# Vandermondes in superspace

## TL;DR

This paper extends Vandermonde determinants to superspace, constructs graded representations of symmetric groups, and connects these to important algebraic structures and conjectures in combinatorics.

## Contribution

It introduces a superspace Vandermonde, develops a framework for graded modules, and links these to key algebraic and combinatorial objects related to the Delta Conjecture.

## Key findings

- Constructed superspace Vandermondes depending on a sequence of integers.
- Revealed connections to hook-shaped Tanisaki quotients and coinvariant rings.
- Proposed a conjectural module related to the Delta Conjecture.

## 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 Poincar\'e 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.