Gleb Nenashev: Commutative algebras counting spanning trees and forests of a graph.

Abstract

In 2004 A.Postnikov and B.Shapiro introduced a new class of commutative algebras generated by powers of linear forms coming from an arbitrary non-directed graph. They proved a number of interesting properties including that total dimension is number of spanning forests/trees and an explicit formula for their Hilbert series (they are specialisations of Tutte polynomial).

My main result is that Postnikov-Shapiro algebra counting forests depends only on  the graphical matroid of $G$. And conversely, we can reconstruct this matroid from the latter algebra. Furthermore, I present "$K$-theoretical"  analog of a such algebra. It has nice property:  two such filtered algebras are isomorphic if and only if their graphs are isomorphic.

In the end I present a generalization of the original algebra to hypergraphs and the definition of a hypergraphical matroid, whose Tutte polynomial allows us to calculate Hilbert series. And furthermore, I present combinatorial definitions of cycles and spanning forests/trees of a hypergraph.