# The separating variety for 2 x 2 matrix invariants

Article

Elmer, J. 2024. 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 |

Publisher | Taylor and Francis |

Journal | Linear and Multilinear Algebra |

ISSN | 0308-1087 |

Electronic | 1563-5139 |

Publication dates | |

Online | 16 Jan 2023 |

Print | 11 Feb 2024 |

Publication process dates | |

Deposited | 01 Dec 2022 |

Accepted | 06 Nov 2022 |

Submitted | 03 Oct 2022 |

Output status | Published |

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 |

Language | English |

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

## Download files

###### Publisher's version

The_separating_variety_for_2x2_matrix_invariants.pdf | ||

License: CC BY 4.0 | ||

File access level: Open |

##### 55

total views##### 13

total downloads##### 0

views this month##### 0

downloads this month

## Export as

## Related outputs

##### The separating variety for matrix semi-invariants

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

##### Modular covariants of cyclic groups of order p

Elmer, J. 2022. Modular covariants of cyclic groups of order p.*Journal of Algebra.*598, pp. 134-155. https://doi.org/10.1016/j.jalgebra.2022.01.015

##### The relative Heller operator and relative cohomology for the Klein 4-group

Elmer, J. 2022. The relative Heller operator and relative cohomology for the Klein 4-group.*Communications in Algebra.*50 (4), pp. 1518-1534. https://doi.org/10.1080/00927872.2021.1984496

##### Degree bounds for modular covariants

Elmer, J. and Sezer, M. 2020. Degree bounds for modular covariants.*Forum Mathematicum.*32 (4), pp. 905-910. https://doi.org/10.1515/forum-2019-0196

##### Locally finite derivations and modular coinvariants

Elmer, J. and Sezer, M. 2018. Locally finite derivations and modular coinvariants.*Quarterly Journal of Mathematics.*69 (3), pp. 1053-1062. https://doi.org/10.1093/qmath/hay013

##### Symmetric powers and modular invariants of elementary abelian p-groups

Elmer, J. 2017. Symmetric powers and modular invariants of elementary abelian p-groups.*Journal of Algebra.*492, pp. 157-184. https://doi.org/10.1016/j.jalgebra.2017.07.020

##### On separating a fixed point from zero by invariants

Elmer, J. and Kohls, M. 2017. On separating a fixed point from zero by invariants.*Communications in Algebra.*45 (1), pp. 371-375. https://doi.org/10.1080/00927872.2016.1175465

##### Zero-separating invariants for finite groups

Elmer, J. and Kohls, M. 2014. Zero-separating invariants for finite groups.*Journal of Algebra.*411, pp. 92-113. https://doi.org/10.1016/j.jalgebra.2014.03.044

##### Separating invariants for arbitrary linear actions of the additive group

Dufresne, E., Elmer, J. and Sezer, M. 2014. Separating invariants for arbitrary linear actions of the additive group.*Manuscripta Mathematica.*143 (1), pp. 207-219. https://doi.org/10.1007/s00229-013-0625-y

##### Separating Invariants for the Basic G_a actions

Elmer, J. and Kohls, M. 2012. Separating Invariants for the Basic G_a actions.*Proceedings of the American Mathematical Society.*140 (1), pp. 135-146.

##### On the depth of separating invariants for finite groups

Elmer, J. 2012. On the depth of separating invariants for finite groups.*Beitrage zur Algebra und Geometrie.*53 (1), pp. 31-39.

##### The Cohen-Macaulay property of separating invariants of finite groups

Dufresne, E., Elmer, J. and Kohls, M. 2009. The Cohen-Macaulay property of separating invariants of finite groups.*Transformation Groups.*14 (4), pp. 771-785.

##### Depth and detection in modular invariant theory

Elmer, J. 2009. Depth and detection in modular invariant theory.*Journal of Algebra.*322 (5), pp. 1653-1666. https://doi.org/10.1016/j.jalgebra.2009.04.036

##### On the depth of modular invariant rings for the groups C_p x C_p

Elmer, J. and Fleischmann, P. 2009. On the depth of modular invariant rings for the groups C_p x C_p. in: Symmetry and Spaces: in honour of Gerry Schwarz Birkhauser Boston.##### Associated primes for cohomology modules

Elmer, J. 2008. Associated primes for cohomology modules.*Archiv der Mathematik.*91 (6), pp. 481-485. https://doi.org/10.1007/s00013-008-2902-7

##### Zero-separating invariants for linear algebraic groups

Elmer, J. and Kohls, M. 2016. Zero-separating invariants for linear algebraic groups.*Proceedings of the Edinburgh Mathematical Society.*59 (4), pp. 911-924. https://doi.org/10.1017/S0013091515000322