Classification of the invariant subspaces of the Lawrence-Krammer representation

TL;DR

This paper classifies the invariant subspaces of the Lawrence-Krammer representation of the braid group, detailing their structure and dimensions, which advances understanding of its reducibility and algebraic properties.

Contribution

It provides a complete classification of invariant subspaces of the Lawrence-Krammer representation in terms of Specht modules, including their dimensions and spanning vectors.

Findings

Invariant subspaces are classified as Specht modules.

Full description of subspace dimensions and spanning vectors.

Clarification of reducibility conditions for the representation.

Abstract

The Lawrence-Krammer representation was used in $2000$ to show the linearity of the braid group. The problem had remained open for many years. The fact that the Lawrence-Krammer representation of the braid group is reducible for some complex values of its two parameters is now known, as well as the complete description of these values under some restrictions on one of the parameters. It is also known that when the representation is reducible, the action on a proper invariant subspace is an Iwahori-Hecke algebra action. In this paper, we prove a theorem of classification for the invariant subspaces of the Lawrence-Krammer space. We classify the proper invariant subspaces in terms of Specht modules. We fully describe them in terms of dimension and spanning vectors in the Lawrence-Krammer space.