Cyclic Sieving for Staircase Plane Partitions via Crystals and Electrical Networks
arXiv:2607.14028
The paper proves a cyclic sieving phenomenon for the promotion action on height‑two staircase plane partitions, using spin crystal representations and the bush basis of the electrical network coordinate ring, and discusses how a full electrical canonical basis could extend the result to all heights.
Abstract
We prove a cyclic sieving result for the action of promotion on the staircase plane partitions of height two. Our proof has two major algebraic inputs: an interpretation of this promotion action in terms of tensor powers of the spin crystal that was recently studied by Pappe--Pfannerer--Schilling--Simone, and the bush basis of the degree two part of the coordinate ring of the space of electrical networks that was recently introduced by Gao--Lam--Xu. Moreover, we explain how the existence of an electrical canonical basis in all degrees would yield cyclic sieving for promotion of staircase plane partitions of all heights.
23 pages, 8 figures