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Grassmannian Cluster Algebras and Combinatorics

Kayla Wright (University of Minnesota)
Wednesday 18 October 2023 16:15–17:15 Koll. G4 (1532-222)
Mathematics Seminar

In this talk, we will discuss joint work with Moriah Elkin and Gregg Musiker about a combinatorial model for certain Grassmannian cluster algebras. The Grassmannian Gr(k,n) of k-planes in C^n, , has a cluster structure that is not well-understood for k>2. In these algebras, Plücker coordinates ∆I give us a subset of the cluster variables and have lovely combinatorial descriptions. However, most cluster variables are more complicated expressions in Plücker coordinates and lack such a combinatorial description. In our work, we give a graph theoretic interpretation for the Laurent expansion of cluster variables of low degree in terms of higher dimer models. This work employs SLk web combinatorics and we conjecture these webs are the key ingredient to understanding Grassmannian cluster algebras. If time permits, I would like to also pose an open problem I hope to work on (possibly with the algebraic power of Aarhus postdocs) relating our dimer combinatorics to the categorification of Grassmannian cluster algebras.

Organised by: AarHomAlg
Contact: Raphael, Amit Revised: 09.10.2023