17th February – Joe: Cluster’s first stand

Special types of cluster algebra arise from quiver mutation. In general these have infinitely many generators. Fomin and Zelevinsky provided a criterion for the finite case in terms of simply laced Dynkin diagrams, also constructing a bijection from the “almost positive” roots of the associated root system to the set of cluster variables.   If one looks at representations  of (some orientation of) the ADE diagrams, Gabriel’s theorem gives a bijection from the isomorphism classes of irreducible representations to the positive roots of the root system.  This can be combined with Fomin and Zelevinsky’s bijection. I’ll explain some of the basics of cluster algebras and demonstrate these results with a simple example.