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 $r\le n$. 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 ${\Delta }_{e_{k-1}}^{\prime }{e}_n$ appearing in the Delta Conjecture of Haglund, Remmel, and Wilson.

32 pages