Growth Diagrams from Polygons in the Affine Grassmannian
| dc.contributor.author | Akhmejanov, Tair | |
| dc.contributor.chair | Knutson, Allen | |
| dc.contributor.committeeMember | Kozen, Dexter Campbell | |
| dc.contributor.committeeMember | Stillman, Michael Eugene | |
| dc.date.accessioned | 2018-10-23T13:33:38Z | |
| dc.date.available | 2018-10-23T13:33:38Z | |
| dc.date.issued | 2018-08-30 | |
| dc.description.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 $c_{\vec\lambda}$ 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, so our results can be viewed as a geometric interpretation of this combinatorial rule. Similar diagrams appeared in the work of Speyer on osculating flags. | |
| dc.identifier.doi | https://doi.org/10.7298/X4MK6B3G | |
| dc.identifier.other | Akhmejanov_cornellgrad_0058F_10894 | |
| dc.identifier.other | http://dissertations.umi.com/cornellgrad:10894 | |
| dc.identifier.other | bibid: 10489669 | |
| dc.identifier.uri | https://hdl.handle.net/1813/59573 | |
| dc.language.iso | en_US | |
| dc.subject | Mathematics | |
| dc.title | Growth Diagrams from Polygons in the Affine Grassmannian | |
| dc.type | dissertation or thesis | |
| dcterms.license | https://hdl.handle.net/1813/59810 | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Cornell University | |
| thesis.degree.level | Doctor of Philosophy | |
| thesis.degree.name | Ph. D., Mathematics |
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