Hilbert series of algebras associated to directed graphs

Abstract

We compute the Hilbert series of some algebras associated to directed graphs and related to factorizations of noncommutative polynomials.

Representations of Aut(A(Γ)) acting on homogeneous components of A

“…We may replace σ by r i σ for some i and assume σ (1) = 1. Then σ (12) is either (12) or (1n), which implies either σ (2) = 2 (and thus σ = id) or σ (2) = n. In the latter case σ = (2n)(3n − 1)(4n − 2) · · · = s. Thus r and s generate the automorphism group of Γ D n ; this is the dihedral group on n elements, D n . Note that these automorphisms may be viewed as reflections and rotations of the n-gon.…”

“…This will give us a recursion that can then be written as a generating function. The second method of proof is much shorter and uses a theorem from [12] that gives a formula for the Hilbert series of an algebra associated to a directed, layered graph. Method 1.…”

“…The automorphism s acting on the algebra when n is even fixes the top vertex, the minimal vertex, and two vertices on level two (v 12 and v n/2+1n/2+2 ) (see Fig. 4).…”

Koszulity of splitting algebras associated with cell complexes☆☆Some of our results were later generalized by T. Cassidy, C. Phan, and B. Shelton in their paper “Noncommutative Koszul algebras from combinatorial topology” in arXiv:0811.3450.

“…We begin by recalling (from [9]) that a complete layered graph is a layered graph Γ = (V , E) with Proof. It is sufficient to prove the result for gr A(Γ ).…”

Möbius Polynomials of Face Posets of Convex Polytopes

“…Our interest in the Möbius polynomial stems from previous work on the splitting algebra or universal labeling algebra of a poset, introduced by Gelfand, Retakh, Serconek, and Wilson in their 2005 paper [2]. In their 2007 paper [3], they showed that the Hilbert series of the algebra A(P ) associated to a finite ranked poset P could be calculated using the formula:…”