Relationship of Mutation Dimer Graphs at 4-Faces and Quiver Mutations at 4-Valent Vertices

Abstract (written for general audience)

We introduce quivers, a mathematical abstraction of a network with a sense of direction. These arose naturally from triangulations of polygons of \(n\)-sides with special interest in mathematical problems involved in counting (combinatorics). Today, they are popular as areas of research in combinatorics, representation theory, and theoretical physics.

Part of our main focus is on the concept of restricted mutations of a quiver. In the case of triangulated surfaces this means we are interested in how many distinct triangulations we can find by modifying only one diagonal of a triangulation at a time. In the more general case, this corresponds to applying the mutation transformation on a vertex with 2 arrows incoming and 2 arrows outgoing.

The conjecture this project focused on was to show that infinitely many restricted mutation transformations lead to only finitely many unique quivers. While unable to give a proof/disproof of this conjecture, we did compute many examples which lead finitely many unique quivers. To achieve this, we modified a pre-existing program by Dr. Bernhard Keller and ran it on many different example quivers.

The other portion this project focused on was to relate quiver mutations to another construction known as dimer graphs. Dimer graphs can be thought of as undirected networks where all of the vertices are either black or white with the additional restriction that one can only connect black vertices to white vertices and vice versa. To each dimer graph, there is also an associated dimer quiver. Provided certain conditions met, we were able to define and prove a new transformation on the dimer graph that corresponds exactly to a modified version of quiver mutation at vertices with 2 arrows incoming and 2 arrows outgoing. We refer to this transformation as a 4-face mutation on the dimer graph.