Complexity of matrix problems

TL;DR

This paper explores the complexity of matrix classification problems in representation theory, establishing their equivalence for quivers and posets, and contrasting them with the more complex three-valent tensor classification problem.

Contribution

It explicitly shows the equivalence of classification problems for quivers and posets and situates the tensor problem's complexity relative to these.

Findings

Matrix classification problems for quivers and posets are equally complex.

All wild classification problems for quivers and posets are mutually reducible.

Classifying three-valent tensors is more complex than quiver and poset problems.

Abstract

In representation theory, the problem of classifying pairs of matrices up to simultaneous similarity is used as a measure of complexity; classification problems containing it are called wild problems. We show in an explicit form that this problem contains all classification matrix problems given by quivers or posets. Then we prove that it does not contain (but is contained in) the problem of classifying three-valent tensors. Hence, all wild classification problems given by quivers or posets have the same complexity; moreover, a solution of any one of these problems implies a solution of each of the others. The problem of classifying three-valent tensors is more complicated.