braces of size $pq$, the authors classify $\Lambda_A$ up to isomorphism for all such braces, identify isomorphism relations among the resulting groups, and then compute the irreducible representations using standard finite-group theory. The main contributions include explicit counts of 1-, $q$-, and $q^2$-dimensional irreducible representations for each brace case, revealing how the arithmetic of $p$ and $q$ governs the representation theory via the associated $p^2q^2$-groups. The results demonstrate that skew brace theory provides a tractable framework for addressing representation-theoretic questions that are otherwise challenging for general groups of order $p^2q^2$, with concrete examples illustrating the constructions.

Abstract

In this paper, we study the irreducible representations of skew braces of order \( pq \), which is equivalent to studying the representation theory of groups of order \( p^2q^2 \) arising from skew left braces, where \( p > q \) are primes. To achieve this, we classify all semidirect product groups \( Λ_A \) associated with skew left braces $A$ of order \( pq \), up to isomorphism.

Theorems & Definitions (25)

• Theorem 1.1
• Remark 2.1
• Theorem 2.2
• Lemma 2.3
• Definition 2.4
• Lemma 2.5
• Remark 2.6
• Theorem 3.1
• Lemma 3.2
• proof