Cyclic Sieving for Plane Partitions and Symmetry

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.

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

“…Remark 4.16. Corollary 4.15 has been noted in the literature, for example, by Hopkins [22] and Frieden [15]. Note the fact that the order of rowmotion on A ([n] × [q − n]) divides q (implicit in the statement of cyclic sieving) also follows from the order of birational rowmotion on the poset [n] × [q − n].…”

“…Our bijection is dual to theirs, but this is an artifact of our conventions, not a substantive difference. See also [22] (Appendix A, especially Proposition A.7) and [15].…”

Promotion and cyclic sieving on families of SSYT

“…We then use the fact that plane partitions together with such toggles exhibit cyclic sieving, see [SW20] (the case of order ideals of 2×[n] posets was considered earlier in [RS12]). We refer the reader to [AKLM05] and [Hop20a] for some related open problems on plane partitions and cyclic sieving.…”

“…Later in [SW20] it is proved that that plane par-titions in an a×b-box with max size n exhibit CSP under piecewise linear toggles, without mentioning the connection with promotion. The connection between promotion, toggles and cyclic sieving is made explicit in S. Hopkins [Hop20a], where he studies plane partitions with additional symmetry (e.g., symmetric under transposition). He also discusses the connection with promotion and rowmotion on posets considered in [SW12].…”

Symmetry of Narayana Numbers and Rowvacuation of Root Posets

“…for which: [5], they used Theorem 3.1 to show that Row acting on A(Φ + ) satisfies a cyclic sieving phenomenon [26], where the sieving polynomial is a natural q-analogue of Cat( ). Recently there has been interest in extending sieving phenomena to dihedral group actions as well [16,24,30]. Hence, it might be interesting to explore sieving phenomenona for Row, Rvac acting on A(Φ + ).…”