On the commutative quotient of Fomin-Kirillov algebras

TL;DR

This paper establishes a connection between the commutative quotients of Fomin-Kirillov algebras associated with graphs and Orlik-Terao algebras, providing explicit formulas for their Hilbert series and a combinatorial reduction algorithm.

Contribution

It proves the isomorphism between the commutative quotient of $ ext{FK}_G$ and the Orlik-Terao algebra, and introduces a reduction algorithm linked to noncrossing forests.

Findings

The commutative quotient of $ ext{FK}_G$ is isomorphic to the Orlik-Terao algebra of $G$.

The Hilbert series of the quotient is expressed via the chromatic polynomial of $G$.

A reduction algorithm describes the structure of non-vanishing graded components.

Abstract

The Fomin-Kirillov algebra $\mathcal E_n$ is a noncommutative algebra with a generator for each edge in the complete graph on $n$ vertices. For any graph $G$ on $n$ vertices, let $\mathcal E_G$ be the subalgebra of $\mathcal E_n$ generated by the edges in $G$. We show that the commutative quotient of $\mathcal E_G$ is isomorphic to the Orlik-Terao algebra of $G$. As a consequence, the Hilbert series of this quotient is given by $(-t)^n \chi_G(-t^{-1})$, where $\chi_G$ is the chromatic polynomial of $G$. We also give a reduction algorithm for the graded components of $\mathcal E_G$ that do not vanish in the commutative quotient and show that their structure is described by the combinatorics of noncrossing forests.