On cyclic descents for tableaux

TL;DR

This paper generalizes the concept of cyclic descent sets for standard Young tableaux beyond rectangles, establishing their existence and uniqueness for most shapes using advanced algebraic tools.

Contribution

It introduces a general framework for cyclic descent sets for tableaux and proves their existence and uniqueness for nearly all shapes, extending prior work limited to rectangles.

Findings

Established existence and uniqueness of cyclic descent sets for most shapes of tableaux.

Connected cyclic descent sets to nonnegativity of Postnikov's toric Schur polynomials.

Provided a new interpretation of certain Gromov-Witten invariants.

Abstract

The notion of descent set, for permutations as well as for standard Young tableaux (SYT), is classical. Cellini introduced a natural notion of {\em cyclic descent set} for permutations, and Rhoades introduced such a notion for SYT --- but only for rectangular shapes. In this work we define {\em 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.