# Boolean product polynomials, Schur positivity, and Chern plethysm

## TL;DR

This paper introduces Boolean product polynomials, proves their Schur positivity using geometric methods involving vector bundles and Chern plethysm, and explores their connections to various combinatorial objects and symmetric group actions.

## Contribution

It presents a novel geometric approach to establish Schur positivity of Boolean product polynomials through Chern plethysm, expanding the toolkit beyond combinatorial proofs.

## Key findings

- Boolean product polynomials are Schur positive.
- Established connections to derangements, positroids, and other combinatorial objects.
- Linked certain polynomials to symmetric group actions on superspace.

## Abstract

Let $1\leq k \leq n$ and let $X_n = (x_1, \dots, x_n)$ be a list of $n$ variables. The {\em Boolean product polynomial} $B_{n,k}(X_n)$ is the product of the linear forms $\sum_{i \in S} x_i$ where $S$ ranges over all $k$-element subsets of $\{1, 2, \dots, n\}$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call {\em Chern plethysm}. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $B_{n,k}(X_n)$ for certain $k$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $B_{n,n-1}(X_n)$ to a bigraded action of the symmetric group $\mathfrak{S}_n$ on a divergence free quotient of superspace.