$K$-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.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2020
- DOI:
- 10.48550/arXiv.2010.01759
- arXiv:
- arXiv:2010.01759
- Bibcode:
- 2020arXiv201001759B
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Algebraic Geometry;
- 05E05;
- 14N15
- E-Print:
- 24 pages