Catalan functions and 𝑘-Schur positivity

Abstract

We prove that graded k k -Schur functions are G 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 k -Schur functions and resolve the Schur positivity and k k -branching conjectures in the strongest possible terms by providing direct combinatorial formulas using strong marked tableaux.