$K$theoretic Catalan functions
Abstract
We prove that the $K$$k$Schur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. LamSchillingShimozono identified the $K$$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 shiftinvariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of IkedaIwaoMaeno, we conjecture that this second basis gives the images of the LenartMaeno quantum Grothendieck polynomials under a $K$theoretic analog of the Peterson isomorphism.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.01759
 arXiv:
 arXiv:2010.01759
 Bibcode:
 2020arXiv201001759B
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Geometry;
 05E05;
 14N15
 EPrint:
 24 pages