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K-theoretic Catalan functions
Journal article   Open access   Peer reviewed

K-theoretic Catalan functions

Jonah Blasiak, Jennifer Morse and George H. Seelinger
Advances in mathematics (New York. 1965), v 404, 108421
06 Aug 2022
DOI: https://doi.org/10.1016/j.aim.2022.108421
url
https://doi.org/10.1016/j.aim.2022.108421View
SubmittedOpen Access (License Unspecified),  Open

Abstract

Affine grassmannian K-k-Schur Katalan function Symmetric functions
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 SLk+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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Domestic collaboration
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Mathematics
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