The separating variety for matrix semi-invariants

Article


Elmer, J. 2023. The separating variety for matrix semi-invariants. Linear Algebra and its Applications. 674, pp. 466-492. https://doi.org/10.1016/j.laa.2023.06.012
TypeArticle
TitleThe separating variety for matrix semi-invariants
AuthorsElmer, J.
Abstract

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V, and let k[V]^G be the corresponding algebra of invariant polynomial functions. A separating set S ⊆ k[V]^G is a set of polynomials with the property that for all v, w ∈ V, if there exists f ∈ k[V]^G separating v and w, then there exists f ∈ S separating v and w. In this article we consider the action of G = SL2(C) × SL2(C) on the C-vector space M of n-tuples of 2 × 2 matrices by multiplication on the left and the right. Minimal generating sets S_n of C[M]^G are known, and |S_n| = 1/24 (n^4 − 6n^3 + 23n^2 + 6n). In recent work, Domokos showed that for all n ≥ 1, S_n is a minimal separating set by inclusion, i.e. that no proper subset of S_n is a separating set. This does not necessarily mean that S_n has minimum cardinality among all separating sets for C[M]^G. Our main result shows that any separating set for C[M]^G has cardinality ≥ 5n−9. In particular, there is no separating set of size dim(C[M]^G) = 4n − 6 for n ≥ 4. Further, S_4 has indeed minimum cardinality as a separating set, but for n ≥ 5 there may exist a smaller separating set than S_n.

KeywordsInvariant theory; Matrix semi-invariants; Separating set; Separating variety; Similarity; Quivers
Sustainable Development Goals9 Industry, innovation and infrastructure
Middlesex University ThemeCreativity, Culture & Enterprise
LanguageEnglish
PublisherElsevier
JournalLinear Algebra and its Applications
ISSN0024-3795
Electronic1873-1856
Publication dates
Online15 Jun 2023
Print01 Oct 2023
Publication process dates
Deposited16 Jun 2023
Accepted12 Jun 2023
Submitted01 Dec 2022
Output statusPublished
Publisher's version
License
File Access Level
Open
Copyright Statement

© 2023 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)

Digital Object Identifier (DOI)https://doi.org/10.1016/j.laa.2023.06.012
Web of Science identifierWOS:001030600400001
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