Speaker: Christopher Voll (Bielefeld)
Title: Representation zeta functions of integral quiver representations
Abstract: An integral representation of a finite quiver (i.e. directed graph) is an assignment of a $\mathbb{Z}$-module (of finite rank, say) to each vertex of the quiver and an $\mathbb{Z}$-module homomorphism to each arrow. A subrepresentation is just an assignment of submodules of the modules for each vertex, which are compatible with the homomorphisms.
The representation zeta function associated with an integral representation is the Dirichlet generating series enumerating the representation's subrepresentations. In this generality, these functions arise naturally in various contexts calling for the enumeration of (tuples of) lattices invariant under linear operators. Examples and applications include Dedekind and other ideal zeta functions, Solomon's zeta functions and P-partition generating functions.
I will report on recent joint work with Seungjai Lee (Seoul National University) on representation zeta functions associated with integral nilpotent representations. Results I shall explain include a self-reciprocity result for representation zeta functions associated with "homogeneous" such representations, as well as connections with combinatorial algebra.
For more details on the BiPb-Seminar see the seminar home page.