Let $Lambda$ be an Artin algebra and let $e$ be an idempotent in $Lambda$. We study certain functors which preserve the singularity categories. Suppose $mathrm{pd}Lambda e_{eLambda e}
{"title":"A note on singular equivalences and idempotents","authors":"Dawei Shen","doi":"10.1090/PROC/15604","DOIUrl":"https://doi.org/10.1090/PROC/15604","url":null,"abstract":"Let $Lambda$ be an Artin algebra and let $e$ be an idempotent in $Lambda$. We study certain functors which preserve the singularity categories. Suppose $mathrm{pd}Lambda e_{eLambda e}<infty$ and $mathrm{id}_Lambdatfrac{Lambda/langle erangle}{mathrm{rad}Lambda/langle erangle} < infty$, we show that there is a singular equivalence between $eLambda e$ and $Lambda$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131208767","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}
We compute all sections of the finite Weyl group, that satisfy the braid relations, in the case that G is an almost-simple connected reductive group defined over an algebraically closed field. We then demonstrate that this set of sections has an interesting partially ordered structure, and also give some applications.
{"title":"The Sections of the Weyl Group","authors":"Moshe Adrian","doi":"10.1093/IMRN/RNAA319","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA319","url":null,"abstract":"We compute all sections of the finite Weyl group, that satisfy the braid relations, in the case that G is an almost-simple connected reductive group defined over an algebraically closed field. We then demonstrate that this set of sections has an interesting partially ordered structure, and also give some applications.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"123 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124186263","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}
We give a presentation of Feynman categories from a representation--theoretical viewpoint. Feynman categories are a special type of monoidal categories and their representations are monoidal functors. They can be viewed as a far reaching generalization of groups, algebras and modules. Taking a new algebraic approach, we provide more examples and more details for several key constructions. This leads to new applications and results. The text is intended to be a self--contained basis for a crossover of more elevated constructions and results in the fields of representation theory and Feynman categories, whose applications so far include number theory, geometry, topology and physics.
{"title":"Feynman categories and representation\u0000 theory","authors":"R. Kaufmann","doi":"10.1090/CONM/769/15419","DOIUrl":"https://doi.org/10.1090/CONM/769/15419","url":null,"abstract":"We give a presentation of Feynman categories from a representation--theoretical viewpoint. \u0000Feynman categories are a special type of monoidal categories and their representations are monoidal functors. They can be viewed as a far reaching generalization of groups, algebras and modules. Taking a new algebraic approach, we provide more examples and more details for several key constructions. This leads to new applications and results. \u0000The text is intended to be a self--contained basis for a crossover of more elevated constructions and results in the fields of representation theory and Feynman categories, whose applications so far include number theory, geometry, topology and physics.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134442978","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}
In the 40s, Mayer introduced a construction of (simplicial) $p$-complex by using the unsigned boundary map and taking coefficients of chains modulo $p$. We look at such a $p$-complex associated to an $(n-1)$-simplex; in which case, this is also a $p$-complex of representations of the symmetric group of rank $n$ - specifically, of permutation modules associated to two-row compositions. In this article, we calculate the so-called slash homology - a homology theory introduced by Khovanov and Qi - of such a $p$-complex. We show that every non-trivial slash homology group appears as an irreducible representation associated two-row partitions, and how this calculation leads to a basis of these irreduicble representations given by the so-called $p$-standard tableaux.
{"title":"Irreducible representations of the symmetric groups from slash homologies of p-complexes","authors":"Aaron Chan, William Wong","doi":"10.5802/ALCO.153","DOIUrl":"https://doi.org/10.5802/ALCO.153","url":null,"abstract":"In the 40s, Mayer introduced a construction of (simplicial) $p$-complex by using the unsigned boundary map and taking coefficients of chains modulo $p$. We look at such a $p$-complex associated to an $(n-1)$-simplex; in which case, this is also a $p$-complex of representations of the symmetric group of rank $n$ - specifically, of permutation modules associated to two-row compositions. In this article, we calculate the so-called slash homology - a homology theory introduced by Khovanov and Qi - of such a $p$-complex. We show that every non-trivial slash homology group appears as an irreducible representation associated two-row partitions, and how this calculation leads to a basis of these irreduicble representations given by the so-called $p$-standard tableaux.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129144967","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}
Pub Date : 2019-11-11DOI: 10.1007/978-3-030-68506-5_9
E. Lapid
{"title":"Explicit Decomposition of Certain Induced Representations of the General Linear Group","authors":"E. Lapid","doi":"10.1007/978-3-030-68506-5_9","DOIUrl":"https://doi.org/10.1007/978-3-030-68506-5_9","url":null,"abstract":"","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121527685","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}
Pub Date : 2019-11-05DOI: 10.7546/CRABS.2020.02.02
Elitza Hristova
In this paper, we consider the exterior algebra $Lambda(W)$ of a polynomial $mathrm{GL}(n)$-module $W$ and use previously developed methods to determine the Hilbert series of the algebra of invariants $Lambda(W)^G$, where $G$ is one of the classical complex subgroups of $mathrm{GL}(n)$, namely $mathrm{SL}(n)$, $mathrm{O}(n)$, $mathrm{SO}(n)$, or $mathrm{Sp}(2d)$ (for $n=2d$). Since $Lambda(W)^G$ is finite dimensional, we apply the described method to compute a lot of explicit examples. For $Lambda(S^3mathbb{C}^3)^{mathrm{SL}(3)}$, using the computed Hilbert series, we obtain an explicit set of generators.
