Mutation Invariant Functions On Cluster Algebras

Abstract: Examples of functions of cluster variables which remain unchanged after mutation arise naturally when studying cluster algebras. They appear as nontrivial elements of upper cluster algebras, elements of a theta basis, trace functions, cluster characters, and Diophantine equations whose solutions are parameterized by a cluster algebra. Interestingly, one often finds that the same mutation invariant function can be interpreted in several distinct ways, but it is not immediately clear why this would be. I will give a concise definition of a mutation invariant function in terms of an action of the cluster modular group, and give many more interesting examples. I will also discuss a classification of invariants for Dehn twists on surface cluster algebras, and more generally for "cluster Dehn twists" on mutation finite cluster algebras. This is the primary result of my recent PhD thesis. It is my hope that this classification allows us to begin to see why the same types functions appear in many distinct guises; each of these constructions (theta basis, trace functions, cluster characters, etc.) produce functions which are manifestly mutation invariant.

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)