Balanced labellings of affine permutations
We study the $\textit{diagrams}$ of affine permutations and their $\textit{balanced}$ labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2013-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/2342/pdf |
| _version_ | 1797270255018442752 |
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| author | Hwanchul Yoo Taedong Yun |
| author_facet | Hwanchul Yoo Taedong Yun |
| author_sort | Hwanchul Yoo |
| collection | DOAJ |
| description | We study the $\textit{diagrams}$ of affine permutations and their $\textit{balanced}$ labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the $\textit{column strict}$ balanced labellings is the affine Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we give a necessary and sufficient condition for a diagram to be an affine permutation diagram. Finally, we conjecture that if two affine permutations are $\textit{diagram equivalent}$ then their affine Stanley symmetric functions coincide. |
| first_indexed | 2024-04-25T02:01:21Z |
| format | Article |
| id | doaj.art-04ea014795164f0e8abc07895050e349 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:21Z |
| publishDate | 2013-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-04ea014795164f0e8abc07895050e3492024-03-07T14:52:36ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502013-01-01DMTCS Proceedings vol. AS,...Proceedings10.46298/dmtcs.23422342Balanced labellings of affine permutationsHwanchul Yoo0Taedong Yun1Korea Institute for Advanced StudyMassachusetts Institute of TechnologyWe study the $\textit{diagrams}$ of affine permutations and their $\textit{balanced}$ labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the $\textit{column strict}$ balanced labellings is the affine Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we give a necessary and sufficient condition for a diagram to be an affine permutation diagram. Finally, we conjecture that if two affine permutations are $\textit{diagram equivalent}$ then their affine Stanley symmetric functions coincide.https://dmtcs.episciences.org/2342/pdfaffine permutationspermutation diagramsbalanced labellingsreduced wordsstanley symmetric functions[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Hwanchul Yoo Taedong Yun Balanced labellings of affine permutations Discrete Mathematics & Theoretical Computer Science affine permutations permutation diagrams balanced labellings reduced words stanley symmetric functions [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Balanced labellings of affine permutations |
| title_full | Balanced labellings of affine permutations |
| title_fullStr | Balanced labellings of affine permutations |
| title_full_unstemmed | Balanced labellings of affine permutations |
| title_short | Balanced labellings of affine permutations |
| title_sort | balanced labellings of affine permutations |
| topic | affine permutations permutation diagrams balanced labellings reduced words stanley symmetric functions [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2342/pdf |
| work_keys_str_mv | AT hwanchulyoo balancedlabellingsofaffinepermutations AT taedongyun balancedlabellingsofaffinepermutations |