Moment curves and cyclic symmetry for positive Grassmannians

Abstract

We show that for each k and n, the cyclic shift map on the Grassmannian Gr k , n ( C ) has exactly 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 ∈ 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 F q on Gr k , n ( C ) . This follows from results of Rietsch about the quantum cohomology ring of Gr k , n ( C ) . We survey many other diverse contexts which feature moment curves and the cyclic shift map.

Locations

• Bulletin of the London Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• Deep Blue (University of Michigan) - View - PDF
• DataCite API - View