Image of the Burau Representation at $d$-th Roots of unity

TL;DR

This paper proves that the image of the Burau representation of the braid group at primitive $d$-th roots of unity is arithmetic when the number of strands exceeds or equals $d$, advancing understanding of braid group representations.

Contribution

It establishes the arithmetic nature of the Burau representation's image at roots of unity for a broad class of braid groups, a new result in the field.

Findings

The image of the Burau representation is arithmetic for $n \\geq d$ at primitive $d$-th roots of unity.

Provides new insights into the structure of braid group representations at roots of unity.

Advances the theory of braid group images and their algebraic properties.

Abstract

We prove that the image of the Full braid group $B_{n+1}$ on $n+1$ strands under the Burau representation, evaluated at a primitive $d$-th root of unity is arithmetic provided $n\geq d$.