Catalan functions and $k$-Schur positivity

Jonah Blasiak, Jennifer Morse, Anna Pun, Daniel Summers

Published 2018-04-10Version 1

We prove that graded $k$-Schur functions are $G$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded $k$-Schur functions and resolve the Schur positivity and $k$-branching conjectures in the strongest possible terms by providing direct combinatorial formulas using strong marked tableaux.