Cyclic descents of a permutation were defined by Cellini, by allowing the permutation to wrap around as if the last entry was followed by the first. A natural question is whether a similar, well-behaved notion of cyclic descents exists for standard Young tableaux (SYT). We conjectured that such a notion exists for SYT of any straight shape other than a hook. Recently, this conjecture has been proved by Adin, Reiner and Roichman using nonnegativity properties of Postnikov’s toric Schur polynomials. Unfortunately, the proof does not provide an explicit definition of the cyclic descent set for a specific tableau.

In this talk, I will present explicit descriptions of cyclic descent sets of SYT of rectangular shape (given by Rhoades), two-row SYT (both straight and skew), SYT consisting of a hook plus an additional cell, and certain skew shapes. In some cases, we also describe an action on SYT that shifts the cyclic descent set.

This is joint work with Ron Adin and Yuval Roichman.