The classification problem for unitary R-Matrices with two eigenvalues

TL;DR

This paper classifies all finite-dimensional unitary R-matrices with exactly two eigenvalues, up to an equivalence, providing a comprehensive understanding of their structure in the context of braid group representations.

Contribution

It offers a full classification theorem for such R-matrices, addressing a previously open problem in the theory of braid group representations.

Findings

Complete classification of unitary R-matrices with two eigenvalues

Identification of a potential exceptional class in even dimensions

Clarification of the structure of these R-matrices in relation to braid representations

Abstract

The problem of classifying all unitary R-matrices of arbitrary finite dimension that have precisely two distinct eigenvalues is described, working up to a natural equivalence relation given by the characters of their braid group representations. Up to one class that might or might not exist in even dimension larger than two, a full classification theorem is obtained.