Most of the finite and affine Coxeter groups fall into well-behaved infinite families. On the level of Coxeter-Dynkin diagrams, each family is constructed by taking a small graph and inserting a path of variable length into it. If we take an arbitrary Coxeter diagram and stretch it out by inserting a path, does the resulting family of Coxeter groups and related objects behave analogously to the finite and affine families? Do attributes of this family admit a uniform description once we stretch far enough?

In this talk, we'll look at two constructions attached to root systems for a Coxeter group -- the root poset, and Reading's theory of shards -- and see how they grow and stabilize when we stretch. As time permits, we'll talk about connections to the study of preprojective algebras. Based on arXiv:2010.10582. Speaker(s): Will Dana (University of Michigan)