Self-dual intervals in the Bruhat order
Björner and Ekedahl (Ann Math (2) 170(2):799–817, 2009) prove that general intervals [e, w] in Bruhat order are “top-heavy”, with at least as many elements in the i-th corank as the i-th rank. Well-known results of Carrell (in: Algebraic groups and their generalizations: classical methods (Universit...
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| Format: | Article |
| Language: | English |
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Springer Science and Business Media LLC
2021
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| Online Access: | https://hdl.handle.net/1721.1/129828 |
| _version_ | 1811068579105210368 |
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| author | Gaetz, Christian Gao, Yibo |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Gaetz, Christian Gao, Yibo |
| author_sort | Gaetz, Christian |
| collection | MIT |
| description | Björner and Ekedahl (Ann Math (2) 170(2):799–817, 2009) prove that general intervals [e, w] in Bruhat order are “top-heavy”, with at least as many elements in the i-th corank as the i-th rank. Well-known results of Carrell (in: Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), volume 56 of proceedings of symposium on pure mathematics, pp 53–61. American Mathematical Society, Providence, RI, 1994) and of Lakshmibai and Sandhya (Proc Indian Acad Sci Math Sci 100(1):45–52, 1990) give the equality case: [e, w] is rank-symmetric if and only if the permutation w avoids the patterns 3412 and 4231 and these are exactly those w such that the Schubert variety X[subscript w] is smooth. In this paper we study the finer structure of rank-symmetric intervals [e, w], beyond their rank functions. In particular, we show that these intervals are still “top-heavy” if one counts cover relations between different ranks. The equality case in this setting occurs when [e, w] is self-dual as a poset; we characterize these w by pattern avoidance and in several other ways. |
| first_indexed | 2024-09-23T07:58:00Z |
| format | Article |
| id | mit-1721.1/129828 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T07:58:00Z |
| publishDate | 2021 |
| publisher | Springer Science and Business Media LLC |
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| spelling | mit-1721.1/1298282022-09-30T01:23:08Z Self-dual intervals in the Bruhat order Gaetz, Christian Gao, Yibo Massachusetts Institute of Technology. Department of Mathematics Björner and Ekedahl (Ann Math (2) 170(2):799–817, 2009) prove that general intervals [e, w] in Bruhat order are “top-heavy”, with at least as many elements in the i-th corank as the i-th rank. Well-known results of Carrell (in: Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), volume 56 of proceedings of symposium on pure mathematics, pp 53–61. American Mathematical Society, Providence, RI, 1994) and of Lakshmibai and Sandhya (Proc Indian Acad Sci Math Sci 100(1):45–52, 1990) give the equality case: [e, w] is rank-symmetric if and only if the permutation w avoids the patterns 3412 and 4231 and these are exactly those w such that the Schubert variety X[subscript w] is smooth. In this paper we study the finer structure of rank-symmetric intervals [e, w], beyond their rank functions. In particular, we show that these intervals are still “top-heavy” if one counts cover relations between different ranks. The equality case in this setting occurs when [e, w] is self-dual as a poset; we characterize these w by pattern avoidance and in several other ways. 2021-02-18T21:47:53Z 2021-02-18T21:47:53Z 2020-11 2020-10 2020-12-05T04:25:01Z Article http://purl.org/eprint/type/JournalArticle 1022-1824 1420-9020 https://hdl.handle.net/1721.1/129828 Gaetz, Christian and Yibo Gao. "Self-dual intervals in the Bruhat order." Selecta Mathematica 26, 5 (November 2020) : 77 © 2020 Springer Nature Switzerland AG en https://doi.org/10.1007/s00029-020-00608-z Selecta Mathematica Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Nature Switzerland AG application/pdf Springer Science and Business Media LLC Springer International Publishing |
| spellingShingle | Gaetz, Christian Gao, Yibo Self-dual intervals in the Bruhat order |
| title | Self-dual intervals in the Bruhat order |
| title_full | Self-dual intervals in the Bruhat order |
| title_fullStr | Self-dual intervals in the Bruhat order |
| title_full_unstemmed | Self-dual intervals in the Bruhat order |
| title_short | Self-dual intervals in the Bruhat order |
| title_sort | self dual intervals in the bruhat order |
| url | https://hdl.handle.net/1721.1/129828 |
| work_keys_str_mv | AT gaetzchristian selfdualintervalsinthebruhatorder AT gaoyibo selfdualintervalsinthebruhatorder |