Abstract:

The preprojective algebra is an algebra associated to a graph G. When G is a finite type Dynkin diagram, this algebra is closely connected to the geometry of the associated finite Coxeter group. In particular, its brick modules correspond to shards, a combinatorially useful partition of the group's hyperplanes of symmetry into cones. Recent work with David Speyer and Hugh Thomas extends this correspondence to arbitrary G by constructing "shard modules" which correspond to shards of an infinite Coxeter group.

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 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 consistent structure in the shards as the path grows. We use this structure to generalize patterns appearing in the shard modules for the type A and D families of diagrams to any family of graphs with tails.

HYBRID: https://umich.zoom.us/j/91433812942 (passcode: shards)

The preprojective algebra is an algebra associated to a graph G. When G is a finite type Dynkin diagram, this algebra is closely connected to the geometry of the associated finite Coxeter group. In particular, its brick modules correspond to shards, a combinatorially useful partition of the group's hyperplanes of symmetry into cones. Recent work with David Speyer and Hugh Thomas extends this correspondence to arbitrary G by constructing "shard modules" which correspond to shards of an infinite Coxeter group.

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 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 consistent structure in the shards as the path grows. We use this structure to generalize patterns appearing in the shard modules for the type A and D families of diagrams to any family of graphs with tails.

HYBRID: https://umich.zoom.us/j/91433812942 (passcode: shards)

Building: | East Hall |
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Event Type: | Presentation |

Tags: | Dissertation, Graduate Students, Mathematics |

Source: | Happening @ Michigan from Dissertation Defense - Department of Mathematics, Department of Mathematics |