Higher-distance commuting varieties
Abstract
The commuting variety of matrices over a given field is a well-studied object in linear algebra and algebraic geometry. As a set, it consists of all pairs of square matrices with entries in that field that commute with one another. In this paper we generalise the commuting variety by using the commuting distance of matrices. We show that over an algebraically closed field, each of our sets does indeed form a variety. We compute the dimension of the distance- commuting variety and characterize its irreducible components. We also work over other fields, showing that the distance- commuting set is a variety but that the higher distance commuting sets may or may not be varieties, depending on the field and on the size of the matrices.
Cite
@article{arxiv.1811.09553,
title = {Higher-distance commuting varieties},
author = {Madeleine Elyze and Alexander Guterman and Ralph Morrison and Klemen Šivic},
journal= {arXiv preprint arXiv:1811.09553},
year = {2020}
}
Comments
21 pages; updated to include new results on: the decomposition of the distance-2 variety into irreducibles, as well as a computation of the dimension; instances of higher-distance commuting sets that are not varieties; and the distance-3 variety over the reals, correcting previous results with erroneous proofs