Higher-distance commuting varieties

Madeleine Elyze; Alexander Guterman; Ralph Morrison; Klemen Šivic

doi:10.1080/03081087.2020.1834493

Higher-distance commuting varieties

Madeleine Elyze, Alexander Guterman, Ralph Morrison, Klemen Šivic

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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 generalize 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-2 commuting variety and characterize its irreducible components. We also work over other fields, showing that the distance-2 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.

Original language	English
Pages (from-to)	3248-3270
Number of pages	23
Journal	Linear and Multilinear Algebra
Volume	70
Issue number	17
DOIs	https://doi.org/10.1080/03081087.2020.1834493
State	Published - 2022
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2020 Informa UK Limited, trading as Taylor & Francis Group.

Funding

The first author was supported by an Undergraduate Scholarship from the Clare Boothe Luce (CBL) Program (https://www.hluce.org/programs/clare-boothe-luce-program/). The investigations of the second author are financially supported by RSF grant 17-11-01124. The fourth author is partially supported by Slovenian Research Agency (ARRS) [grant numbers N1-0103 and P1-0222]. The authors would like to thank Ngoc Tran, who organized the discussions that led to this project; as well as Spencer Backman and Martin Ulirsch, who took part in those discussions. They also thank Jan Draisma, Tom Garrity, Jake Levinson, and Eyal Markman for many helpful discussions and ideas about higher distance commuting varieties.

Funders	Funder number
Spencer Backman and Martin Ulirsch
American Roentgen Ray Society	N1-0103, P1-0222
Javna Agencija za Raziskovalno Dejavnost RS
Russian Science Foundation	17-11-01124

Keywords

14M12
15A27
Commuting variety
commuting distance
commuting graph

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10.1080/03081087.2020.1834493

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Higher-distance commuting varieties

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Keywords

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