Cyclic sieving for plane partitions and symmetry

Sam Hopkins

Published 2019-07-22Version 1

The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count fixed points for 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. Actually, Rhoades has already studied the way that promotion and complementation interact. Our contribution is to study how promotion and transposition, and promotion and transpose-complementation, interact, and to give cyclic sieving-like formulas in this context. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews. We then go on to consider the way these same symmetries interact with rowmotion, another operator acting on plane partitions closely related to promotion. Rowmotion, unlike promotion, is defined for any poset. Our original motivation for studying the way that rowmotion interacts with these symmetries was a series of cyclic sieving conjectures we made in a previous paper concerning rowmotion acting on $P$-partitions for certain triangular posets $P$. As we explain, these conjectures can be rephrased in terms of counting the number of plane partitions fixed by various subgroups of the group generated by rowmotion and transposition. Our results in this paper do not directly imply anything about these conjectures, but they are morally very similar.