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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