Zeta functions of integral nilpotent quiver representations

TL;DR

This paper introduces multivariate zeta functions for integral nilpotent quiver representations, establishing conditions for their local functional equations and unifying previous work on submodule zeta functions.

Contribution

It generalizes and unifies the theory of submodule zeta functions for nilpotent structures, providing new conditions for functional equations in this context.

Findings

Established a homogeneity condition for local functional equations.

Unified previous results on ideal zeta functions and graded analogues.

Extended the theory to multivariate zeta functions of quiver representations.

Abstract

We introduce and study multivariate zeta functions enumerating subrepresentations of integral quiver representations. For nilpotent such representations defined over number fields, we exhibit a homogeneity condition that we prove to be sufficient for local functional equations of the generic Euler factors of these zeta functions. This generalizes and unifies previous work on submodule zeta functions including, specifically, ideal zeta functions of nilpotent (Lie) rings and their graded analogues.