Indecomposable solutions of the Yang-Baxter equation of square-free cardinality

2022/12/13 by Ferran Cedó, Cedó, Ferran, Jan Okniński +1 · 3 citations
Mathematics · #16T25 (Primary) 20B15 #20F16 (Secondary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2212.06753

openalex publication_date 2022/12/13 · openalex created_date 2022/12/27 · openalex updated_date 2026/07/28

Abstract

Indecomposable involutive non-degenerate set-theoretic solutions (X,r) of the Yang-Baxter equation of cardinality p₁⋯ p_n, for different prime numbers p₁,…, p_n, are studied. It is proved that they are multipermutation solutions of level ≤ n. In particular, there is no simple solution of a non-prime square-free cardinality. This solves a problem stated in [F. Cedó, J. Okniński, Constructing finite simple solutions of the Yang-Baxter equation, Adv. Math. 391 (2021), 107968] and provides a far reaching extension of several earlier results on indecomposability of solutions. The proofs are based on a detailed study of the brace structure on the permutation group \mathcal G(X,r) associated to such a solution. It is proved that p₁,…, p_n are the only primes dividing the order of G(X,r). Moreover, the Sylow p_i-subgroups of G(X,r) are elementary abelian p_i-groups and if P_i denotes the Sylow p_i-subgroup of the additive group of the left brace G(X,r), then there exists a permutation σ∈ S_n such that P_σ(1), P_σ(1)P_σ(2), … , P_σ(1)P_σ(2)⋯ P_σ(n) are ideals of the left brace G(X,r) and G(X,r)=P₁P₂⋯ P_n. In addition, indecomposable solutions of cardinality p₁⋯ p_n that are multipermutation of level n are constructed, for every nonnegative integer n.

Cited by

• On cardinalities whose arithmetical properties determine the structure of solutions of the Yang--Baxter equation
• One-generator skew braces and indecomposable set-theoretic solutions to the Yang-Baxter equation
• Endocabling of involutive solutions to the Yang-Baxter equation, with an application to solutions whose diagonal is a cyclic permutation

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