Graph products of spheres, associative graded algebras and Hilbert series

TL;DR

This paper constructs a graded associative algebra from a graph, linking its Hilbert series to the graph's clique polynomial, and explores applications in toric topology and loop-space homology.

Contribution

It introduces a new algebraic construction based on graphs and relates its Hilbert series to the clique polynomial, with implications for toric topology.

Findings

Hilbert series equals the inverse of the clique polynomial

Recognition of inert ideals is simplified using the main result

Application to loop-space homology of generalized moment-angle complexes

Abstract

Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Non-commutative Grobner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment-angle complex. We apply our result to the loop-space homology of this space.