Growth Diagrams from Polygons in the Affine Grassmannian

Tair Akhmejanov

Séminaire lotharingien de combinatoire, 80B (2018)

Résumé

We introduce growth diagrams arising from the geometry of the affine Grassmannian for GL_m. These affine growth diagrams are in bijection with the c_λ many components of the polygon space Poly(λ) for λ a sequence of minuscule weights and c_λ 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 GL_m. 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.