$P$-strict promotion and $B$-bounded rowmotion, with applications to tableaux of many flavors

TL;DR

This paper introduces P-strict labelings as a generalization of semistandard Young tableaux, linking promotion to toggle actions and rowmotion, and applies these concepts to various tableaux types to derive new combinatorial conjectures.

Contribution

It defines P-strict labelings, establishes their connection with toggle actions and rowmotion, and applies these to various tableaux to produce new cyclic sieving and homomesy conjectures.

Findings

Promotion is in equivariant bijection with toggle actions on B-bounded Q-partitions.

Toggle actions are conjugate to rowmotion in many cases.

New cyclic sieving and homomesy conjectures are proposed for flagged, Gelfand-Tsetlin, and symplectic tableaux.

Abstract

We define P-strict labelings for a finite poset P as a generalization of semistandard Young tableaux and show that promotion on these objects is in equivariant bijection with a toggle action on B-bounded Q-partitions of an associated poset Q. In many nice cases, this toggle action is conjugate to rowmotion. We apply this result to flagged tableaux, Gelfand-Tsetlin patterns, and symplectic tableaux, obtaining new cyclic sieving and homomesy conjectures. We also show P-strict promotion can be equivalently defined using Bender-Knuth and jeu de taquin perspectives.