Cyclic sieving, rotation, and geometric representation theory

TL;DR

This paper explores the geometric and combinatorial aspects of rotation actions on invariant vectors in tensor products of minuscule representations, connecting them to the cyclic sieving phenomenon through advanced geometric representation theory.

Contribution

It introduces a combinatorial notion of rotation for minuscule Littelmann paths and links it to geometric rotation in affine Grassmannians, extending cyclic sieving results.

Findings

Rotation action permutes a basis of invariant spaces (up to sign)

Eigenspaces of rotation are described by intersection homology of quiver varieties

Generalizes Rhoades' cyclic sieving phenomenon to new settings

Abstract

We study rotation of invariant vectors in tensor products of minuscule representations. We define a combinatorial notion of rotation of minuscule Littelmann paths. Using affine Grassmannians, we show that this rotation action is realized geometrically as rotation of components of the Satake fibre. As a consequence, we have a basis for invariant spaces which is permuted by rotation (up to global sign). Finally, we diagonalize the rotation operator by showing that its eigenspaces are given by intersection homology of quiver varieties. As a consequence, we generalize Rhoades' work on the cyclic sieving phenomenon.