On the commutative quotient of Fomin–Kirillov algebras

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 algebr...

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Bibliographic Details
Published in:European journal of combinatorics 2016-05, Vol.54, p.65-75
Main Author: Liu, Ricky Ini
Format: Article
Language:English
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Summary: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.
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2015.12.003