Claire Levaillant, USC

Reducibility criterion for the Cohen-Wales representation of the Artin group of type E6

Abstract: We introduce tangles of type En and relations on these tangles. We use this novel diagrammatic algebra to build a representation of the Birman-Murakami-Wenzl algebra of type E6. As a representation of the Artin group of type E6, this representation is equivalent to the faithful representation of Cohen and Wales introduced by them in 2000 as a generalization to the Artin groups of the faithful Lawrence-Krammer representation of the braid group. The latter representation became famous as it is the only representation of the braid group that is known to be faithful. We use our newly built representation to find a reducibility criterion for the Cohen-Wales representation of the Artin group of type E6. Our method generalizes to types E7 and E8. This talk is given to honor the memory of my late Ph.D. advisor David Wales (1939-2023).