Cyclic Sieving for Staircase Plane Partitions via Crystals and Electrical Networks

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.