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Shift system and its applications 移位系统及其应用
arXiv - MATH - Representation Theory Pub Date : 2024-09-11 DOI: arxiv-2409.07381
Hao Liwith an appendix by Myungbo Shim, Shoma Sugimotowith an appendix by Myungbo Shim
{"title":"Shift system and its applications","authors":"Hao Liwith an appendix by Myungbo Shim, Shoma Sugimotowith an appendix by Myungbo Shim","doi":"arxiv-2409.07381","DOIUrl":"https://doi.org/arxiv-2409.07381","url":null,"abstract":"We introduce a new concept named shift system. This is a purely Lie algebraic\u0000setting to develop the geometric representation theory of Feigin-Tipunin\u0000construction. After reformulating the discussion in past works of the second\u0000author under this new setting, as an application, we extend almost all the main\u0000results of these works to the (multiplet) principal W-algebra at positive\u0000integer level associated with a simple Lie algebra $mathfrak{g}$ and Lie\u0000superalgebra $mathfrak{osp}(1|2n)$, respectively. This paper also contains an\u0000appendix by Myungbo Shim on the relationship between Feigin-Tipunin\u0000construction and recent quantum field theories.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187292","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
Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras 简单排列的 Kac-Moody 算法的最大紧凑子代数的高自旋表征
arXiv - MATH - Representation Theory Pub Date : 2024-09-11 DOI: arxiv-2409.07247
Robin Lautenbacher, Ralf Köhl
{"title":"Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras","authors":"Robin Lautenbacher, Ralf Köhl","doi":"arxiv-2409.07247","DOIUrl":"https://doi.org/arxiv-2409.07247","url":null,"abstract":"Given the maximal compact subalgebra $mathfrak{k}(A)$ of a split-real\u0000Kac-Moody algebra $mathfrak{g}(A)$ of type $A$, we study certain\u0000finite-dimensional representations of $mathfrak{k}(A)$, that do not lift to\u0000the maximal compact subgroup $K(A)$ of the minimal Kac-Moody group $G(A)$\u0000associated to $mathfrak{g}(A)$ but only to its spin cover $Spin(A)$.\u0000Currently, four elementary of these so-called spin representations are known.\u0000We study their (ir-)reducibility, semi-simplicity, and lift to the group level.\u0000The interaction of these representations with the spin-extended Weyl-group is\u0000used to derive a partial parametrization result of the representation matrices\u0000by the real roots of $mathfrak{g}(A)$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187294","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
Macdonald Identities: revisited 重新审视麦克唐纳身份
arXiv - MATH - Representation Theory Pub Date : 2024-09-11 DOI: arxiv-2409.07317
K. Iohara, Y. Saito
{"title":"Macdonald Identities: revisited","authors":"K. Iohara, Y. Saito","doi":"arxiv-2409.07317","DOIUrl":"https://doi.org/arxiv-2409.07317","url":null,"abstract":"In this note, after recalling a proof of the Macdonald identities for\u0000untwisted affine root systems, we derive the Macdonald identities for twisted\u0000affine root systems.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187293","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
Wakamatsu tilting subcategories and weak support tau-tilting subcategories in recollement 若松倾斜子范畴和弱支持头倾斜子范畴的再置换
arXiv - MATH - Representation Theory Pub Date : 2024-09-11 DOI: arxiv-2409.07026
Yongduo Wang, Hongyang Luo, Jian He, Dejun Wu
{"title":"Wakamatsu tilting subcategories and weak support tau-tilting subcategories in recollement","authors":"Yongduo Wang, Hongyang Luo, Jian He, Dejun Wu","doi":"arxiv-2409.07026","DOIUrl":"https://doi.org/arxiv-2409.07026","url":null,"abstract":"In this article, we prove that if (A, B, C) is a recollement of abelian\u0000categories, then wakamatsu tilting (resp. weak support tau-tilting)\u0000subcategories in A and C can induce wakamatsu tilting (resp. weak support\u0000tau-tilting) subcategories in B, and the converses hold under natural\u0000assumptions. As an application, we mainly consider the relationship of\u0000tau-cotorsion torsion triples in (A, B, C).","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187295","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
On Character Variety of Anosov Representations 论阿诺索夫表征的特征多样性
arXiv - MATH - Representation Theory Pub Date : 2024-09-11 DOI: arxiv-2409.07316
Krishnendu Gongopadhyay, Tathagata Nayak
{"title":"On Character Variety of Anosov Representations","authors":"Krishnendu Gongopadhyay, Tathagata Nayak","doi":"arxiv-2409.07316","DOIUrl":"https://doi.org/arxiv-2409.07316","url":null,"abstract":"Let $Gamma$ be the free group $F_n$ of $n$ generators, resp. the fundamental\u0000group $pi_1(Sigma_g)$ of a closed, connnected, orientatble surface of genus\u0000$g geq 2$. We show that the charater variety of irreducible, resp. Zariski\u0000dense, Anosov representations of $Gamma$ into $SL(n, C)$ is a complex\u0000manifold of (complex) dimension $(n-1)(n^2-1)$, resp. $(2g-2) (n^2-1)$. For\u0000$Gamma=pi_1(Sigma_g)$, we also show that these character varieties are\u0000holomorphic symplectic manifolds.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187303","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
Linear Reedy categories, quasi-hereditary algebras and model structures 线性里迪范畴、准遗传代数和模型结构
arXiv - MATH - Representation Theory Pub Date : 2024-09-10 DOI: arxiv-2409.06823
Georgios Dalezios, Jan Stovicek
