B 型双联赫克代数

Pub Date : 2024-01-10 DOI:10.1142/s021949882550166x
Yu Xie, An Zhang, Bin Shu
{"title":"B 型双联赫克代数","authors":"Yu Xie, An Zhang, Bin Shu","doi":"10.1142/s021949882550166x","DOIUrl":null,"url":null,"abstract":"<p>As a sequel to [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in <i>Algebra Colloq.</i>], in this article we first introduce a so-called duplex Hecke algebras of type <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>B</mi></mstyle></math></span><span></span> which is a <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℚ</mi><mo stretchy=\"false\">(</mo><mi>q</mi><mo stretchy=\"false\">)</mo></math></span><span></span>-algebra associated with the Weyl group <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"script\">𝒲</mi><mo stretchy=\"false\">(</mo><mstyle mathvariant=\"sans-serif\"><mi>B</mi></mstyle><mo stretchy=\"false\">)</mo></math></span><span></span> of type <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>B</mi></mstyle></math></span><span></span>, and symmetric groups <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>𝔖</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span><span></span> for <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi></math></span><span></span>, satisfying some Hecke relations (see Definition 3.1). This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur–Weyl duality of Levi-type (see [B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]), extending the duplex Hecke algebra of type <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>A</mi></mstyle></math></span><span></span> arising from the related <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>q</mi></math></span><span></span>-Schur–Weyl duality of Levi-type (see [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in <i>Algebra Colloq.</i>]). A duplex Hecke algebra of type <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>B</mi></mstyle></math></span><span></span> admits natural representations on certain tensor spaces. We then establish a Levi-type <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>q</mi></math></span><span></span>-Schur–Weyl duality of type <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>B</mi></mstyle></math></span><span></span>, which reveals the double centralizer property between such duplex Hecke algebras and <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>ı</mi></math></span><span></span>quantum groups studied by Bao and Wang in [H. Bao and W. Wang, A new approach to Kazhdan–Lusztig theory of type B via quantum symmetric pairs, <i>Astérisque</i><b>402</b> (2018)].</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Duplex Hecke algebras of type B\",\"authors\":\"Yu Xie, An Zhang, Bin Shu\",\"doi\":\"10.1142/s021949882550166x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>As a sequel to [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in <i>Algebra Colloq.</i>], in this article we first introduce a so-called duplex Hecke algebras of type <span><math altimg=\\\"eq-00003.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mstyle mathvariant=\\\"sans-serif\\\"><mi>B</mi></mstyle></math></span><span></span> which is a <span><math altimg=\\\"eq-00004.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>ℚ</mi><mo stretchy=\\\"false\\\">(</mo><mi>q</mi><mo stretchy=\\\"false\\\">)</mo></math></span><span></span>-algebra associated with the Weyl group <span><math altimg=\\\"eq-00005.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi mathvariant=\\\"script\\\">𝒲</mi><mo stretchy=\\\"false\\\">(</mo><mstyle mathvariant=\\\"sans-serif\\\"><mi>B</mi></mstyle><mo stretchy=\\\"false\\\">)</mo></math></span><span></span> of type <span><math altimg=\\\"eq-00006.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mstyle mathvariant=\\\"sans-serif\\\"><mi>B</mi></mstyle></math></span><span></span>, and symmetric groups <span><math altimg=\\\"eq-00007.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><msub><mrow><mi>𝔖</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span><span></span> for <span><math altimg=\\\"eq-00008.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi></math></span><span></span>, satisfying some Hecke relations (see Definition 3.1). This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur–Weyl duality of Levi-type (see [B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]), extending the duplex Hecke algebra of type <span><math altimg=\\\"eq-00009.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mstyle mathvariant=\\\"sans-serif\\\"><mi>A</mi></mstyle></math></span><span></span> arising from the related <span><math altimg=\\\"eq-00010.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>q</mi></math></span><span></span>-Schur–Weyl duality of Levi-type (see [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in <i>Algebra Colloq.</i>]). A duplex Hecke algebra of type <span><math altimg=\\\"eq-00011.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mstyle mathvariant=\\\"sans-serif\\\"><mi>B</mi></mstyle></math></span><span></span> admits natural representations on certain tensor spaces. We then establish a Levi-type <span><math altimg=\\\"eq-00012.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>q</mi></math></span><span></span>-Schur–Weyl duality of type <span><math altimg=\\\"eq-00013.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mstyle mathvariant=\\\"sans-serif\\\"><mi>B</mi></mstyle></math></span><span></span>, which reveals the double centralizer property between such duplex Hecke algebras and <span><math altimg=\\\"eq-00014.gif\\\" display=\\\"inline\\\" overflow=\\\"scroll\\\"><mi>ı</mi></math></span><span></span>quantum groups studied by Bao and Wang in [H. Bao and W. Wang, A new approach to Kazhdan–Lusztig theory of type B via quantum symmetric pairs, <i>Astérisque</i><b>402</b> (2018)].</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-01-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/s021949882550166x\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s021949882550166x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

作为 [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021, arXiv:2108.07587[math.RT]] 的续篇,已被接受。Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.在本文中,我们首先介绍一种所谓的 B 型双工 Hecke 代数,它是一个与 B 型韦尔群𝒲(B) 和对称群𝔖l(l=0,1,...,m)相关联的ℚ(q)代数,满足一些 Hecke 关系(见定义 3.1)。这一概念源于对一种 Levi 型舒尔-韦尔对偶性的研究过程中产生的退化双工 Hecke 代数(见 [B. Shu and Y. Yao, On enhanced Hecke algebra] [中文版])。Shu and Y. Yao, On enhanced reductive groups (I):见[B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]], 扩展了由相关的 Levi 型 q-Schur-Weyl 对偶性产生的 A 型 duplex Hecke 代数(见[C. Xue and A. Zhang, Doulex Hecke algebra of type A arising from the related q-Schur-Weyl duality of Levi-type])。Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.)B 型双工赫克代数在某些张量空间上有自然表示。Bao and W. Wang, A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs, Astérisque402 (2018)].
本文章由计算机程序翻译,如有差异,请以英文原文为准。
分享
查看原文
Duplex Hecke algebras of type B

As a sequel to [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.], in this article we first introduce a so-called duplex Hecke algebras of type B which is a (q)-algebra associated with the Weyl group 𝒲(B) of type B, and symmetric groups 𝔖l for l=0,1,,m, satisfying some Hecke relations (see Definition 3.1). This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur–Weyl duality of Levi-type (see [B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]), extending the duplex Hecke algebra of type A arising from the related q-Schur–Weyl duality of Levi-type (see [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.]). A duplex Hecke algebra of type B admits natural representations on certain tensor spaces. We then establish a Levi-type q-Schur–Weyl duality of type B, which reveals the double centralizer property between such duplex Hecke algebras and ıquantum groups studied by Bao and Wang in [H. Bao and W. Wang, A new approach to Kazhdan–Lusztig theory of type B via quantum symmetric pairs, Astérisque402 (2018)].

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信