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Quiver Bases of Cartan Squares of Minuscule Representations

  • David B Rush
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Abstract

We consider the Cartan square V2λ of a minuscule representation Vλ of a simply laced complex simple Lie algebra g. We construct for V2λ a family of bases, which we call quiver bases, each indexed by the set RPP2(Pλ) of reverse plane partitions of height two on the minuscule poset Pλ of Vλ. Let Q be a quiver on the Dynkin diagram of g, and let cQ be the corresponding Coxeter element. The quiver basis ℬQ is distinguished by the following property: Up to sign, the action of the Tits representative ċQ on ℬQ lifts the action of cQ, via piecewise-linear toggles, on RPP2(Pλ). This proves uniformly that, for any minuscule poset P, piecewise-linear Coxeter-motion and rowmotion on RPP2(P) exhibit the cyclic sieving phenomenon. In type~A, the quiver basis for the standard orientation recovers, up to rescaling, the canonical basis, whose compatibility with the long cycle was established by Rhoades. In other types, however, we show the canonical basis is not compatible with any Coxeter element.

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Cite

BibTeXRIS
BibTeX
@misc{rush2026quiver,
  title = {{Quiver Bases of Cartan Squares of Minuscule Representations}},
  author = {David B Rush},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29475},
  eprint = {2609.29475},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29475},
}
RIS
TY  - UNPB
TI  - Quiver Bases of Cartan Squares of Minuscule Representations
AU  - David B Rush
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29475
UR  - https://arxiv.org/abs/2609.29475
AB  - We consider the Cartan square V^2λ of a minuscule representation V^λ of a simply laced complex simple Lie algebra g. We construct for V^2λ a family of bases, which we call quiver bases, each indexed by the set RPP_2(P_λ) of reverse plane partitions of height two on the minuscule poset P_λ of V^λ. Let Q be a quiver on the Dynkin diagram of g, and let c_Q be the corresponding Coxeter element. The quiver basis ℬ^Q is distinguished by the following property: Up to sign, the action of the Tits representative ċ_Q on ℬ^Q lifts the action of c_Q, via piecewise-linear toggles, on RPP_2(P_λ). This proves uniformly that, for any minuscule poset P, piecewise-linear Coxeter-motion and rowmotion on RPP_2(P) exhibit the cyclic sieving phenomenon. In type~A, the quiver basis for the standard orientation recovers, up to rescaling, the canonical basis, whose compatibility with the long cycle was established by Rhoades. In other types, however, we show the canonical basis is not compatible with any Coxeter element.
ER  -