Promotion and Rowmotion

TL;DR

This paper establishes a unified equivariant bijection linking promotion and rowmotion actions on posets, extending known results and applying to various combinatorial objects like alternating sign matrices and plane partitions.

Contribution

It introduces a novel equivariant bijection connecting promotion and rowmotion, generalizing previous results and applying to multiple combinatorial structures.

Findings

Bijection between promotion and rowmotion on certain posets

Extension of Stanley's promotion results to new posets

Equivariant bijections for alternating sign matrices and plane partitions

Abstract

We present an equivariant bijection between two actions--promotion and rowmotion--on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and recent work of D. Armstrong, C. Stump, and H. Thomas on root posets and noncrossing partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation. We extend the same idea to give an equivariant bijection between alternating sign matrices under rowmotion and under B. Wieland's gyration. Finally, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions.