Regular Schur labeled skew shape posets and their 0-Hecke modules
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866912192749633536 |
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| author | Kim, Young-Hun Lee, So-Yeon Oh, Young-Tak |
| author_facet | Kim, Young-Hun Lee, So-Yeon Oh, Young-Tak |
| contents | Assuming Stanley's $P$-partition conjecture holds, the regular Schur labeled skew shape posets with underlying set $\{1,2,\ldots, n\}$ are precisely the posets $P$ such that the $P$-partition generating function is symmetric and the set of linear extensions of $P$, denoted $Σ_L(P)$, is a left weak Bruhat interval in the symmetric group $\mathfrak{S}_n$. We describe the permutations in $Σ_L(P)$ in terms of reading words of standard Young tableaux when $P$ is a regular Schur labeled skew shape poset, and classify $Σ_L(P)$'s up to descent-preserving isomorphism as $P$ ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the $0$-Hecke modules $\mathsf{M}_P$ associated with regular Schur labeled skew shape posets $P$ up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the posets whose linear extensions form a dual plactic-closed subset of $\mathfrak{S}_n$. Using this characterization, we construct distinguished filtrations of $\mathsf{M}_P$ with respect to the Schur basis when $P$ is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the $0$-Hecke modules $\mathsf{M}_P$ are also discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20571 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Regular Schur labeled skew shape posets and their 0-Hecke modules Kim, Young-Hun Lee, So-Yeon Oh, Young-Tak Representation Theory Combinatorics 20C08, 06A07, 05E10, 05E05 Assuming Stanley's $P$-partition conjecture holds, the regular Schur labeled skew shape posets with underlying set $\{1,2,\ldots, n\}$ are precisely the posets $P$ such that the $P$-partition generating function is symmetric and the set of linear extensions of $P$, denoted $Σ_L(P)$, is a left weak Bruhat interval in the symmetric group $\mathfrak{S}_n$. We describe the permutations in $Σ_L(P)$ in terms of reading words of standard Young tableaux when $P$ is a regular Schur labeled skew shape poset, and classify $Σ_L(P)$'s up to descent-preserving isomorphism as $P$ ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the $0$-Hecke modules $\mathsf{M}_P$ associated with regular Schur labeled skew shape posets $P$ up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the posets whose linear extensions form a dual plactic-closed subset of $\mathfrak{S}_n$. Using this characterization, we construct distinguished filtrations of $\mathsf{M}_P$ with respect to the Schur basis when $P$ is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the $0$-Hecke modules $\mathsf{M}_P$ are also discussed. |
| title | Regular Schur labeled skew shape posets and their 0-Hecke modules |
| topic | Representation Theory Combinatorics 20C08, 06A07, 05E10, 05E05 |
| url | https://arxiv.org/abs/2310.20571 |