Tutte polynomials in superspace

Abstract

We associate a quotient of superspace to any hyperplane arrangement by considering the differential closure of an ideal generated by powers of certain homogeneous linear forms. This quotient is a superspace analogue of the external zonotopal algebra, and it further contains the central zonotopal algebra in the appropriate grading. We show that an evaluation of the bivariate Tutte polynomial is the bigraded Hilbert series of this quotient. We then use this fact to construct an explicit basis for the Macaulay inverse. These results generalize those of Ardila-Postnikov and Holtz-Ron. We also discuss enumerative consequences of our results in the setting of hyperplane arrangements.