Alladi Ramakrishnan Hall

Counting quiver representations over finite fields

Amritanshu Prasad

IMSc

The classification of representations of a quiver is a generalization of the similarity problem for matrices to families of matrices, possibly of different rectangular shapes.

How many representations (up to isomorphism) does a given quiver have over a field of order q for a fixed dimension vector? I will explain why the answer turns out to be a polynomial function of q. The proof is due to Victor Kac, and is based on Burnside's lemma and the notion of matrix type. I will also work out some simple but interesting examples.