On the commutative quotient of Fomin-Kirillov algebras

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.

arXiv comment11 pages, 3 figures

References in corpus (1)

• Subalgebras of the Fomin-Kirillov algebra

Cited by in corpus (1)

• The Orlik-Terao algebra and the cohomology of configuration space