Bizonotopal Graphical Algebras

Zonotopal algebras (external, central, and internal) of an undirected graph G introduced by Postnikov-Shapiro and Holtz-Ron, are finite-dimensional commutative graded algebras whose Hilbert series contain a wealth of combinatorial information about G. In this paper, we associate to G a new family of algebras, which we call bizonotopal, because their definition involves doubling the set of edges of G. These algebras are monomial and have intricate properties related, among other things, to the combinatorics of graphical parking functions and their polytopes. Unlike the case of usual zonotopal algebras, the Hilbert series of bizonotopal algebras are not specializations of the Tutte polynomial of G. Still, we show that in the external and central cases these Hilbert series satisfy a modified deletion-contraction relation. In addition, we prove that the external bizonotopal algebra is a complete graph invariant.