Categorification of $k$-Schur functions and refined Macdonald positivity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915523559686144 |
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| author | Kato, Syu |
| author_facet | Kato, Syu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_23202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Categorification of $k$-Schur functions and refined Macdonald positivity Kato, Syu Representation Theory Algebraic Geometry Combinatorics 05E05, 20G05 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. |
| title | Categorification of $k$-Schur functions and refined Macdonald positivity |
| topic | Representation Theory Algebraic Geometry Combinatorics 05E05, 20G05 |
| url | https://arxiv.org/abs/2505.23202 |