The Deligne-Simpson Problem
Rings and Algebras
2026-04-16 v5
Abstract
Given k similarity classes of invertible matrices, the Deligne-Simpson problem asks to determine whether or not one can find matrices in these classes whose product is the identity and with no common invariant subspace. The first author conjectured an answer in terms of an associated root system, and proved one implication in joint work with Shaw. In this paper we prove the other implication, thus confirming the conjecture.
Cite
@article{arxiv.2509.11998,
title = {The Deligne-Simpson Problem},
author = {William Crawley-Boevey and Andrew Hubery},
journal= {arXiv preprint arXiv:2509.11998},
year = {2026}
}
Comments
Minor improvements to the exposition