Quiver semi-invariants and SAGBI bases
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.
Cite
@article{arxiv.2312.17224,
title = {Quiver semi-invariants and SAGBI bases},
author = {Liana Heuberger and Elana Kalashnikov},
journal= {arXiv preprint arXiv:2312.17224},
year = {2025}
}
Comments
Revised following referee's suggestions. To appear in JACO. 40 pages, many figures, comments welcome!