University of Connecticut

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Algebra Seminar
Combinatorics Of Cluster Algebras
Veronique Bazier-Matte

Wednesday, September 23, 2020
11:15am – 12:05pm

Storrs Campus

The first part of this talk will briefly introduce cluster algebras. Cluster algebras are Laurent polynomial algebras whose generators are obtained by a recursive process called mutation. We start with a seed (a pair formed with a set a n variables called a cluster) and with a quiver (a directed graph with n vertices). The mutation of a seed replaces one variable at a time and modifies the quiver, giving therefore a new seed. The cluster algebra is generated by all variables obtained by successive mutations, called cluster variables.

The exchange graph of a cluster algebra allows to visualize relations between its clusters. In this graph, vertices correspond to cluster of the cluster algebras and edges correspond to mutations: two vertices are joined by an edge if the associated clusters are obtained one from the other by a mutation. An intuitive way to construct the exchange graph of a cluster algebra is to compute one at a time every mutation. In the second part of this talk, we will show a way to realize the exchange graph of a cluster algebra with a finite number of cluster variables in $$\mathbb R^n$$ where n represents the number of vertices in the quiver. We compute this realization directly rather than recursively.

Finally, in the last part of this talk, we prove the unistructurality of certain type of algebras. A cluster algebra is unistructural if the set of its cluster variables determine uniquely its cluster. In other words, a cluster algebra is unistructural if another cluster algebra with exactly the same cluster variables must have also the same clusters.

Contact the organizer for the WebEx link.


Algebra Seminar (primary), College of Liberal Arts and Sciences, UConn Master Calendar

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