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We prove that the K-k-Schur 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. Lam-Schilling-Shimozono identified the K-k-Schur functions as Schubert representatives for K-homology of the affine Grassmannian for SL_{k+1}. Our perspective reveals that the K-k-Schur 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 K-k-Schur functions produces a second shift-invariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of Ikeda-Iwao-Maeno, we conjecture that this second basis gives the images of the Lenart-Maeno quantum Grothendieck polynomials under a K-theoretic analog of the Peterson isomorphism.
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Boolean Product Polynomials, Schur Positivity, and Chern Plethysm
Abstract 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 $$B_{n,n-1}(X_n)$$ to a bigraded action of the symmetric group $${\mathfrak{S}}_n$$ on a divergence free quotient of superspace.
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- Award ID(s):
- 1764012
- PAR ID:
- 10125811
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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