Return to Colloquia & Seminar listing

### 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.