Descents of permutations have been studied for more than a century. This concept was vastly generalized, in particular to standard Young tableaux (SYT). More recently, cyclic descents of permutations were introduced by Cellini and further studied by Dilks, Petersen and Stembridge. Looking for a corresponding concept for SYT, Rhoades found a very elegant solution for rectangular shapes.
In an attempt to extend the concept of cyclic descents, explicit combinatorial definitions for two-row and certain other shapes have been found. Consequently, the existence of cyclic descents was conjectured for all shapes.
This talk will report on the surprising resolution of this conjecture: Cyclic descent sets exist for nearly all skew shapes, with an interesting small set of exceptions. The proof applies nonnegativity properties of Postnikov's toric Schur polynomials and a new combinatorial interpretation of certain Gromov-Witten invariants.
If time allows we will discuss applications and implications to quasi-symmetric functions.
Based on joint works with Ron Adin, Sergi Elizalde and Vic Reiner.