Moment curves and cyclic symmetry for positive Grassmannians

Abstract

We show that for each k and n, the cyclic shift map on the complex Grassmannian Gr(k,n) has exactly $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ 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.

18 pages. v2: Minor changes

References in corpus (2)

• Total positivity, Grassmannians, and networks
• Regularity theorem for totally nonnegative flag varieties

Cited by in corpus (4)

• Ising model and the positive orthogonal Grassmannian
• Critical varieties in the Grassmannian
• Gradient flows, adjoint orbits, and the topology of totally nonnegative flag varieties
• Fukaya category of Grassmannians: rectangles