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Multiplicity One Theorem for General Spin Groups: The Archimedean Case 一般自旋群的多重性一定理：阿基米德情况
arXiv - MATH - Representation Theory Pub Date : 2024-09-14 DOI: arxiv-2409.09320
Melissa Emory, Yeansu Kim, Ayan Maiti
{"title":"Multiplicity One Theorem for General Spin Groups: The Archimedean Case","authors":"Melissa Emory, Yeansu Kim, Ayan Maiti","doi":"arxiv-2409.09320","DOIUrl":"https://doi.org/arxiv-2409.09320","url":null,"abstract":"Let $GSpin(V)$ (resp. $GPin(V)$) be a general spin group (resp. a general\u0000Pin group) associated with a nondegenerate quadratic space $V$ of dimension $n$\u0000over an Archimedean local field $F$. For a nondegenerate quadratic space $W$ of\u0000dimension $n-1$ over $F$, we also consider $GSpin(W)$ and $GPin(W)$. We prove\u0000the multiplicity-at-most-one theorem in the Archimedean case for a pair of\u0000groups ($GSpin(V), GSpin(W)$) and also for a pair of groups ($GPin(V),\u0000GPin(W)$); namely, we prove that the restriction to $GSpin(W)$ (resp.\u0000$GPin(W)$) of an irreducible Casselman-Wallach representation of $GSpin(V)$\u0000(resp. $GPin(V)$) is multiplicity free.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
New techniques for calculation of Jordan-Kronecker invariants for Lie algebras and Lie algebra representations 计算李代数和李代数表示的乔丹-克朗内克不变式的新技术
arXiv - MATH - Representation Theory Pub Date : 2024-09-14 DOI: arxiv-2409.09535
I. K. Kozlov
{"title":"New techniques for calculation of Jordan-Kronecker invariants for Lie algebras and Lie algebra representations","authors":"I. K. Kozlov","doi":"arxiv-2409.09535","DOIUrl":"https://doi.org/arxiv-2409.09535","url":null,"abstract":"We introduce two novel techniques that simplify calculation of\u0000Jordan-Kronecker invariants for a Lie algebra $mathfrak{g}$ and for a Lie\u0000algebra representation $rho$. First, the stratification of matrix pencils\u0000under strict equivalence puts restrictions on the Jordan-Kronecker invariants.\u0000Second, we show that the Jordan-Kronecker invariants of a semi-direct sum\u0000$mathfrak{g} ltimes_{rho} V$ are sometimes determined by the\u0000Jordan-Kronecker invariants of the dual Lie algebra representation $rho^*$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Crystals for Kostant-Kumar modules of $widehat{mathfrak{sl}_2}$ $widehatmathfrak{sl}_2}$ 的 Kostant-Kumar 模块晶体
arXiv - MATH - Representation Theory Pub Date : 2024-09-14 DOI: arxiv-2409.09328
Mrigendra Singh Kushwaha, K. N. Raghavan, Sankaran Viswanath
{"title":"Crystals for Kostant-Kumar modules of $widehat{mathfrak{sl}_2}$","authors":"Mrigendra Singh Kushwaha, K. N. Raghavan, Sankaran Viswanath","doi":"arxiv-2409.09328","DOIUrl":"https://doi.org/arxiv-2409.09328","url":null,"abstract":"We consider the affine Lie algebra $widehat{mathfrak{sl}_2}$ and the\u0000Kostant-Kumar submodules of tensor products of its level 1 highest weight\u0000integrable representations. We construct crystals for these submodules in terms\u0000of the charged partitions model and describe their decomposition into\u0000irreducibles.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Almost Commutative Terwilliger Algebras of Group Association Schemes II: Primitive Idempotents 群联方案的几乎交换 Terwilliger 算法 II：原始等价物
arXiv - MATH - Representation Theory Pub Date : 2024-09-14 DOI: arxiv-2409.09482
Nicholas L. Bastian
{"title":"Almost Commutative Terwilliger Algebras of Group Association Schemes II: Primitive Idempotents","authors":"Nicholas L. Bastian","doi":"arxiv-2409.09482","DOIUrl":"https://doi.org/arxiv-2409.09482","url":null,"abstract":"This paper is a continuation of Almost Commutative Terwilliger Algebras of\u0000Group Association Schemes I: Classification [1]. In that paper, we found all\u0000groups G for which the Terwilliger algebra of the group association scheme,\u0000denoted T (G), is almost commutative. We also found the primitive idempotents\u0000for T (G) for three of the four types of such groups. In this paper, we\u0000determine the primitive idempotents for the fourth type.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Non-vanishing condition on Mogelin-Renard's parametrization for Arthur packets of $mathrm U(p,q)$ $mathrm U(p,q)$ 阿瑟包的莫格林-勒纳参数化的非消失条件
arXiv - MATH - Representation Theory Pub Date : 2024-09-14 DOI: arxiv-2409.09358
Chang Huang
{"title":"Non-vanishing condition on Mogelin-Renard's parametrization for Arthur packets of $mathrm U(p,q)$","authors":"Chang Huang","doi":"arxiv-2409.09358","DOIUrl":"https://doi.org/arxiv-2409.09358","url":null,"abstract":"Mogelin-Renard parametrize A-packet of unitary group through cohomological\u0000induction in good parity case. Each parameter gives rise to an $A_{mathfrak\u0000q}(lambda)$ which is either $0$ or irreducible. Trapa proposed an algorithm to\u0000determine whether a mediocre $A_{mathfrak q}(lambda)$ of $mathrm U(p, q)$ is\u0000non-zero. Based on his result, we present a further understanding of the\u0000non-vanishing condition of Mogelin-Renard's parametrization. Our criterion come\u0000out to be a system of linear constraints, and very similiar to the $p$-adic\u0000case.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"187 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The stack of spherical Langlands parameters 球形朗兰兹参数堆栈
