Inspired by a talk of S. Hopkins, we (Gaetz–Pechenik–Pfannerer–Striker–Swanson) introduced the notion of promotion permutations. Given a rectangular standard Young tableau T with r rows, this is a tuple of r−1 permutations that records its orbit structure of Schützenberger promotion. For r=4, these objects played a key role in our construction of the first rotation-invariant web bases for the SL4 invariant spaces.

In this talk, however, I will focus on different directions, highlighting some of the combinatorial properties of promotion permutations and applications of related constructions. For example, in the situation where T is obtained by stacking two rectangular tableaux P and Q, one recovers the Robinson–Schensted correspondence.

In Type C, a related construction yields an alternative description of Sundaram’s bijection between the set of r-symplectic oscillating tableaux of empty shape and the set of (r+1)-noncrossing perfect matchings.

In Type B, similar ideas allow us to prove a cyclic sieving phenomenon for the action of promotion on staircase plane partitions of height two, using the bush basis of the degree-two part of the coordinate ring of the space of electrical networks, due to Gao–Lam–Xu.

This talk is based on joint work with various subsets of: Gaetz, Hopkins, Kim, Pappe, Pechenik, Rubey, Schilling, Simone, Striker, Swanson, and Westbury.