Multipermutation solutions of the Yang–Baxter equation

Set-theoretic solutions of the Yang–Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set X and a function r : X × X → X × X which satisfies the braid relation.We examine solutions here mainly from the point of view of permutation groups: a solution gives rise to a map from X to the symmetric group Sym(X) on X satisfying certain conditions, whose image we call a Yang–Baxter permutation group. Our results include new constructions based on strong twisted unions, with an investigation of retracts and the multipermutation level and the solvable length of the groups defined by the solutions; new results about decompositions of solutions of arbitrary cardinality into invariant subsets and decompositions and factorisations of the associated Yang–Baxter group as a product of groups of the solutions defined by these invariant subsets. In particular, we obtain strong decomposition results if the Yang–Baxter permutation group is abelian or the solution is of finite multipermutation level.