Poincaré series of some hypergraph algebras

Abstract

A hypergraph H = (V, E), where V = {x1,⋯,xn} and E ⊆ 2V defines a hypergraph algebra RH = k[x 1,⋯, xn]/(xi1 ··· xik; {i1,⋯, ik} ∈ E). All our hypergraphs are d-uniform, i.e., |ei| = d for all ei ∈ E. We determine the Poincaré series PRH (t) = Σi=1∞ dimk ToriRH (k, k)ti for some hypergraphs generalizing lines, cycles, and stars. We finish by calculating the graded Betti numbers and the Poincaré series of the graph algebra of the wheel graph.