On the commutative quotient of Fomin–Kirillov algebras

Abstract

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