Vandermondes in superspace

Superspace of rank n is a Q-algebra with n commuting generators x(1), ..., x(n) and n anticommuting generators theta(1), ..., theta(n). We present an extension of the Vandermonde determinant to superspace which depends on a sequence a = (a(1), ..., a(r)) of nonnegative integers of length r <= n. We use superspace Vandermondes to construct graded representations of the symmetric group. This construction recovers hook-shaped Tanisaki quotients, the coinvariant ring for the Delta Conjecture constructed by Haglund, Rhoades, and Shimozono, and a superspace quotient related to positroids and Chern plethysm constructed by Billey, Rhoades, and Tewari. We define a notion of partial differentiation with respect to anticommuting variables to construct doubly graded modules from superspace Vandermondes. These doubly graded modules carry a natural ring structure which satisfies a 2-dimensional version of Poincare duality. The application of polarization operators gives rise to other bigraded modules which give a conjectural module for the symmetric function Delta'(ek-1) e(n) appearing in the Delta Conjecture of Haglund, Remmel, and Wilson.