On Macdonald expansions of $q$-chromatic symmetric functions and the Stanley-Stembridge Conjecture
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912318253694976 |
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| author | Griffin, Sean T. Mellit, Anton Romero, Marino Weigl, Kevin Wen, Joshua Jeishing |
| author_facet | Griffin, Sean T. Mellit, Anton Romero, Marino Weigl, Kevin Wen, Joshua Jeishing |
| contents | The Stanley-Stembridge conjecture asserts that the chromatic symmetric function of a $(3+1)$-free graph is $e$-positive. Recently, Hikita proved this conjecture by giving an explicit $e$-expansion of the Shareshian-Wachs $q$-chromatic refinement for unit interval graphs. Using the $\mathbb{A}_{q,t}$ algebra, we give an expansion of these $q$-chromatic symmetric functions into Macdonald polynomials. Upon setting $t=1$, we obtain another proof of the Stanley-Stembridge conjecture and rederive Hikita's formula. Upon setting $t=0$, we obtain an expansion into Hall-Littlewood symmetric functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06936 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Macdonald expansions of $q$-chromatic symmetric functions and the Stanley-Stembridge Conjecture Griffin, Sean T. Mellit, Anton Romero, Marino Weigl, Kevin Wen, Joshua Jeishing Combinatorics Representation Theory The Stanley-Stembridge conjecture asserts that the chromatic symmetric function of a $(3+1)$-free graph is $e$-positive. Recently, Hikita proved this conjecture by giving an explicit $e$-expansion of the Shareshian-Wachs $q$-chromatic refinement for unit interval graphs. Using the $\mathbb{A}_{q,t}$ algebra, we give an expansion of these $q$-chromatic symmetric functions into Macdonald polynomials. Upon setting $t=1$, we obtain another proof of the Stanley-Stembridge conjecture and rederive Hikita's formula. Upon setting $t=0$, we obtain an expansion into Hall-Littlewood symmetric functions. |
| title | On Macdonald expansions of $q$-chromatic symmetric functions and the Stanley-Stembridge Conjecture |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2504.06936 |