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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2212.09419 |
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| _version_ | 1866917252644732928 |
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| author | Kim, Donghyun Lee, Seung Jin Oh, Jaeseong |
| author_facet | Kim, Donghyun Lee, Seung Jin Oh, Jaeseong |
| contents | For a partition $ν$, let $λ,μ\subseteq ν$ be two distinct partitions such that $|ν/λ|=|ν/μ|=1$. Butler conjectured that the divided difference $\operatorname{I}_{λ,μ}[X;q,t]=(T_λ\widetilde{H}_μ[X;q,t]-T_μ\widetilde{H}_λ[X;q,t])/(T_λ-T_μ)$ of modified Macdonald polynomials of two partitions $λ$ and $μ$ is Schur positive. By introducing a new LLT equivalence called column exchange rule, we give a combinatorial formula for $\operatorname{I}_{λ,μ}[X;q,t]$, which is a positive monomial expansion. We also prove Butler's conjecture for some special cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_09419 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Toward Butler's conjecture Kim, Donghyun Lee, Seung Jin Oh, Jaeseong Combinatorics Representation Theory 05E05, 05E10, 05A05 For a partition $ν$, let $λ,μ\subseteq ν$ be two distinct partitions such that $|ν/λ|=|ν/μ|=1$. Butler conjectured that the divided difference $\operatorname{I}_{λ,μ}[X;q,t]=(T_λ\widetilde{H}_μ[X;q,t]-T_μ\widetilde{H}_λ[X;q,t])/(T_λ-T_μ)$ of modified Macdonald polynomials of two partitions $λ$ and $μ$ is Schur positive. By introducing a new LLT equivalence called column exchange rule, we give a combinatorial formula for $\operatorname{I}_{λ,μ}[X;q,t]$, which is a positive monomial expansion. We also prove Butler's conjecture for some special cases. |
| title | Toward Butler's conjecture |
| topic | Combinatorics Representation Theory 05E05, 05E10, 05A05 |
| url | https://arxiv.org/abs/2212.09419 |