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Cyclic sieving, rotation, and geometric representation theory

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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 fiber. 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.

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Acknowledgments

We would like to thank Alexander Braverman, Dennis Gaitsgory, Greg Kuperberg, Anthony Licata, Hiraku Nakajima, and Oded Yacobi for helpful conversations.

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Authors and Affiliations

  1. Cornell University, Ithaca, NY, USA

    Bruce Fontaine

  2. University of Toronto, Toronto, Canada

    Joel Kamnitzer

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  1. Bruce Fontaine
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  2. Joel Kamnitzer
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Correspondence to Joel Kamnitzer.

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Fontaine, B., Kamnitzer, J. Cyclic sieving, rotation, and geometric representation theory. Sel. Math. New Ser. 20, 609–625 (2014). https://doi.org/10.1007/s00029-013-0144-4

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