AI 中文总结
本文为极小表示的Cartan平方构造箭图基，统一证明任意极小偏序集上分段线性Coxeter运动与行运动满足循环筛选现象，并指出典范基仅与A型长循环相容。
AI 中文摘要
我们考虑单连通复单李代数$\mathfrak g$的极小表示$V^\lambda$的Cartan平方$V^{2\lambda}$。我们为$V^{2\lambda}$构造一族基，称为箭图基，每个基由$V^{\lambda}$的极小偏序集$P_{\lambda}$上的高度为2的逆平面分拆集合$\operatorname{RPP}_2(P_\lambda)$中的元素索引。设$Q$为$\mathfrak g$的Dynkin图上的箭图，$c_Q$为对应的Coxeter元素。箭图基$\mathcal B^Q$由以下性质区分：在符号意义下，Tits代表元$\dot c_Q$在$\mathcal B^Q$上的作用，通过分段线性切换，提升了$c_Q$在$\operatorname{RPP}_2(P_\lambda)$上的作用。这统一证明了：对于任何极小偏序集$P$，$\operatorname{RPP}_2(P)$上的分段线性Coxeter运动和行运动表现出循环筛选现象。在$A$型中，标准方向的箭图基在重新标度下恢复了典范基，其与长循环的相容性由Rhoades建立。然而在其他类型中，我们证明典范基与任何Coxeter元素都不相容。
英文摘要
We consider the Cartan square $V^{2λ}$ of a minuscule representation $V^λ$ of a simply laced complex simple Lie algebra $\mathfrak g$. We construct for $V^{2λ}$ a family of bases, which we call quiver bases, each indexed by the set $\operatorname{RPP}_2(P_λ)$ 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 $\mathfrak g$, and let $c_Q$ be the corresponding Coxeter element. The quiver basis $\mathcal B^Q$ is distinguished by the following property: Up to sign, the action of the Tits representative $\dot c_Q$ on $\mathcal B^Q$ lifts the action of $c_Q$, via piecewise-linear toggles, on $\operatorname{RPP}_2(P_λ)$. This proves uniformly that, for any minuscule poset $P$, piecewise-linear Coxeter-motion and rowmotion on $\operatorname{RPP}_2(P)$ 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.
Comments38 pages
