Koszul Graded M\"obius Algebras and Strongly Chordal Graphs

TL;DR

This paper investigates the conditions under which graded M"obius algebras of matroids are Koszul, linking algebraic properties to graph theory, and characterizes strongly chordal graphs through these algebraic insights.

Contribution

It provides a characterization of Koszul graded M"obius algebras for cycle matroids and introduces a new graph-theoretic characterization of strongly chordal graphs.

Findings

Koszul property of graded M"obius algebras depends on graph properties.

Characterization of strongly chordal graphs via algebraic conditions.

Connection between algebraic structures and graph orderings.

Abstract

The graded M\"{o}bius algebra of a matroid is a commutative graded algebra which encodes the combinatorics of the lattice of flats of the matroid. As a special subalgebra of the augmented Chow ring of the matroid, it plays an important role in the recent proof of the Dowling-Wilson Top Heavy Conjecture. Recently, Mastroeni and McCullough proved that the Chow ring and the augmented Chow ring of a matroid are Koszul. We study when graded M\"obius algebras are Koszul. We characterize the Koszul graded M\"obius algebras of cycle matroids of graphs in terms of properties of the graphs. Our results yield a new characterization of strongly chordal graphs via edge orderings.