Claire Levaillant, USC

Abstract:  We generalize Cohen-Gijsbers-Wales tangles of type Dn to Coxeter type En and introduce some relations on these tangles. Using the tangles, we build a representation of the parameters based Birman-Murakami-Wenzl algebra of type E6.

We show that as a representation of the Artin group, this representation is equivalent to the parameters based representation which was introduced by Cohen and Wales as a generalization of the Lawrence-Krammer representation of the braid group. The latter representation became famous as it is the only known representation of the braid group that is faithful. Likewise, the Cohen-Wales representation of the Artin group of type E6 is the only known faithful representation of the Artin group of type E6.

We use our representation to find a reducibility criterion for the faithful Cohen-Wales representation, depending on the values of its two parameters. We further derive values of the parameters for which the Birman-Murakami-Wenzl algebra of type E6 is not semisimple.

This work was achieved after the completion of my Ph.D. at Caltech under David Wales (1939-2023). This talk is given to honor his memory.