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 then-cyclec=(1 2 ... n). This twisting is needed becausecdoes not 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 ofSgenerated by_{n}candw_{0}, the longest element, and show that this induces an action of this subgroup on the torus-invariant prime ideals of the quantum Grassmannian. We 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.

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