{"title":"Linear Reedy categories, quasi-hereditary algebras and model structures","authors":"Georgios Dalezios, Jan Stovicek","doi":"arxiv-2409.06823","DOIUrl":"https://doi.org/arxiv-2409.06823","url":null,"abstract":"We study linear versions of Reedy categories in relation with finite\u0000dimensional algebras and abelian model structures. We prove that, for a linear\u0000Reedy category $mathcal{C}$ over a field, the category of left\u0000$mathcal{C}$--modules admits a highest weight structure, which in case\u0000$mathcal{C}$ is finite corresponds to a quasi-hereditary algebra with an exact\u0000Borel subalgebra. We also lift complete cotorsion pairs and abelian model\u0000structures to certain categories of additive functors indexed by linear Reedy\u0000categories, generalizing analogous results from the hereditary case.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187296","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
A classification of $n$-representation infinite algebras of type à $n$表示型无限代数的分类
arXiv - MATH - Representation Theory Pub Date : 2024-09-10 DOI: arxiv-2409.06553
Darius Dramburg, Oleksandra Gasanova
{"title":"A classification of $n$-representation infinite algebras of type Ã","authors":"Darius Dramburg, Oleksandra Gasanova","doi":"arxiv-2409.06553","DOIUrl":"https://doi.org/arxiv-2409.06553","url":null,"abstract":"We classify $n$-representation infinite algebras $Lambda$ of type ~A. This\u0000type is defined by requiring that $Lambda$ has higher preprojective algebra\u0000$Pi_{n+1}(Lambda) simeq k[x_1, ldots, x_{n+1}] ast G$, where $G leq\u0000operatorname{SL}_{n+1}(k)$ is finite abelian. For the classification, we group\u0000these algebras according to a more refined type, and give a combinatorial\u0000characterisation of these types. This is based on so-called height functions,\u0000which generalise the height function of a perfect matching in a Dimer model. In\u0000terms of toric geometry and McKay correspondence, the types form a lattice\u0000simplex of junior elements of $G$. We show that all algebras of the same type\u0000are related by iterated $n$-APR tilting, and hence are derived equivalent. By\u0000disallowing certain tilts, we turn this set into a finite distributive lattice,\u0000and we construct its maximal and minimal elements.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187304","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
S-dual of Hamiltonian $mathbf G$ spaces and relative Langlands duality 哈密顿$mathbf G$空间的S对偶性和相对朗兰兹对偶性
arXiv - MATH - Representation Theory Pub Date : 2024-09-10 DOI: arxiv-2409.06303
Hiraku Nakajima
{"title":"S-dual of Hamiltonian $mathbf G$ spaces and relative Langlands duality","authors":"Hiraku Nakajima","doi":"arxiv-2409.06303","DOIUrl":"https://doi.org/arxiv-2409.06303","url":null,"abstract":"The S-dual $(mathbf G^veecurvearrowrightmathbf M^vee)$ of the pair\u0000$(mathbf Gcurvearrowrightmathbf M)$ of a smooth affine algebraic symplectic\u0000manifold $mathbf M$ with hamiltonian action of a complex reductive group\u0000$mathbf G$ was introduced implicitly in [arXiv:1706.02112] and explicitly in\u0000[arXiv:1807.09038] under the cotangent type assumption. The definition was a\u0000modification of the definition of Coulomb branches of gauge theories in\u0000[arXiv:1601.03586]. It was motivated by the S-duality of boundary conditions of\u00004-dimensional $mathcal N=4$ super Yang-Mills theory, studied by Gaiotto and\u0000Witten [arXiv:0807.3720]. It is also relevant to the relative Langlands duality\u0000proposed by Ben-Zvi, Sakellaridis and Venkatesh. In this article, we review the\u0000definition and properties of S-dual.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187124","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
Disconnected Reductive Groups: Classification and Representations 断开的还原基团：分类与表示
arXiv - MATH - Representation Theory Pub Date : 2024-09-10 DOI: arxiv-2409.06375
Dylan Johnston, Diego Martín Duro, Dmitriy Rumynin
{"title":"Disconnected Reductive Groups: Classification and Representations","authors":"Dylan Johnston, Diego Martín Duro, Dmitriy Rumynin","doi":"arxiv-2409.06375","DOIUrl":"https://doi.org/arxiv-2409.06375","url":null,"abstract":"In this article, we classify disconnected reductive groups over an\u0000algebraically closed field with a few caveats. Internal parts of our result are\u0000both a classification of finite groups and a classification of integral\u0000representations of a fixed finite group. Modulo these classifications - which\u0000are impossible in different senses - our main result explicitly tabulates the\u0000groups with an efficient algorithm. Besides this, we obtain new results about\u0000the representation theory of disconnected reductive groups in characteristic\u0000zero. We give two descriptions of their representation rings and prove that\u0000their Knutson Index is finite.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187306","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 axioms for right (n+2)-angulated categories 右 (n+2)-angulated 类别公理
arXiv - MATH - Representation Theory Pub Date : 2024-09-09 DOI: arxiv-2409.05561
Jing He, Jiangsha Li
{"title":"The axioms for right (n+2)-angulated categories","authors":"Jing He, Jiangsha Li","doi":"arxiv-2409.05561","DOIUrl":"https://doi.org/arxiv-2409.05561","url":null,"abstract":"Drawing inspiration from the works of Beligiannis-Marmaridis and Lin, we\u0000refine the axioms for a right $(n+2)$-angulated category and give some examples\u0000of such categories. Interestingly, we show that the morphism axiom for a right\u0000$(n+2)$-angulated category is actually redundant. Moreover, we prove that the\u0000higher octahedral axiom is equivalent to the mapping cone axiom for a right\u0000$(n+2)$-angulated category.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187123","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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