arXiv - MATH - Representation Theory Pub Date : 2024-09-14 DOI: arxiv-2409.09522
Thibaud van den Hove
{"title":"The stack of spherical Langlands parameters","authors":"Thibaud van den Hove","doi":"arxiv-2409.09522","DOIUrl":"https://doi.org/arxiv-2409.09522","url":null,"abstract":"For a reductive group over a nonarchimedean local field, we define the stack\u0000of spherical Langlands parameters, using the inertia-invariants of the\u0000Langlands dual group. This generalizes the stack of unramified Langlands\u0000parameters in case the group is unramified. We then use this stack to deduce\u0000the Eichler--Shimura congruence relations for Hodge type Shimura varieties,\u0000without restrictions on the ramification.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Kac Diagrams for Elliptic Weyl Group Elements 椭圆韦尔群元素的 Kac 图
arXiv - MATH - Representation Theory Pub Date : 2024-09-14 DOI: arxiv-2409.09255
Stephen DeBacker, Jacob Haley
{"title":"Kac Diagrams for Elliptic Weyl Group Elements","authors":"Stephen DeBacker, Jacob Haley","doi":"arxiv-2409.09255","DOIUrl":"https://doi.org/arxiv-2409.09255","url":null,"abstract":"Suppose $mathfrak{g}$ is a semisimple complex Lie algebra and $mathfrak{h}$\u0000is a Cartan subalgebra of $mathfrak{g}$. To the pair\u0000$(mathfrak{g},mathfrak{h})$ one can associate both a Weyl group and a set of\u0000Kac diagrams. There is a natural map from the set of elliptic conjugacy classes\u0000in the Weyl group to the set of Kac diagrams. In both this setting and the\u0000twisted setting, this paper (a) shows that this map is injective and (b)\u0000explicitly describes this map's image.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dunkl and Cherednik operators Dunkl 和 Cherednik 算子
arXiv - MATH - Representation Theory Pub Date : 2024-09-13 DOI: arxiv-2409.09005
Oleg Chalykh
{"title":"Dunkl and Cherednik operators","authors":"Oleg Chalykh","doi":"arxiv-2409.09005","DOIUrl":"https://doi.org/arxiv-2409.09005","url":null,"abstract":"This survey article, written for the Encyclopedia of Mathematical Physics,\u00002nd edition, is devoted to the remarkable family of operators introduced by\u0000Charles Dunkl and to their $q$-analogues discovered by Ivan Cherednik. The main\u0000focus is on the r^ole of these operators in studying integrable many-body\u0000systems such as the Calogero-Moser and the Ruijsenaars systems. To put these\u0000constructions into a wider context, we indicate their relationship with the\u0000theory of the rational Cherednik algebras and double affine Hecke algebras.\u0000While we do not include proofs, references to the original research articles\u0000are provided, accompanied by brief historical comments.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Contravariant Koszul duality between non-positive and positive dg algebras 非正和正 dg 结构之间的科斯祖尔共变对偶性
arXiv - MATH - Representation Theory Pub Date : 2024-09-13 DOI: arxiv-2409.08842
Riku Fushimi
{"title":"Contravariant Koszul duality between non-positive and positive dg algebras","authors":"Riku Fushimi","doi":"arxiv-2409.08842","DOIUrl":"https://doi.org/arxiv-2409.08842","url":null,"abstract":"We characterize locally finite non-positive dg algebras that arise as Koszul\u0000duals of locally finite non-positive dg algebras. Moreover, we show that the\u0000Koszul dual functor induces contravariant derived equivalnces. As a\u0000consequence, we prove that every functorially finite bounded heart of $pvd A$\u0000of a locally finite non-positive dg algebra is a length category.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Kac-Moody Quaternion Lie Algebra 卡-莫迪四元数列代数
arXiv - MATH - Representation Theory Pub Date : 2024-09-13 DOI: arxiv-2409.10396
Ferdi, Amir Kamal Amir, Andi Muhammad Anwar
{"title":"Kac-Moody Quaternion Lie Algebra","authors":"Ferdi, Amir Kamal Amir, Andi Muhammad Anwar","doi":"arxiv-2409.10396","DOIUrl":"https://doi.org/arxiv-2409.10396","url":null,"abstract":"This research aims to define Kac-Moody Lie algebra in Quaternion by using the\u0000concept of Quaternification of Lie algebra. The results of this research\u0000obtained the definition of Universal Kac-Moody Quaternion Lie algebra, Standard\u0000Kac-Moody Quaternion Lie algebra, and Reduced Kac-Moody Quaternion Lie algebra","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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