This is a brief survey of the research accomplishments of Bob Griess, focusing on the work primarily related to simple groups and their classification. It does not attempt to also cover Bob’s many contributions to the theory of vertex operator algebras. (This is only because I am unqualified to survey that VOA material.) For background references on simple groups and their classification, I’ll mainly use the “outline” book [1] Over half of Bob Griess’s 85 papers on MathSciNet are more or less directly concerned with simple groups. Obviously I can only briefly describe the contents of so much work. (And I have left his work on vertex algebras etc to articles in this volume by more expert authors.) Background: Quasisimple components and the list of simple groups We first review some standard material from the early part of [1, Sec 0.3]. (More experienced readers can skip ahead to the subsequent subsection on the list of simple groups.) Received September 12, 2016. AMS Subject Classification: 20D05, 20D06, 20D08, 20E32, 20E42, 20J06.
{"title":"A Survey: Bob Griess's Work on Simple Groups and Their Classification","authors":"Stephen D. Smith","doi":"10.21915/bimas.2018401","DOIUrl":"https://doi.org/10.21915/bimas.2018401","url":null,"abstract":"This is a brief survey of the research accomplishments of Bob Griess, focusing on the work primarily related to simple groups and their classification. It does not attempt to also cover Bob’s many contributions to the theory of vertex operator algebras. (This is only because I am unqualified to survey that VOA material.) For background references on simple groups and their classification, I’ll mainly use the “outline” book [1] Over half of Bob Griess’s 85 papers on MathSciNet are more or less directly concerned with simple groups. Obviously I can only briefly describe the contents of so much work. (And I have left his work on vertex algebras etc to articles in this volume by more expert authors.) Background: Quasisimple components and the list of simple groups We first review some standard material from the early part of [1, Sec 0.3]. (More experienced readers can skip ahead to the subsequent subsection on the list of simple groups.) Received September 12, 2016. AMS Subject Classification: 20D05, 20D06, 20D08, 20E32, 20E42, 20J06.","PeriodicalId":43960,"journal":{"name":"Bulletin of the Institute of Mathematics Academia Sinica New Series","volume":"30 8 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74037765","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 study the quantizations of the algebras of regular functions on nilpo- tent orbits. We show that such a quantization always exists and is unique if the orbit is birationally rigid. Further we show that, for special birationally rigid orbits, the quan- tization has integral central character in all cases but four (one orbit in E7 and three orbits in E8). We use this to complete the computation of Goldie ranks for primitive ideals with integral central character for all special nilpotent orbits but one (in E8). Our main ingredient are results on the geometry of normalizations of the closures of nilpotent orbits by Fu and Namikawa.
{"title":"Quantizations of Regular Functions on Nilpotent Orbits","authors":"Ivan Loseu","doi":"10.21915/BIMAS.2018202","DOIUrl":"https://doi.org/10.21915/BIMAS.2018202","url":null,"abstract":"We study the quantizations of the algebras of regular functions on nilpo- tent orbits. We show that such a quantization always exists and is unique if the orbit is birationally rigid. Further we show that, for special birationally rigid orbits, the quan- tization has integral central character in all cases but four (one orbit in E7 and three orbits in E8). We use this to complete the computation of Goldie ranks for primitive ideals with integral central character for all special nilpotent orbits but one (in E8). Our main ingredient are results on the geometry of normalizations of the closures of nilpotent orbits by Fu and Namikawa.","PeriodicalId":43960,"journal":{"name":"Bulletin of the Institute of Mathematics Academia Sinica New Series","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75492357","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}
Let G' be a connected reductive group over the complex numbers. We show that the set of conjugacy classes of G' is in natural bijection with the set of two-sided cells associated to a certain algebra.
设G'是复数上的连通约化群。证明了G'的共轭类集与与某代数相关的双面元集是自然双射的。
{"title":"Conjugacy Classes in Reductive Groups and Two-sided Cells","authors":"G. Lusztig","doi":"10.21915/bimas.2019301","DOIUrl":"https://doi.org/10.21915/bimas.2019301","url":null,"abstract":"Let G' be a connected reductive group over the complex numbers. We show that the set of conjugacy classes of G' is in natural bijection with the set of two-sided cells associated to a certain algebra.","PeriodicalId":43960,"journal":{"name":"Bulletin of the Institute of Mathematics Academia Sinica New Series","volume":"98 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2017-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81176864","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 the elementary divisors of the adjacency and Laplacian matrices of the Grassmann graph on $2$-dimensional subspaces in a finite vector space. We also compute the corresponding invariants of the complementary graphs.
