Moment curves and cyclic symmetry for positive Grassmannians

TL;DR

This paper explores fixed points of cyclic shift maps on Grassmannians, revealing unique nonnegative fixed points related to moment curves and connecting these to quantum cohomology via a q-deformation.

Contribution

It introduces a q-deformation of the cyclic shift map on Grassmannians and links its fixed points to critical points of a superpotential, extending the understanding of moment curves and symmetry.

Findings

Exactly $inom{n}{k}$ fixed points for each cyclic shift map

Unique totally nonnegative fixed point on moment curves

Fixed points of q-deformed map correspond to superpotential critical points

Abstract

We show that for each k and n, the cyclic shift map on the complex Grassmannian Gr(k,n) has exactly $\binom{n}{k}$ fixed points. There is a unique totally nonnegative fixed point, given by taking n equally spaced points on the trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is even). We introduce a parameter q, and show that the fixed points of a q-deformation of the cyclic shift map are precisely the critical points of the mirror-symmetric superpotential $\mathcal{F}_q$ on Gr(k,n). This follows from results of Rietsch about the quantum cohomology ring of Gr(k,n). We survey many other diverse contexts which feature moment curves and the cyclic shift map.