Séminaire Lotharingien de Combinatoire, 80B.40 (2018), 12 pp.

Tair Akhmejanov

Growth Diagrams from Polygons in the Affine Grassmannian

Abstract. We introduce growth diagrams arising from the geometry of the affine Grassmannian for GLm. These affine growth diagrams are in bijection with the cλ many components of the polygon space Poly(λ) for λ a sequence of minuscule weights and cλ the Littlewood-Richardson coefficient. Unlike Fomin growth diagrams, they are infinite periodic on a staircase shape, and each vertex is labeled by a dominant weight of GLm. Letting m go to infinity, a dominant weight can be viewed as a pair of partitions, and we recover the RSK correspondence and Fomin growth diagrams within affine growth diagrams. The main combinatorial tool used in the proofs is the n-hive of Knutson-Tao-Woodward. The local growth rule satisfied by the diagrams previously appeared in van Leeuwen's work on Littelmann paths, so our results can be viewed as a geometric interpretation of this combinatorial rule.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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