Positivity for toric Kac polynomials in higher depth

TL;DR

This paper proves non-negativity of certain polynomials counting indecomposable quiver representations over truncated power series rings, extending positivity results for toric Kac polynomials to higher depth and exploring their cohomological and asymptotic properties.

Contribution

It generalizes positivity of toric Kac polynomials to higher depth and establishes new plethystic identities relating counts of quiver representations and jet spaces.

Findings

Proved non-negativity of polynomials counting indecomposable representations in higher depth.

Established plethystic identities linking counts of quiver representations and jet spaces.

Proved conjectures on asymptotic behavior of these counts as depth increases.

Abstract

We prove that the polynomials counting locally free, absolutely indecomposable, rank 1 representations of quivers over rings of truncated power series have non-negative coefficients. This is a generalisation to higher depth of positivity for toric Kac polynomials. The proof goes by inductively contracting/deleting arrows of the quiver and is inspired from a previous work of Abdelgadir, Mellit and Rodriguez-Villegas on toric Kac polynomials. We also relate counts of absolutely indecomposable quiver representations in higher depth and counts of jets over fibres of quiver moment maps. This is expressed in a plethystic identity involving generating series of these counts. In rank 1, we prove a cohomological upgrade of this identity, by computing the compactly supported cohomology of jet spaces over preprojective stacks. This is reminiscent of PBW isomorphisms for preprojective cohomological Hall algebras. Finally, our plethystic identity allows us to prove two conjectures by Wyss on the asymptotic behaviour of both counts, when depth goes to infinity.