2013/10/15 by Jonah Blasiak, Blasiak, Jonah, Ricky Ini Liu +3 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #math.CO #math.QA #msc:05E15 #msc:16T30 #msc:20F55
paper · pdf · doi:10.48550/arxiv.1310.4112
38 pages, 10 figures + appendix
arxiv created 2014/03/12 · arxiv updated 2014/03/13
The Fomin-Kirillov algebra \mathcal En 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 EG to be the subalgebra of \mathcal En 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 EG is a free \mathcal EH-module for any H⊆ G, and if \mathcal EG is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for \mathcal EG when G is a simply-laced finite Dynkin diagram or a cycle, in particular showing that \mathcal EG is finite-dimensional in these cases. We also present conjectures for the Hilbert series of \mathcal E_Dn, \mathcal E_E6, and \mathcal E_E7, as well as for which graphs G on six vertices \mathcal EG is finite-dimensional.