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Promotion and affine crystal combinatorics via the Grassmannian

Algebra & Discrete Mathematics

Speaker: Gabriel Frieden, University of Michigan
Location: 1147 MSB
Start time: Mon, Oct 23 2017, 4:15PM

Many combinatorial maps, such as the RSK bijection and the Schutzenberger involution, can be described by piecewise-linear formulas. These formulas can then be ``de-tropicalized,'' or ``lifted,'' to subtraction-free rational functions on an algebraic variety, and certain properties of the combinatorial maps become more transparent in the algebro-geometric setting.

Let Gr(k,n) be the Grassmannian of k-planes in an n-dimensional vector space. I will explain how the cyclic shift map on the Grassmannian that comes from rotating a basis of the n-dimensional vector space is (essentially) a geometric lift of promotion on semistandard tableaux of rectangular shape. This gives an algebro-geometric proof of the combinatorial result that promotion on rectangular tableaux has order n. Furthermore, this is the key to lifting the combinatorics of tensor products of affine type A Kirillov-Reshetikhin crystals.