Twisted Lawrence-Krammer representations

TL;DR

This paper extends Lawrence-Krammer representations to twisted versions for certain Artin-Tits monoids and groups, proving faithfulness and irreducibility under specific conditions, thus broadening understanding of their linear representations.

Contribution

It introduces twisted Lawrence-Krammer representations for fixed point submonoids of Artin-Tits monoids under automorphisms, establishing their faithfulness and irreducibility in many cases.

Findings

Twisted representations are faithful under specified conditions.

Irreducibility is proven in all but one spherical case.

Formulas are computed for automorphisms of order two or three.

Abstract

Lawrence-Krammer representations are an important family of linear representations of Artin-Tits groups of small type, which are known, under some assumptions on the parameters, to be faithful when the type is spherical (or more generally when they are restricted to the Artin-Tits monoid) and irreducible when the type is connected. Here, we investigate an analogue of these representations --- introduced by Digne in the spherical cases --- for every Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under a group of graph automorphisms, and for the corresponding Artin-Tits group. Under the same assumptions on the parameters as in the small type cases, we first show that these so-called "twisted Lawrence-Krammer representations" are faithful, and we then prove, by computing their formulas when the group of graph automorphisms is of order two or three, their irreducibility in all the spherical and connected cases but one.