Reflection Groups and Quiver Mutation: Diagrammatics

We extend Carter's notion of admissible diagrams and attach a Dynkin-like diagram to each reduced reflection factorization of an element in a finite Weyl group. We give a complete classification for the diagrams attached to reduced reflection factorizations. We show that such a diagram is cyclically orientable if and only if it is isomorphic to the underlying graph of a quiver which is mutation-equivalent to a Dynkin quiver. Furthermore, these diagrams encode a natural presentation of the Weyl group as reflection group, as shown by Cameron-Seidel-Tsaranov (1994) as well as Barot-Marsh (2015).