{"title":"Hilbert Series and Invariants in Exterior Algebras","authors":"Elitza Hristova","doi":"10.7546/CRABS.2020.02.02","DOIUrl":"https://doi.org/10.7546/CRABS.2020.02.02","url":null,"abstract":"In this paper, we consider the exterior algebra $Lambda(W)$ of a polynomial $mathrm{GL}(n)$-module $W$ and use previously developed methods to determine the Hilbert series of the algebra of invariants $Lambda(W)^G$, where $G$ is one of the classical complex subgroups of $mathrm{GL}(n)$, namely $mathrm{SL}(n)$, $mathrm{O}(n)$, $mathrm{SO}(n)$, or $mathrm{Sp}(2d)$ (for $n=2d$). Since $Lambda(W)^G$ is finite dimensional, we apply the described method to compute a lot of explicit examples. For $Lambda(S^3mathbb{C}^3)^{mathrm{SL}(3)}$, using the computed Hilbert series, we obtain an explicit set of generators.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130114464","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}
Pub Date : 2019-10-30DOI: 10.1142/s0129167x20500494
Haian He
Let $G$ be a noncompact connected simple Lie group, and $(G,G^Gamma)$ a Klein four symmetric pair. In this paper, the author shows a necessary condition for the discrete decomposability of unitarizable simple $(mathfrak{g},K)$-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for $(G,G^Gamma)$, there does not exist any unitarizable simple $(mathfrak{g},K)$-module that is discretely decomposable as a $(mathfrak{g}^Gamma,K^Gamma)$-module. As an application, for $G=mathrm{E}_{6(-14)}$, the author obtains a complete classification of Klein four symmetric pairs $(G,G^Gamma)$ with $G^Gamma$ noncompact, such that there exists at least one nontrivial unitarizable simple $(mathfrak{g},K)$-module that is discretely decomposable as a $(mathfrak{g}^Gamma,K^Gamma)$-module and is also discretely decomposable as a $(mathfrak{g}^sigma,K^sigma)$-module for some nonidentity element $sigmainGamma$.
{"title":"A criterion for discrete branching laws for Klein four symmetric pairs and its application to E6(−14)","authors":"Haian He","doi":"10.1142/s0129167x20500494","DOIUrl":"https://doi.org/10.1142/s0129167x20500494","url":null,"abstract":"Let $G$ be a noncompact connected simple Lie group, and $(G,G^Gamma)$ a Klein four symmetric pair. In this paper, the author shows a necessary condition for the discrete decomposability of unitarizable simple $(mathfrak{g},K)$-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for $(G,G^Gamma)$, there does not exist any unitarizable simple $(mathfrak{g},K)$-module that is discretely decomposable as a $(mathfrak{g}^Gamma,K^Gamma)$-module. As an application, for $G=mathrm{E}_{6(-14)}$, the author obtains a complete classification of Klein four symmetric pairs $(G,G^Gamma)$ with $G^Gamma$ noncompact, such that there exists at least one nontrivial unitarizable simple $(mathfrak{g},K)$-module that is discretely decomposable as a $(mathfrak{g}^Gamma,K^Gamma)$-module and is also discretely decomposable as a $(mathfrak{g}^sigma,K^sigma)$-module for some nonidentity element $sigmainGamma$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132607902","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}
We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D(Mod-A) of a ring A. To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in A, which is a natural extension of the construction of compactly generated t-structures from chains of subsets of the Zariski spectrum known for the commutative noetherian case. We also provide constructions of silting and cosilting objects in D(Mod-A). This leads us to classification results over some classes of commutative rings and over finite dimensional hereditary algebras.
{"title":"Parametrizing torsion pairs in derived categories","authors":"Lidia Angeleri Hugel, Michal Hrbek","doi":"10.1090/ert/579","DOIUrl":"https://doi.org/10.1090/ert/579","url":null,"abstract":"We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D(Mod-A) of a ring A. To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in A, which is a natural extension of the construction of compactly generated t-structures from chains of subsets of the Zariski spectrum known for the commutative noetherian case. We also provide constructions of silting and cosilting objects in D(Mod-A). This leads us to classification results over some classes of commutative rings and over finite dimensional hereditary algebras.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131551163","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}
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