$K$-theoretic Catalan functions

TL;DR

This paper introduces a new family of inhomogeneous symmetric functions related to $K$-$k$-Schur functions, revealing their shift invariance, positivity properties, and conjectural connections to quantum Grothendieck polynomials within algebraic geometry and $K$-theory.

Contribution

It establishes the shift invariance of $K$-$k$-Schur functions, demonstrates positivity of their branching coefficients, and proposes a new basis conjecturally linked to quantum Grothendieck polynomials.

Findings

$K$-$k$-Schur functions are part of a family with Catalan functions as top components.

Proved shift invariance and positivity of branching coefficients.

Conjectured a second basis with positive branching and rectangle factorization.

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.