Let $k \leq n$ be positive integers, and let $X_n = (x_1, \dots , x_n)$ be a list of $n$ variables. The 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 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 $B_{n,n1}(X_n)$ to a bigraded action of the symmetric group ${\mathfrak{S}}_n$ on a divergence free quotient of superspace.
 Award ID(s):
 1802139
 NSFPAR ID:
 10338318
 Date Published:
 Journal Name:
 Journal of the Australian Mathematical Society
 Volume:
 111
 Issue:
 1
 ISSN:
 14467887
 Page Range / eLocation ID:
 17 to 36
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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