# A Combinatorial Approach to the set theoretic solutions of the Yang-Baxter equation

## ABSTRACT

A bijective map $r: X^2 \longrightarrow X^2$, where $X = \{x_1, \cdots , x_n \}$ is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter equation} (YBE) if the braid relation $r_{12}r_{23}r_{12} = r_{23}r_{12}r_{23}$ holds in $X^3.$ Each such a solution (we denote it by $(X;r)$) determines a semigroup $S(r)$ and a group $G(r)$ with a set of generators $X$ and a set of quadratic relations $R(r)$ determined by $r$. The problem of studying the set-theoretic solutions of YBE was posed by Drinfeld in 1990. In this talk I shall discuss the relation between the set-theoretic solutions of YBE, and a special class of standard finitely presented semigroups $S_0$, called \emph{semigroups of skew polynomial type} which I introduced in 1990. In a joint work with Michel Van den Bergh we prove that the set of relations $R_0$ of each semigroup $S_0$ of skew polynomial type determines a (\emph{non-degenerate involutive square-free}) solution of YBE. The corresponding semigroup rings also present a new class of Noetherian Artin-Schelter regular domains. In connection with this, in 1996, I made the conjecture that for each such solution $(X;r)$ the set $X$ can be ordered so, that the relations determined by $r$ form a Groebner basis, and the semigroup $S(r)$ is of skew polynomial type. Using combinatorial methods, recently I verified the conjecture for $n \leq 31$. I shall also discuss the relation between my conjecture and a conjecture of Etingof and Schedler, and various cases in which I verify their conjecture for large $n$.

Speaker's Contact Info: tatiana(at-sign)math.havard.edu