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

Syu Kato

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arXiv · 2505.23202

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

Syu Kato

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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.

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Syu Kato. 2025-05-29. Categorification of $k$-Schur functions and refined Macdonald positivity. https://arxiv.org/abs/2505.23202

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