Hilbert series of algebras associated to direct graphs and order homology

TL;DR

This paper provides a homological perspective on the Hilbert series of algebras linked to directed graphs, offering conditions for Koszulity and methods to construct algebras with specific Hilbert series.

Contribution

It introduces a homological interpretation of Hilbert series coefficients and derives necessary conditions for Koszulity based on graph properties.

Findings

Homological interpretation of Hilbert series coefficients

Necessary conditions for Koszulity in graph-associated algebras

Construction of algebras with prescribed Hilbert series

Abstract

We give a homological interpretation of the coefficients of the Hilbert series for an algebra associated with a directed graph and its dual algebra. This allows us to obtain necessary conditions for Koszulity of such algebras in terms of homological properties of the graphs. We use our results to construct algebras with a prescribed Hilbert series.