Quiver semi-invariants and SAGBI bases
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914160574464000 |
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| author | Heuberger, Liana Kalashnikov, Elana |
| author_facet | Heuberger, Liana Kalashnikov, Elana |
| contents | We introduce a new combinatorial structure of linked tableaux, which generalize the semi-standard tableaux that index a SAGBI basis of the Plücker coordinate ring of a flag variety. We show that linked tableaux index Domokos-Zubkov semi-invariants, which span the semi-invariant ring of a quiver. The semi-invariant ring of a quiver coincides in many cases with the Cox ring of an associated quiver moduli space. We show that these semi-invariants satisfy straightening laws. In the case of the generalized Kronecker quiver, we prove that the semi-invariants associated to semi-standard linked tableaux are a (possibly infinite) SAGBI basis. For the generalized Kronecker quiver with dimension vector (2,2), we show that the SAGBI basis is finite and describe it explicitly. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_17224 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quiver semi-invariants and SAGBI bases Heuberger, Liana Kalashnikov, Elana Commutative Algebra Algebraic Geometry Combinatorics 13A50, 14L24 We introduce a new combinatorial structure of linked tableaux, which generalize the semi-standard tableaux that index a SAGBI basis of the Plücker coordinate ring of a flag variety. We show that linked tableaux index Domokos-Zubkov semi-invariants, which span the semi-invariant ring of a quiver. The semi-invariant ring of a quiver coincides in many cases with the Cox ring of an associated quiver moduli space. We show that these semi-invariants satisfy straightening laws. In the case of the generalized Kronecker quiver, we prove that the semi-invariants associated to semi-standard linked tableaux are a (possibly infinite) SAGBI basis. For the generalized Kronecker quiver with dimension vector (2,2), we show that the SAGBI basis is finite and describe it explicitly. |
| title | Quiver semi-invariants and SAGBI bases |
| topic | Commutative Algebra Algebraic Geometry Combinatorics 13A50, 14L24 |
| url | https://arxiv.org/abs/2312.17224 |