Categorification of $k$-Schur functions and refined Macdonald positivity

Abstract: We characterize $k$-Schur functions as the graded characters of simple modules of an additive category. This implies conjectures in Chen [thesis, UCBerkeley], and therefore establishes the algebraic framework proposed there. As a consequence, we demonstrate that transformed Macdonald polynomials are $k$-Schur positive, achieving the original motivation behind the definition of $k$-Schur functions by Lapointe-Lascoux-Morse [Duke Math. J. (2003)].