Complete invariants for simultaneous similarity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910244190289920 |
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| author | Bongartz, Klaus Friedland, Shmuel |
| author_facet | Bongartz, Klaus Friedland, Shmuel |
| contents | Always dealing with an arbitrary field we consider the variety $(k^{n\times n})^{p}$ under the action of $GL_{n}$ by simultaneous similarity. We define discrete and continuous invariants which completely determine the orbits. The discrete invariants induce a disjoint decomposition of the variety into finitely many locally closed $GL_{n}$-stable subsets and for each of these we construct finitely many invariant morphisms to $k$ separating the orbits. The complicated action of $GL_{n}$ by similarity is reduced to left multiplication of a product of $GL_{l_{i}}$'s on a product of $k^{l_{i}\times m_{i}}$'s. An analogous result holds for the left-right action of $GL_{m}\times GL_{n}$ on $(k^{m\times n })^{p}$ and more generally for all varieties of finite dimensional modules over some finitely generated algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_00379 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complete invariants for simultaneous similarity Bongartz, Klaus Friedland, Shmuel Representation Theory 14L24, 14L30, 15A21, 16G20, 32S60 Always dealing with an arbitrary field we consider the variety $(k^{n\times n})^{p}$ under the action of $GL_{n}$ by simultaneous similarity. We define discrete and continuous invariants which completely determine the orbits. The discrete invariants induce a disjoint decomposition of the variety into finitely many locally closed $GL_{n}$-stable subsets and for each of these we construct finitely many invariant morphisms to $k$ separating the orbits. The complicated action of $GL_{n}$ by similarity is reduced to left multiplication of a product of $GL_{l_{i}}$'s on a product of $k^{l_{i}\times m_{i}}$'s. An analogous result holds for the left-right action of $GL_{m}\times GL_{n}$ on $(k^{m\times n })^{p}$ and more generally for all varieties of finite dimensional modules over some finitely generated algebra. |
| title | Complete invariants for simultaneous similarity |
| topic | Representation Theory 14L24, 14L30, 15A21, 16G20, 32S60 |
| url | https://arxiv.org/abs/2601.00379 |