One class of wild but brick-tame matrix problems

TL;DR

This paper introduces a class of complex matrix problems that are 'brick-tame', meaning they have a manageable set of simple representations, with applications to vector bundles on elliptic curve degenerations and group actions.

Contribution

It identifies and characterizes a new class of wild matrix problems that are still 'brick-tame', expanding understanding of their structure and examples in algebraic geometry and group theory.

Findings

Class of wild matrix problems shown to be brick-tame

Includes problems related to vector bundles on elliptic curves

Applies to coadjoint actions of linear groups

Abstract

We present a class of wild matrix problems (representations of boxes), which are "brick-tame," i.e. only have one-parameter families of \emph{bricks} (representations with trivial endomorphism algebra). This class includes several boxes that arise in study of simple vector bundles on degenerations of elliptic curves, as well as those arising from the coadjoint action of some linear groups.