Positivity for toric Kac polynomials in higher depth

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.