A SPECIAL CASE OF POSTNIKOV-SHAPIRO CONJECTURE

Abstract. For a graph G, Postnikov-Shapiro [21] construct two ideals IG and JG. IG is a monomial ideal and JG is generated by powers of linear forms. They proved the equality of their Hilbert series and conjectured that the graded Betti numbers are equal. When G = Kl,kn+1 is the complete graph on the vertices {0, 1, · · · , n} with the edges ei,j, i, j 6 = 0, of multiplicity k and the edges e0,i of multiplicity l, for two non-negative integers k and l, they gave an explicit formula for the graded Betti numbers of IG, which are conjecturally the same for JG. We prove this conjecture in the case n = 3, which was also conjectured by Schenck [22]. 1