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.
Ktheoretic Catalan functions
We prove that the KkSchur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. LamSchillingShimozono identified the KkSchur functions as Schubert representatives for Khomology of the affine Grassmannian for SL_{k+1}. Our perspective reveals that the KkSchur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of Baldwin and Kumar. We further show that a slight adjustment of our formulation for KkSchur functions produces a second shiftinvariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of IkedaIwaoMaeno, we conjecture that this second basis gives the images of the LenartMaeno quantum Grothendieck polynomials under a Ktheoretic analog of the Peterson isomorphism.
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 Award ID(s):
 1840234
 NSFPAR ID:
 10530113
 Publisher / Repository:
 Elsevier
 Date Published:
 Journal Name:
 Advances in Mathematics
 Volume:
 404
 Issue:
 PB
 ISSN:
 00018708
 Page Range / eLocation ID:
 108421
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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