# Multipermutation solutions of the Yang-Baxter equation - Mathematics > Quantum Algebra

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Abstract: Set-theoretic solutions of the Yang-Baxter equation form a meeting-ground ofmathematical physics, algebra and combinatorics. Such a solution consists of aset $X$ and a function r:X x X -> X x X which satisfies the braid relation.We examine solutions here mainly from the point of view of finite permutationgroups: a solution gives rise to a map from $X$ to the symmetric group $SymX$on $X$ satisfying certain conditions.Our results include many new constructions based on strong twisted union andwreath product, with an investigation of retracts and the multipermutationlevel and the solvable length of the groups defined by the solutions and newresults about decompositions and factorisations of the groups defined byinvariant subsets of the solution.

Author: ** Tatiana Gateva-Ivanova, Peter Cameron**

Source: https://arxiv.org/