Classification of the invariant subspaces of the Cohen-Wales representation of the Artin group of type $D_n$

TL;DR

This paper classifies the proper invariant subspaces of the Cohen-Wales representation of the Artin group of type D_n, linking them to Specht modules and extending previous work on the representation's reducibility.

Contribution

It provides a detailed classification of invariant subspaces in terms of Specht modules, advancing understanding of the representation's structure and reducibility conditions.

Findings

Invariant subspaces correspond to Specht modules indexed by double partitions.

The classification clarifies when the representation is reducible based on parameter values.

The work extends previous classifications to a broader set of parameters.

Abstract

Recently, Cohen and Wales built a faithful linear representation of the Artin group of type $D_n$, hence showing the linearity of this group. It was later discovered that this representation is reducible for some complex values of its two parameters. It was also shown that when the representation is reducible, the action on a proper invariant subspace is a Hecke algebra action of type $D_n$. The goal of this paper is to classify these proper invariant subspaces in terms of Specht modules indexed by double partitions of the integer $n$. This work is the continuation of arXiv:1103.5673