Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 130, 40 pages      arXiv:1907.09337      https://doi.org/10.3842/SIGMA.2020.130

Cyclic Sieving for Plane Partitions and Symmetry

Sam Hopkins
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Received May 05, 2020, in final form December 06, 2020; Published online December 09, 2020

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.

Key words: plane partitions; cyclic sieving phenomena; promotion; rowmotion; canonical bases.

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