Categorification of $k$-Schur functions and refined Macdonald positivity
Abstract
We characterize the -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 -Schur positive, thus realizing the original motivation behind the definition of the -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 -Schur functions and the Hall--Littlewood functions.
Keywords
Cite
@article{arxiv.2505.23202,
title = {Categorification of $k$-Schur functions and refined Macdonald positivity},
author = {Syu Kato},
journal= {arXiv preprint arXiv:2505.23202},
year = {2025}
}
Comments
v2, 57 pages. Extensively revised; notation has been updated to align with arXiv:2301.00862, with many additional details and examples. The proof of Theorem 7.1 (prepared through \S5--6) is now more direct. We have also added \S1.1 and expanded \S9--10. Another round of revisions is expected before journal submission. Comments are most welcome