Vandermondes in superspace

Abstract

Superspace of rank n n is a Q \mathbb {Q} -algebra with n n commuting generators x 1 , … , x n x_1, \dots , x_n and n n anticommuting generators θ 1 , … , θ n \theta _1, \dots , \theta _n . We present an extension of the Vandermonde determinant to superspace which depends on a sequence a = ( a 1 , … , a r ) \mathbf {a} = (a_1, \dots , a_r) of nonnegative integers of length r ≤ n r \leq 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 Poincaré duality. The application of polarization operators gives rise to other bigraded modules which give a conjectural module for the symmetric function Δ e k − 1 ′ e n \Delta ’_{e_{k-1}} e_n appearing in the Delta Conjecture of Haglund, Remmel, and Wilson.