An extended generalization of RSK correspondence via $A$ type quiver representations

TL;DR

This paper introduces a broad generalization of the RSK correspondence using $A$ type quiver representations, unifying previous variants and expanding combinatorial understanding of fillings and plane partitions.

Contribution

It constructs a new bijection linking fillings of a partition to reverse plane partitions via Coxeter elements, generalizing earlier RSK variants based on quiver representations.

Findings

Unified previous RSK generalizations under a new framework

Established bijections for fillings and plane partitions using Coxeter elements

Extended combinatorial applications of type $A$ quiver theory

Abstract

Let $\lambda=(\lambda_1 \geqslant \ldots \geqslant \lambda_k > 0)$. For any $c$ Coxeter element of $\mathfrak{S}_{\lambda_1+k-1}$, we construct a bijection from fillings of $\lambda$ to reverse plane partitions. We recover two previous generalizations of the Robinson--Schensted--Knuth correspondence for particular choices of Coxeter element depending on $\lambda$: one based on the work of, among others, Burge, Hillman, Grassl, Knuth, and uniformly presented by Gansner; the other developed by Garver, Partrias, and Thomas, and independently by Dauvergne, called Scrambled RSK. Our results in this paper develop the combinatorial consequence of our previous work of type $A_{\lambda_1+k}$ quivers.