Unitary and Euclidean representations of a quiver

TL;DR

This paper discusses the classification of unitary and Euclidean representations of quivers, providing algorithms for canonical forms, and describing the parameter spaces of indecomposable representations.

Contribution

It introduces an algorithm for reducing matrices to canonical form and relates Euclidean representations to unitary ones, advancing classification methods.

Findings

Algorithm for canonical form reduction

Description of indecomposable representation dimensions

Parameter count for indecomposable representations

Abstract

A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations in canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension.