On Cyclic Descents for Tableaux

Abstract

The notion of descent set, for permutations as well as for standard Young tableaux (SYT), is classical. Cellini introduced a natural notion of cyclic descent set for permutations, and Rhoades introduced such a notion for SYT - but only for rectangular shapes. In this work we define cyclic extensions of descent sets in a general context and prove existence and essential uniqueness for SYT of almost all shapes. The proof applies nonnegativity properties of Postnikov's toric Schur polynomials, providing a new interpretation of certain Gromov-Witten invariants.