Stretched Root Systems and the Geometry of Shard Modules

Abstract

A classic theorem of Gabriel states that, for a finite type Dynkin diagram G, the indecomposable representations of any quiver orienting G are in bijection with reflecting hyperplanes of the associated Coxeter group W. This is the starting point for a rich web of connections between the representation theory of algebras and the combinatorics and geometry of Coxeter groups.

Recent work of Iyama, Reading, Reiten, and Thomas constructs a similar correspondence between brick modules of the preprojective algebra Pi_G for a finite type Dynkin diagram G and a combinatorially useful partition of the hyperplanes into cones called shards. A paper of the author, Speyer, and Thomas generalizes this beyond finite type Dynkin diagrams by defining a class of bricks of Pi_G called shard modules which correspond to shards for arbitrary diagrams G. Although harder to understand than in the finite type case, shard modules provide a potential categorical tool for studying infinite Coxeter groups and cluster algebras.

In this thesis, we study how the relative position of shards affects the properties of their associated shard modules. We generalize beyond finite type a result of Iyama, Reading, Reiten, and Thomas showing that, when three shards meet in a certain configuration, their shard modules fit into a short exact sequence. We pay specific attention to "stretched" families of graphs obtained by inserting a path into a fixed diagram, describing recurring structure in the shards as the path grows. We use this structure to generalize patterns appearing in the shard modules for the A_n and D_n families of diagrams to any family of diagrams with tails.