# Yang-Baxter equations and the symmetric groups

*by Simon Wood*

*Institution:*Cardiff University

*Date: Tue 23rd May 2017*

*Time: 1:00 PM*

*Location: Peter Hall 213*

*Abstract*: The Yang-Baxter equation is a cubic equation in a linear map R acting on the tensor square of some vector space V. If R is invertible, then the Yang-Baxter equation is equivalent to the braid relations of the braid groups, thus defining a representation of the n-th braid group B_n on the n-th tensor power of V. If, in addition, R squares to 1, then the defining relations of the symmetric groups are satisfied and the n-th tensor power of V is furnished with a representation of the n-th symmetric group. In this talk I will present some recent work on classifying such symmetric group solutions of the Yang-Baxter equation and the interesting structures encountered in this context.