# Mathematics Colloquia and Seminars

### 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.