On m-tuples of nilpotent 2x2 matrices over an arbitrary field
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916565967962112 |
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| author | Lopatin, Artem |
| author_facet | Lopatin, Artem |
| contents | The algebra of ${\rm GL}_n$-invariants of $m$-tuples of $n\times n$ matrices with respect to the action by simultaneous conjugation is a classical topic in case of infinite base field. On the other hand, in case of a finite field generators of polynomial invariants even in case of a pair of $2\times 2$ matrices are not known. Working over an arbitrary field we classified all ${\rm GL}_2$-orbits on $m$-tuples of $2\times 2$ nilpotent matrices for all $m>0$. As a consequence, we obtained a minimal separating set for the algebra of ${\rm GL}_2$-invariant polynomial functions of $m$-tuples of $2\times 2$ nilpotent matrices. We also described the least possible number of elements of a separating set for an algebra of invariant polynomial functions over a finite field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_00477 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On m-tuples of nilpotent 2x2 matrices over an arbitrary field Lopatin, Artem Rings and Algebras Commutative Algebra Representation Theory 16R30, 13A50 The algebra of ${\rm GL}_n$-invariants of $m$-tuples of $n\times n$ matrices with respect to the action by simultaneous conjugation is a classical topic in case of infinite base field. On the other hand, in case of a finite field generators of polynomial invariants even in case of a pair of $2\times 2$ matrices are not known. Working over an arbitrary field we classified all ${\rm GL}_2$-orbits on $m$-tuples of $2\times 2$ nilpotent matrices for all $m>0$. As a consequence, we obtained a minimal separating set for the algebra of ${\rm GL}_2$-invariant polynomial functions of $m$-tuples of $2\times 2$ nilpotent matrices. We also described the least possible number of elements of a separating set for an algebra of invariant polynomial functions over a finite field. |
| title | On m-tuples of nilpotent 2x2 matrices over an arbitrary field |
| topic | Rings and Algebras Commutative Algebra Representation Theory 16R30, 13A50 |
| url | https://arxiv.org/abs/2310.00477 |