# 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
Type Article The separating variety for 2 x 2 matrix invariants Elmer, J. 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. Invariant theory; matrix invariants; separating set; separating variety; orbit closure; conjugation Creativity, Culture & Enterprise English Taylor and Francis Linear and Multilinear Algebra 0308-1087 1563-5139 16 Jan 2023 2024 01 Dec 2022 06 Nov 2022 03 Oct 2022 In press TheSeparatingVarietyFor2x2MatrixInvariants.pdfLicenseCC BY 4.0File Access LevelOpen © 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. https://doi.org/10.1080/03081087.2022.2158300 WOS:000916925300001

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