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arXiv · 1805.06004

Moment curves and cyclic symmetry for positive Grassmannians

Steven N. Karp
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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.

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Steven N. Karp. 2018-05-15. Moment curves and cyclic symmetry for positive Grassmannians. https://doi.org/10.1112/blms.12280

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