A Combinatorial Approach to the set
theoretic solutions of the YangBaxter equation
Tatiana GatevaIvanova
May 17,
4:15pm
refreshments at 3:45pm
2338
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{settheoretic solution of the
YangBaxter 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 settheoretic solutions of YBE
was posed by Drinfeld in 1990.
In this talk I shall discuss the relation
between the settheoretic 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{nondegenerate involutive squarefree}) solution of YBE.
The corresponding semigroup rings also
present a new class of Noetherian ArtinSchelter 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(atsign)math.havard.edu
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