Categorification of $k$-Schur functions and refined Macdonald positivity
Syu Kato
TL;DR
The paper categorifies $k$-Schur functions by identifying them as graded characters of simple objects in a module category tied to current algebras and a geometric Catalan framework. It constructs $k$-Schur modules $s^{(k)}_{\lambda}$ via HL$_{\lambda}^{\Psi[\lambda,k]}$ and Schur–Weyl duality, proving that HL$^{\Psi}_{\lambda}$ has a simple head and, for shallow root ideals, a simple socle, which yields refined Macdonald positivity through filtrations of Garsia–Haiman-type modules by $k$-Schur modules. The approach develops a robust toolkit—affine Demazure functors, promotion functors, geometric Pieri and straightening rules, branching laws, and dual $k$-Schur complexes—tying combinatorial objects to Chen–Haiman modules and the affine Grassmannian geometry. Collectively, the results provide a conceptual mechanism for Macdonald positivity and unify Catalan symmetric-function theory with representation-theoretic categorification, yielding concrete filtrations and Ext-vanishing criteria that govern the $k$-Schur side.
Abstract
We characterize the $k$-Schur functions as the graded characters of simple objects in an additive module category. This confirms a set of conjectures formulated in the Ph.D. thesis of Chen, written under the direction of Mark Haiman, and thereby establishes the algebraic framework proposed therein. As a consequence, we deduce that the modified Macdonald polynomials are $k$-Schur positive, thus realizing the original motivation behind the definition of the $k$-Schur functions by Lapointe, Lascoux, and Morse. Our approach builds on our previous work on the algebraic and geometric realization of Catalan symmetric functions, which encompasses both the $k$-Schur functions and the Hall--Littlewood functions.