Moriah Elkin

Cluster algebras are a class of commutative algebras defined in 2001 by Fomin and Zelevinsky, reliant on “mutating” an initial set of variables in order to create rational functions called cluster variables. A wide variety of mathematical objects can be understood as cluster algebras. One such structure is the Grassmannian Gr(k,n), the space of all k-dimensional subspaces of an n-dimensional space. The cluster algebra structure of the Grassmannian has important applications to physics, especially string theory and scattering amplitudes, as well as statistical mechanics. Gr(3,7) has 42 cluster variables, 28 of which are relatively-well-understood Plücker coordinates, and 14 of which are enigmatic quadratic differences; the fundamental-yet-cryptic twist map permutes these variables. This project defines a beautiful way to interpret the twist map for both Plücker coordinates and quadratic differences via dimers (almost-perfect matchings) on a plabic graph associated to Gr(3,7). Specifically, we give a method to interpret any cluster variable using boundary conditions that define a certain set of dimers; we then provide a method to read off a Laurent polynomial expression for the twist of that variable from the set of dimers. Conjecturally, these methods may be extended to study Gr(k,n) for larger values of k and n, where significantly less is known about the cluster algebra structure.