Coxeter Groups, Scattering Diagrams, and Shards

North Carolina State University, Raleigh NC

Investigators

Abstract

The goal of this project is to better understand certain mathematical objects called Coxeter groups, and use Coxeter groups to make progress on a variety of other mathematical questions. Coxeter groups are collections (usually called "groups") of symmetries of highly symmetric objects, like for example the Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) and higher-dimensional analogues. An important theme of mathematics in the last hundred years or so is that Coxeter groups and related geometric objects provide a very accessible and useful key to unlock many other mathematical theories. This project explores the applications of Coxeter groups to Artin groups (including the "braid group", which describes the mathematics of braided strands), scattering diagrams (a key tool for "mirror symmetry" in string theory), and cluster algebras (a newer theory that also functions as a key to other mathematical theories). The award will also fund graduate students working on this project. One part of the project concerns the combinatorics of cluster scattering diagrams and their theta functions. The goal is to connect theta functions to the mutation fan (a fan that encodes the piecewise-linear geometry of matrix mutation) and also to construct the cluster scattering fan in the surfaces/orbifolds case and show that the theta functions are the bracelets basis in that case. A second part of the project is to extend (from finite type to affine type) the deep connections between the noncrossing partition lattices (which appear in the theory of Artin groups) and generalized associahedra/scattering fans. Closely related to this goal is the project of constructing planar diagrams that realize noncrossing partition lattices of classical affine types. The third part of the project is (in its most ambitious formulation) to extend the weak order on a finite Coxeter group to an infinite lattice (not semilattice) for completely general Coxeter groups and then carry out the Cambrian fan construction of cluster scattering diagrams in complete generality. In finite type, the weak order can be understood in terms of shards and the Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL). In the infinite case, the plan is to construct a lattice from shards and an infinite version of FTFSDL. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.