The separating variety for 2 x 2 matrix invariants

Article


Elmer, J. 2023. The separating variety for 2 x 2 matrix invariants. Linear and Multilinear Algebra. 72 (3), pp. 389-411. https://doi.org/10.1080/03081087.2022.2158300
TypeArticle
TitleThe separating variety for 2 x 2 matrix invariants
AuthorsElmer, J.
Abstract

We study the action of the group G = GL_2(C) of invertible matrices over the complex numbers on the complex vector space V of n-tuples of 2x2 matrices. The algebra of invariants C[V]^G for this action is well-known, and has dimension 4n-3 and minimum generating set E_n with cardinality 1/6(n^3+11n). In recent work, Kaygorodov, Lopatin and Popov showed that this generating set is also a minimal separating set by inclusion, i.e. no proper subset is a separating set. This does not mean it has smallest possible cardinality among all separating sets. We show that if S is a separating set for C[V]^G then |S| is at least 5n-5. In particular for n=3, the set E_n is indeed of minimal cardinality, but for n>3 may not be so. We then show that a smaller separating set does in fact exist for n>4, We also prove similar results for the left-right action of SL_2(C)xSL_2(C) on V.

KeywordsInvariant theory; matrix invariants; separating set; separating variety; orbit closure; conjugation
Middlesex University ThemeCreativity, Culture & Enterprise
LanguageEnglish
PublisherTaylor and Francis
JournalLinear and Multilinear Algebra
ISSN0308-1087
Electronic1563-5139
Publication dates
Online16 Jan 2023
Print2024
Publication process dates
Deposited01 Dec 2022
Accepted06 Nov 2022
Submitted03 Oct 2022
Output statusIn press
Publisher's version
License
File Access Level
Open
Copyright Statement

© 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Digital Object Identifier (DOI)https://doi.org/10.1080/03081087.2022.2158300
Web of Science identifierWOS:000916925300001
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