K-theoretic Catalan Functions

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.

This paper was published in the journal Advances in Mathematics in Volume 404, Part B, 6 August 2022.

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Date published: Wednesday, May 4, 2022