Strong Lefschetz property of the Kirchhoff algebra of the complete tripartite graph $K_{1,1,n}$

Abstract: By Whitney's 2-isomorphism theorem, the cycle matroids of two graphs are isomorphic if and only if one graph can be transformed into the other by finitely many applications of two operations: separation or identification at a cut-vertex, and twisting. The complete tripartite graph whose vertex set is the disjoint union of two singletons and an $n$-element set is invariant, up to isomorphism, under these operations. In this paper we prove that the Macaulay inverse system (the Kirchhoff algebra) determined by the basis generating polynomial of the cycle matroid of such a graph has the strong Lefschetz property over a field of characteristic $0$. The proof reduces the non-degeneracy of the higher Hessians to the ranks of disjointness matrices of families of subsets (the Gottlieb--Wilson theorem) and to the representation theory of the symmetric group in relation to the Johnson scheme.