# 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 |
---|---|

Title | The separating variety for 2 x 2 matrix invariants |

Authors | Elmer, 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. |

Keywords | Invariant theory; matrix invariants; separating set; separating variety; orbit closure; conjugation |

Middlesex University Theme | Creativity, Culture & Enterprise |

Language | English |

Publisher | Taylor and Francis |

Journal | Linear and Multilinear Algebra |

ISSN | 0308-1087 |

Electronic | 1563-5139 |

Publication dates | |

Online | 16 Jan 2023 |

Print | 2024 |

Publication process dates | |

Deposited | 01 Dec 2022 |

Accepted | 06 Nov 2022 |

Submitted | 03 Oct 2022 |

Output status | In 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. |

Digital Object Identifier (DOI) | https://doi.org/10.1080/03081087.2022.2158300 |

Web of Science identifier | WOS:000916925300001 |

https://repository.mdx.ac.uk/item/8q29v

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