Tree modules and counting polynomials

TL;DR

This paper derives a formula for counting tree modules on a specific quiver and shows that their number is polynomial in the number of loops, aligning with known counting polynomials for indecomposables.

Contribution

It introduces a new formula linking tree modules on a quiver to those on its universal cover and connects these counts to existing polynomials for indecomposable modules.

Findings

Number of d-dimensional tree modules is polynomial in g.

The polynomial degree and leading coefficient match the counting polynomial for indecomposables.

Established a connection between tree modules and known counting polynomials.

Abstract

We give a formula for counting tree modules for the quiver S_g with g loops and one vertex in terms of tree modules on its universal cover. This formula, along with work of Helleloid and Rodriguez-Villegas, is used to show that the number of d-dimensional tree modules for S_g is polynomial in g with the same degree and leading coefficient as the counting polynomial A_{S_g}(d, q) for absolutely indecomposables over F_q, evaluated at q=1.