{"title":"The Smith group and the critical group of the Grassmann graph of lines in finite projective space and of its complement","authors":"Joshua E. Ducey, Peter Sin","doi":"10.21915/BIMAS.2018404","DOIUrl":"https://doi.org/10.21915/BIMAS.2018404","url":null,"abstract":"We compute the elementary divisors of the adjacency and Laplacian matrices of the Grassmann graph on $2$-dimensional subspaces in a finite vector space. We also compute the corresponding invariants of the complementary graphs.","PeriodicalId":43960,"journal":{"name":"Bulletin of the Institute of Mathematics Academia Sinica New Series","volume":"32 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2017-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73749412","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 show the existence of a finite group $G$ having an irreducible character $chi$ with Frobenius-Schur indicator $nu_2(chi){=}{+}1$ such that $chi^2$ has an irreducible constituent $varphi$ with $nu_2(varphi){=}{-}1$. This provides counterexamples to the positivity conjecture in rational CFT and a conjecture of Zhenghan Wang about pivotal fusion categories.
{"title":"A Brief History of the Positivity Conjecture in Tensor Category","authors":"G. Mason","doi":"10.21915/bimas.2019202","DOIUrl":"https://doi.org/10.21915/bimas.2019202","url":null,"abstract":"We show the existence of a finite group $G$ having an irreducible character $chi$ with Frobenius-Schur indicator $nu_2(chi){=}{+}1$ such that $chi^2$ has an irreducible constituent $varphi$ with $nu_2(varphi){=}{-}1$. This provides counterexamples to the positivity conjecture in rational CFT and a conjecture of Zhenghan Wang about pivotal fusion categories.","PeriodicalId":43960,"journal":{"name":"Bulletin of the Institute of Mathematics Academia Sinica New Series","volume":"16 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2017-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89714000","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 previous work the authors introduced a new class of modular quasi-Hopf algebras $D^{omega}(G, A)$ associated to a finite group $G$, a central subgroup $A$, and a $3$-cocycle $omegain Z^3(G, C^x)$. In the present paper we propose a description of the class of orbifold models of rational vertex operator algebras whose module category is tensor equivalent to $D^{omega}(G, A)$-mod. The paper includes background on quasi-Hopf algebras and a discussion of some relevant orbifolds.
在先前的工作中，作者引入了一类新的模拟hopf代数$D^{omega}(G, a)$，它们与Z^3(G, C^x)$中的有限群$G$、中心子群$ a $和$3$-环$omega $相关联。本文给出了一类模范畴张量等价于D^{ ω}(G, a)$-mod的有理顶点算子代数的轨道模型的描述。本文包括拟hopf代数的背景和对一些相关轨道的讨论。
{"title":"Generalized twisted quantum doubles of a finite group and rational orbifolds","authors":"G. Mason, S. Ng","doi":"10.21915/bimas.2019101","DOIUrl":"https://doi.org/10.21915/bimas.2019101","url":null,"abstract":"In previous work the authors introduced a new class of modular quasi-Hopf algebras $D^{omega}(G, A)$ associated to a finite group $G$, a central subgroup $A$, and a $3$-cocycle $omegain Z^3(G, C^x)$. In the present paper we propose a description of the class of orbifold models of rational vertex operator algebras whose module category is tensor equivalent to $D^{omega}(G, A)$-mod. The paper includes background on quasi-Hopf algebras and a discussion of some relevant orbifolds.","PeriodicalId":43960,"journal":{"name":"Bulletin of the Institute of Mathematics Academia Sinica New Series","volume":"6 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2017-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87488899","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}
For the simple Lie algebra $ frak{so}_m$, we study the commutant vertex operator algebra of $ L_{widehat{frak{so}}_{m}}(n,0)$ in the $n$-fold tensor product $ L_{widehat{frak{so}}_{m}}(1,0)^{otimes n}$. It turns out that this commutant vertex operator algebra can be realized as a fixed point subalgebra of $L_{widehat{frak{so}}_{n}}(m,0)$ (or its simple current extension) associated with a certain abelian group. This result may be viewed as a version of level-rank duality.
