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Growth Diagrams from the Affine Grassmannian

Algebra & Discrete Mathematics

Speaker: Tair Akhmejanov, UC Davis
Location: 1147 MSB
Start time: Mon, Oct 8 2018, 11:00AM

We introduce growth diagrams arising from the geometry of the affine 
Grassmannian for $GL_m$. These affine growth diagrams are in bijection 
with the $c_{\vec\lambda}$ many components of the polygon space 
Poly($\vec\lambda$) for $\vec\lambda$ a sequence of minuscule weights 
and $c_{\vec\lambda}$ 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. Similar diagrams appeared in the work of Speyer on 
osculating flags, and that of Westbury on coboundary categories.