Quiver Bases of Cartan Squares of Minuscule Representations

Source abstract

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.