Minuscule reverse plane partitions via quiver representations

TL;DR

This paper introduces a new invariant for quiver representations using Jordan forms of nilpotent endomorphisms, establishing a bijection with reverse plane partitions and connecting promotion actions to Coxeter theory, with applications in type A.

Contribution

It defines a novel invariant for certain quiver representations and links it to reverse plane partitions, extending classical combinatorial correspondences and Coxeter group actions.

Findings

Invariant classifies representations up to isomorphism

Bijection with reverse plane partitions of minuscule posets

Promotion order equals Coxeter number, with classical cases included

Abstract

A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector space at each vertex. Generically among all nilpotent endomorphisms, there is a well-defined Jordan form for these linear transformations, which is an interesting new invariant of a quiver representation. If $Q$ is a Dynkin quiver and $m$ is a minuscule vertex, we show that representations consisting of direct sums of indecomposable representations all including $m$ in their support, the category of which we denote by $\mathcal{C}_{Q,m}$, are determined up to isomorphism by this invariant. We use this invariant to define a bijection from isomorphism classes of representations in $\mathcal{C}_{Q,m}$ to reverse plane partitions whose shape is the minuscule poset corresponding to $Q$ and $m$. By relating the piecewise-linear promotion action on reverse plane partitions to Auslander-Reiten translation in the derived category, we give a uniform proof that the order of promotion equals the Coxeter number. In type $A_n$, we show that special cases of our bijection include the Robinson-Schensted-Knuth and Hillman-Grassl correspondences.

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