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

TL;DR

This paper provides a categorification of $k$-Schur functions, confirming conjectures and establishing an algebraic framework that demonstrates the positivity of modified Macdonald polynomials in terms of $k$-Schur functions.

Contribution

It characterizes $k$-Schur functions as graded characters of simple objects in a module category, confirming conjectures and linking to Macdonald positivity.

Findings

Confirmed conjectures from Chen's Ph.D. thesis.

Established $k$-Schur functions as graded characters.

Proved modified Macdonald polynomials are $k$-Schur positive.

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.