Cyclic sieving, rotation, and geometric representation theory

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.

16 pages

References in corpus (2)

• Quiver varieties and Beilinson-Drinfeld Grassmannians of type A
• Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties