Cyclic Sieving for Plane Partitions and Symmetry

TL;DR

This paper extends the cyclic sieving phenomenon to symmetric plane partitions under various symmetries, connecting these results to rowmotion and promotion operators, and providing new polynomial formulas for counting fixed points.

Contribution

It introduces new cyclic sieving results for symmetric plane partitions considering complementation and transposition symmetries, expanding Rhoades's work.

Findings

Cyclic sieving applies to symmetric plane partitions with new symmetry considerations.

Product formulas for symmetric plane partitions are confirmed and extended.

Connections between symmetry operations and rowmotion are established.

Abstract

The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.