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arXiv · 2312.17224

Quiver semi-invariants and SAGBI bases

Liana Heuberger·Elana Kalashnikov
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Abstract

We introduce a new combinatorial structure of linked tableaux, which generalize the semi-standard tableaux that index a SAGBI basis of the Pl\"ucker 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.

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Liana Heuberger, Elana Kalashnikov. 2023-12-28. Quiver semi-invariants and SAGBI bases. https://arxiv.org/abs/2312.17224

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