Poincaré Series of some Hypergraph Algebras

Abstract

A hypergraph $H=(V,E)$, where $V=\{x_1,\ldots,x_n\}$ and $E\subseteq 2^V$ defines a hypergraph algebra $R_H=k[x_1,\ldots, x_n]/(x_{i_1}\cdots x_{i_k}; \{i_1,\ldots,i_k\}\in E)$. All our hypergraphs are $d$-uniform, i.e., $|e_i|=d$ for all $e_i\in E$. We determine the Poincaré series $P_{R_H}(t)=\sum_{i=1}^\infty\dim_k\mathrm{Tor}_i^{R_H}(k,k)t^i$ 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.