On the commutative quotient of Fomin–Kirillov algebras

The Fomin–Kirillov algebra EnEn is a noncommutative algebra with a generator for each edge of the complete graph on nn vertices. For any graph GG on nn vertices, let EGEG be the subalgebra of EnEn generated by the edges in GG. We show that the commutative quotient of EGEG is isomorphic to the Orlik–Terao algebra of GG. As a consequence, the Hilbert series of this quotient is given by (−t)nχG(−t−1)(−t)nχG(−t−1), where χGχG is the chromatic polynomial of GG. We also give a reduction algorithm for the graded components of EGEG that do not vanish in the commutative quotient and show that their structure is described by the combinatorics of noncrossing forests.