A dihedral group action on Grassmannians and its quantization

The Grassmannian Gr(k,n) is the space of all k-dimensional subspaces of an n-dimensional space. This very nice geometric object has an action of the symmetric group S_n on it but this action does not respect some of the additional structure we might like to put on the Grassmannian. However a certain dihedral subgroup does and in this talk we will explore which structures this dihedral group preserves and also how this action can even be passed to the quantum version of the Grassmannian, but in a somewhat unexpected way.