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The (twisted) cyclic shift $(v_1,v_2,\ldots,v_n) \mapsto (v_2,v_3,\ldots,v_n,(-1)^{k-1}v_1)$ acting on the Grassmannian $\mathrm{Gr}(\mathbb{C};k,n)$ of $k$-planes in $\mathbb{C}^n$ is an important symmetry with applications to combinatorics, geometry, representation theory, et cetera, especially in the context of total positivity. See for instance this nice paper of Karp which contains some survey of applications of the cyclic shift: https://arxiv.org/abs/1805.06004.

Question: is the orbit structure of the cyclic shift acting on the finite Grassmannian $\mathrm{Gr}(\mathbb{F}_q;k,n)$ understood? E.g., is there a cyclic sieving phenomenon here (see https://arxiv.org/abs/1008.0790 for a survey on the CSP)?

The prototypical cyclic sieving result about $k$-subsets of $\{1,2,\ldots,n\}$ under rotation would be a ``$\mathbb{F}_1$'' version of whatever this is.

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    $\begingroup$ I assume you know the Berget-Huang paper on CSPs for finite Grassmannians? It's not what you are looking for, but it is of a similar flavor. $\endgroup$
    – Oliver
    Commented Aug 8, 2019 at 1:59
  • $\begingroup$ @Oliver: yes, thanks for the reference; although if I remember correctly already in the original RSW paper they show some CSPs for the action of the multiplicative group of $\mathbb{F}_q$ on flag varieties. $\endgroup$
    – Sam Hopkins
    Commented Aug 8, 2019 at 2:28
  • $\begingroup$ This might be relevant: mi.uni-koeln.de/~lbossing/YTGRT/slides/Daping.pdf $\endgroup$
    – Per Alexandersson
    Commented Aug 8, 2019 at 8:23
  • $\begingroup$ Possibly relevant paper to this question: arxiv.org/abs/2410.06177 $\endgroup$
    – Sam Hopkins
    Commented Oct 10 at 13:11

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