Vandermondes in superspace

Abstract:

Superspace of rank $n$ is a $\mathbb {Q}$-algebra with $n$ commuting generators $x_1, \dots , x_n$ and $n$ anticommuting generators $\theta _1, \dots , \theta _n$. We present an extension of the Vandermonde determinant to superspace which depends on a sequence $\mathbf {a} = (a_1, \dots , a_r)$ of nonnegative integers of length $r \leq n$. 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 $\Delta ’_{e_{k-1}} e_n$ appearing in the Delta Conjecture of Haglund, Remmel, and Wilson.