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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2504.09123 |
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| _version_ | 1866916686513307648 |
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| author | Huh, JiSun Hwang, Byung-Hak Kim, Donghyun Kim, Jang Soo Oh, Jaeseong |
| author_facet | Huh, JiSun Hwang, Byung-Hak Kim, Donghyun Kim, Jang Soo Oh, Jaeseong |
| contents | We study the symmetric functions \( g_{\mm,k}(x;q) \), introduced by
Abreu and Nigro for a Hessenberg function \( \mm \) and a positive
integer \( k \), which refine the chromatic symmetric function.
Building on Hikita's recent breakthrough on the Stanley--Stembridge
conjecture, we prove the \( e \)-positivity of \( g_{\mm,k}(x;1) \),
refining Hikita's result. We also provide a Schur expansion of the
sum \( \sum_{k=1}^n e_k(x) g_{\mm,n-k}(x;q) \) in terms of
\( P \)-tableaux with 1 in the upper-left corner. We introduce a
restricted version of the modular law as our main tool. Then, we
show that any function satisfying the restricted modular law is
determined by its values on disjoint unions of path graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09123 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Refinement of Hikita's $e$-positivity theorem via Abreu--Nigro's $g$-functions and restricted modular law Huh, JiSun Hwang, Byung-Hak Kim, Donghyun Kim, Jang Soo Oh, Jaeseong Combinatorics We study the symmetric functions \( g_{\mm,k}(x;q) \), introduced by Abreu and Nigro for a Hessenberg function \( \mm \) and a positive integer \( k \), which refine the chromatic symmetric function. Building on Hikita's recent breakthrough on the Stanley--Stembridge conjecture, we prove the \( e \)-positivity of \( g_{\mm,k}(x;1) \), refining Hikita's result. We also provide a Schur expansion of the sum \( \sum_{k=1}^n e_k(x) g_{\mm,n-k}(x;q) \) in terms of \( P \)-tableaux with 1 in the upper-left corner. We introduce a restricted version of the modular law as our main tool. Then, we show that any function satisfying the restricted modular law is determined by its values on disjoint unions of path graphs. |
| title | Refinement of Hikita's $e$-positivity theorem via Abreu--Nigro's $g$-functions and restricted modular law |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.09123 |