Counting representations of quivers with multiplicities

TL;DR

This thesis develops a plethystic formula linking counts of quiver representations over finite rings with jet schemes and p-adic integrals, revealing new algebraic and geometric properties and conjecturing connections to BPS invariants.

Contribution

It introduces a novel plethystic formula for counting quiver representations over finite rings and explores their geometric and cohomological properties, including conjectures on BPS invariants.

Findings

Proved a plethystic formula relating counts over finite rings and jets.

Showed counts of jets converge to p-adic integrals for totally negative quivers.

Established non-negativity of polynomials counting indecomposable representations.

Abstract

In this thesis, we study counts of quiver representations over finite rings of truncated power series. We prove a plethystic formula relating counts of quiver representations over these rings and counts of jets on fibres of quiver moment maps. This solves a conjecture of Wyss and allows us to compute both counts on additional examples, using local zeta functions. The relation between counts of representations and counts of jets generalises the relation between Kac polynomials and counts of points on preprojective stacks. Pursuing this analogy, we establish further properties of our counts. We show that, for totally negative quivers, counts of jets converge to p-adic integrals on fibres of quiver moment maps. One expects a relation between these p-adic integrals and BPS invariants of preprojective algebras i.e. Kac polynomials. For small rank vectors, we also prove that the polynomials counting indecomposable quiver representations over finite rings have non-negative coefficients. Moreover, we show that jet schemes of fibres of quiver moment maps are cohomologically pure in that setting, so that their Poincar\'e polynomials are given by the former counts. This is reminiscent of the structure of cohomological Hall algebras, which are built from the cohomology of preprojective stacks. Finally, we compute the cohomology of jet spaces of preprojective stacks explicitly for the A2 quiver. Building on the structure of the preprojective cohomological Hall algebra of A2, we propose a candidate analogue of the BPS Lie algebra and conjecture the existence of a Hall product on the cohomology of these jet spaces.