Moment curves and cyclic symmetry for positive Grassmannians.

We show that for each k and n, the cyclic shift map on the Grassmannian Grk,n(C) has exactly nk 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∈C×, 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 Fq on Grk,n(C). This follows from results of Rietsch about the quantum cohomology ring of Grk,n(C). We survey many other diverse contexts which feature moment curves and the cyclic shift map. [ABSTRACT FROM AUTHOR]