A quantum analogue of the dihedral action on Grassmannians

In recent work, Launois and Lenagan have shown how to construct a cocycle twisting of the quantum Grassmannian and an isomorphism of the twisted and untwisted algebras that sends a given quantum minor to the minor whose index set is permuted according to the $n$-cycle $c=(1, 2, ..., n)$, up to a power of $q$. This twisting is needed because $c$ does not naturally induce an automorphism of the quantum Grassmannian, as it does classically and semi-classically. We extend this construction to give a quantum analogue of the action on the Grassmannian of the dihedral subgroup of $S_{n}$ generated by $c$ and $w_{0}$, the longest element, and this analogue takes the form of a groupoid. We show that there is an induced action of this subgroup on the torus-invariant prime ideals of the quantum Grassmannian and also show that this subgroup acts on the totally nonnegative and totally positive Grassmannians. Then we see that this dihedral subgroup action exists classically, semi-classically (by Poisson automorphisms and anti-automorphisms, a result of Yakimov) and in the quantum and nonnegative settings.