Combinatorial Rules for Canonical Decomposition of Quiver Representations, Casey Appleton (UIUC ICLUE alum/UC Berkeley) -- Wednesday, June 4th, 2025

Quiver representation theory allows one to analyze situations arising in linear and homological algebra by treating (not necessarily commutative) diagrams of vector spaces as directed graphs whose vertices and edges are labeled by vector spaces and linear maps between them, respectively. The directed graphs are called quivers, and the diagrams are called quiver representations. One can fix the dimensions of the vector spaces labeling each vertex and ask what happens when the linear maps between them are chosen randomly. It turns out that for a special class of quivers called Dynkin quivers, with probability 1, one always gets the same quiver representation up to a change of bases of the vector spaces, and that its decomposition into “prime” representations has special properties. This decomposition is known as the canonical decomposition associated to the given quiver and sequence of vector space dimensions.

In this talk, we present simple combinatorial rules for computing the canonical decomposition for arbitrary type A_n and D_n quivers for arbitrary sequences of dimensions. In these cases, the rules admit descriptions in terms of the dissection of certain “Arts & Crafts projects” obtained by gluing together colored strips of paper.

If time permits, the combinatorial rule for canonical decompositions of representations of mutation type A_n quivers with potential may also be discussed.

Organizers: Ian Cavey, Andrew Hardt, Alexander Yong