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arXiv · 2010.01759

$K$-theoretic Catalan functions

Jonah Blasiak·Jennifer Morse·George H. Seelinger
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Abstract

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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Jonah Blasiak, Jennifer Morse, George H. Seelinger. 2020-10-05. $K$-theoretic Catalan functions. https://arxiv.org/abs/2010.01759

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