{"title":"Level-Rank Duality for Vertex Operator Algebras of types B and D","authors":"Cuipo Jiang, C. Lam","doi":"10.21915/BIMAS.2019103","DOIUrl":"https://doi.org/10.21915/BIMAS.2019103","url":null,"abstract":"For the simple Lie algebra $ frak{so}_m$, we study the commutant vertex operator algebra of $ L_{widehat{frak{so}}_{m}}(n,0)$ in the $n$-fold tensor product $ L_{widehat{frak{so}}_{m}}(1,0)^{otimes n}$. It turns out that this commutant vertex operator algebra can be realized as a fixed point subalgebra of $L_{widehat{frak{so}}_{n}}(m,0)$ (or its simple current extension) associated with a certain abelian group. This result may be viewed as a version of level-rank duality.","PeriodicalId":43960,"journal":{"name":"Bulletin of the Institute of Mathematics Academia Sinica New Series","volume":"38 11 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2017-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82821922","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}
Let $G$ be a simple, simply-connected algebraic group over the complex numbers with Lie algebra $mathfrak g$. The main result of this article is a proof that each irreducible representation of the fundamental group of the orbit $mathcal O$ through a nilpotent element $e in mathfrak g$ lifts to a representation of a Jacobson-Morozov parabolic subgroup of $G$ associated to $e$. This result was shown in some cases by Barbasch and Vogan in their study of unipotent representations for complex groups and, in general, in an unpublished part of the author's doctoral thesis. In the last section of the article, we state two applications of this result, whose details will appear elsewhere: to answering a question of Lusztig regarding special pieces in the exceptional groups (joint work with Fu, Juteau, and Levy); and to computing the $G$-module structure of the sections of an irreducible local system on $mathcal O$. A key aspect of the latter application is some new cohomological statements that generalize those in earlier work of the author.
{"title":"Irreducible Local Systems on Nilpotent Orbits","authors":"E. Sommers","doi":"10.21915/BIMAS.2018302","DOIUrl":"https://doi.org/10.21915/BIMAS.2018302","url":null,"abstract":"Let $G$ be a simple, simply-connected algebraic group over the complex numbers with Lie algebra $mathfrak g$. The main result of this article is a proof that each irreducible representation of the fundamental group of the orbit $mathcal O$ through a nilpotent element $e in mathfrak g$ lifts to a representation of a Jacobson-Morozov parabolic subgroup of $G$ associated to $e$. This result was shown in some cases by Barbasch and Vogan in their study of unipotent representations for complex groups and, in general, in an unpublished part of the author's doctoral thesis. In the last section of the article, we state two applications of this result, whose details will appear elsewhere: to answering a question of Lusztig regarding special pieces in the exceptional groups (joint work with Fu, Juteau, and Levy); and to computing the $G$-module structure of the sections of an irreducible local system on $mathcal O$. A key aspect of the latter application is some new cohomological statements that generalize those in earlier work of the author.","PeriodicalId":43960,"journal":{"name":"Bulletin of the Institute of Mathematics Academia Sinica New Series","volume":"21 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2016-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82014073","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 consider a series of VOAs generated by 3-dimensional Griess algebras. We will show that these VOAs can be characterized by their 3-dimensional Griess algebras and their structures are uniquely determined. As an application, we will determine the groups generated by the Miyamoto involutions associated to Virasoro vectors of our VOAs.
{"title":"3-dimensional Griess Algebras and Miyamoto Involutions","authors":"C. Lam, H. Yamauchi","doi":"10.21915/bimas.2019201","DOIUrl":"https://doi.org/10.21915/bimas.2019201","url":null,"abstract":"We consider a series of VOAs generated by 3-dimensional Griess algebras. We will show that these VOAs can be characterized by their 3-dimensional Griess algebras and their structures are uniquely determined. As an application, we will determine the groups generated by the Miyamoto involutions associated to Virasoro vectors of our VOAs.","PeriodicalId":43960,"journal":{"name":"Bulletin of the Institute of Mathematics Academia Sinica New Series","volume":"80 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2016-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73666839","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 describe the structure of Sylow {ell}-subgroups of a finite reduc-tive group G(Fq) when q $notequiv$ 0 (mod {ell}) that we find governed by a complex reflection group attached to G and {ell}, which depends on {ell} only through the set of cyclotomic factors of the generic order of G(Fq) whose value at q is divisible by {ell}. We also tackle the more general case G F where F is an isogeny whose a power is a Frobenius morphism.
{"title":"The Sylow Subgroups of a Finite Reductive Group","authors":"Michel Enguehard, J. Michel","doi":"10.21915/BIMAS.2018203","DOIUrl":"https://doi.org/10.21915/BIMAS.2018203","url":null,"abstract":"We describe the structure of Sylow {ell}-subgroups of a finite reduc-tive group G(Fq) when q $notequiv$ 0 (mod {ell}) that we find governed by a complex reflection group attached to G and {ell}, which depends on {ell} only through the set of cyclotomic factors of the generic order of G(Fq) whose value at q is divisible by {ell}. We also tackle the more general case G F where F is an isogeny whose a power is a Frobenius morphism.","PeriodicalId":43960,"journal":{"name":"Bulletin of the Institute of Mathematics Academia Sinica New Series","volume":"16 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2016-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87653277","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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