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<p>We introduce a new combinatorial structure of <i>linked tableaux</i>, 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 inherited from Plücker coordinates. 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,&#xa0;2), we show that the SAGBI basis is finite and describe it explicitly.</p>

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Quiver semi-invariants and SAGBI bases

  • 期刊论文

  • 发表日期：

    2025年12月15日

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  • L. Heuberger,
  • E. Kalashnikov

摘要

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 inherited from Plücker coordinates. 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.