On (twisted) Lawrence-Krammer representations

TL;DR

This paper classifies Lawrence-Krammer representations for small and affine types of Artin-Tits monoids and groups, and extends the twisted LK-representations to non-small cases via automorphisms, advancing understanding of their structure and generalizations.

Contribution

It provides a classification of LK-representations for small and affine types and generalizes twisted LK-representations to broader classes of Artin-Tits monoids.

Findings

Classified LK-representations for small and affine types.

Constructed twisted LK-representations for non-small types.

Extended the construction to fixed point submonoids under automorphisms.

Abstract

Lawrence-Krammer representations (LK-representations for short) are linear representations of Artin-Tits groups of small type, which are of importance since they are known to be faithful when the type is spherical, or more generally when restricted to the monoid. If the construction is essentially unique for a given small and spherical type, the structure of the set of LK-representations for a given small type is not understood in general. Another important question is to ask if there exists an analogue of this construction in the non-small cases ; a first answer is given in [Digne, On the linearity of Artin Braid groups. J. Algebra 268, (2003) 39-57], where is constructed a faithful ``twisted'' LK-representation for the spherical, non-small and crystallographic types. The aim of this paper is to continue the investigations on those two topics. Regarding the first one, we classify the LK-representations of the Artin-Tits monoids and groups of small and affine type. Concerning the second one, we generalize the construction of op.cit. to any Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under the action of graph automorphisms.