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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 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.

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

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