The Cone of Cyclic Sieving Phenomena

TL;DR

This paper introduces a geometric framework for understanding cyclic sieving phenomena (CSP) using a rational polyhedral cone, providing criteria to identify CSPs and applying this to conjecture and prove new instances involving Schur polynomials.

Contribution

It develops a polyhedral cone model for CSPs, offers a criterion to recognize CSPs without explicit combinatorial realizations, and applies this to conjecture and verify new CSP cases.

Findings

Characterized the CSP cone and its geometric properties.

Provided a criterion to determine if a statistic and cyclic action form a CSP.

Conjectured and proved a new CSP involving stretched Schur polynomials.

Abstract

We study cyclic sieving phenomena (CSP) on combinatorial objects from an abstract point of view by considering a rational polyhedral cone determined by the linear equations that define such phenomena. Each lattice point in the cone corresponds to a non-negative integer matrix which jointly records the statistic and cyclic order distribution associated with the set of objects realizing the CSP. In particular we consider a universal subcone onto which every CSP matrix linearly projects such that the projection realizes a CSP with the same cyclic orbit structure, but via a universal statistic that has even distribution on the orbits. Reiner et.al. showed that every cyclic action give rise to a unique polynomial (mod $q^n-1$) complementing the action to a CSP. We give a necessary and sufficient criterion for the converse to hold. This characterization allows one to determine if a combinatorial set with a statistic give rise (in principle) to a CSP without having a combinatorial realization of the cyclic action. We apply the criterion to conjecture a new CSP involving stretched Schur polynomials and prove our conjecture for certain rectangular tableaux. Finally we study some geometric properties of the CSP cone. We explicitly determine its half-space description and in the prime order case we determine its extreme rays.