Subalgebras of the Fomin-Kirillov algebra

Abstract

The Fomin-Kirillov algebra ${\mathcal{E}}_n$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on $n$ vertices. For any graph $G$ on $n$ vertices, we define ${\mathcal{E}}_G$ to be the subalgebra of ${\mathcal{E}}_n$ generated by the edges of $G$. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, ${\mathcal{E}}_G$ is a free ${\mathcal{E}}_H$-module for any $H\subseteq G$, and if ${\mathcal{E}}_G$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for ${\mathcal{E}}_G$ when $G$ is a simply-laced finite Dynkin diagram or a cycle, in particular showing that ${\mathcal{E}}_G$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of ${\mathcal{E}}_{{\tilde{D}}_n}$, ${\mathcal{E}}_{{\tilde{E}}_6}$, and ${\mathcal{E}}_{{\tilde{E}}_7}$, as well as for which graphs $G$ on six vertices ${\mathcal{E}}_G$ is finite-dimensional.

arXiv comment38 pages, 10 figures + appendix

References in corpus (2)

• A factorization theorem for affine Kazhdan-Lusztig basis elements
• Schur times Schubert via the Fomin-Kirillov algebra

Cited by in corpus (2)

• On the commutative quotient of Fomin-Kirillov algebras
• Positive expressions for skew divided difference operators