76 research outputs found
Cyclic sieving, rotation, and geometric representation theory
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.Comment: 16 page
The octahedron recurrence and gl(n) crystals
We study the hive model of gl(n) tensor products, following Knutson, Tao, and
Woodward. We define a coboundary category where the tensor product is given by
hives and where the associator and commutor are defined using a modified
octahedron recurrence. We then prove that this category is equivalent to the
category of crystals for the Lie algebra gl(n). The proof of this equivalence
uses a new connection between the octahedron recurrence and the Jeu de Taquin
and Schutzenberger involution procedures on Young tableaux.Comment: 25 pages, 19 figures, counterexample to Yang-Baxter equation adde
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