Reading the Dual Garside Length of Braids from Homological and Quantum Representations

Abstract

We show that Lawrence’s representation and linear representations of the braid groups from generic highest weight representations of $${U_{q}({\mathfrak{sl}}_2)}$$ detect the dual Garside length of braids in a simple and natural way. That is, by expressing a representation as a matrix over a Laurent polynomial ring using a certain natural basis, the span of the variable is equal to a constant multiple of the dual Garside length.