LEADER 07119nam 22006015 450 001 9910633923903321 005 20240222145123.0 010 $a3-031-13873-2 024 7 $a10.1007/978-3-031-13873-7 035 $a(MiAaPQ)EBC7150313 035 $a(Au-PeEL)EBL7150313 035 $a(CKB)25504480100041 035 $a(DE-He213)978-3-031-13873-7 035 $a(PPN)26634917X 035 $a(EXLCZ)9925504480100041 100 $a20221129d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aRepresentation Theory of Finite Group Extensions $eClifford Theory, Mackey Obstruction, and the Orbit Method /$fby Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli 205 $a1st ed. 2022. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2022. 215 $a1 online resource (347 pages) 225 1 $aSpringer Monographs in Mathematics,$x2196-9922 311 08$aPrint version: Ceccherini-Silberstein, Tullio Representation Theory of Finite Group Extensions Cham : Springer International Publishing AG,c2022 9783031138720 320 $aIncludes bibliographical references and index. 327 $aIntro -- Preface -- Contents -- 1 Preliminaries -- 1.1 Representations of Finite Groups -- 1.2 The Group Algebra and the Left-Regular Representation -- 1.3 Induced Representations -- 1.4 Further Results on Induced Representations -- 1.5 Semidirect Products, Wreath Products, and Group Extensions -- 1.6 Regular Wreath Products and the Kaloujnine-KrasnerTheorem -- 2 Clifford Theory -- 2.1 Preliminaries and Notation -- 2.2 Basic Clifford Theory -- 2.3 First Applications and the Little Group Method -- 2.4 The Case Where AG(?) -1.2mu=IG(?)/N is Abelian -- 2.5 Some Applications of Mackey Theory to Clifford Theory -- 2.6 The G-Action on the N-Conjugacy Classes -- 2.7 Real, Complex, and Quaternionic Representations and Clifford Theory -- 2.8 Semidirect Products with an Abelian Normal Subgroup -- 2.9 Semidirect Products of Abelian Groups -- 2.10 Representation Theory of Wreath Products of Finite Groups -- 2.11 Multiplicity-Free Normal Subgroups -- 3 Abelian Extensions -- 3.1 The Dual Action -- 3.2 The Conjugation Action -- 3.3 The Intermediary Representations -- 3.4 Diagrammatic Summaries -- 4 The Little Group Method for Abelian Extensions -- 4.1 General Theory -- 4.2 Normal Subgroups with the Prime Condition -- 4.3 Normal Subgroups of Prime Index -- 4.4 The Case of Index Two Subgroups -- 5 Examples and Applications -- 5.1 Representation Theory and Conjugacy Classes of the Symmetric Groups Sn -- 5.2 Conjugacy Classes of An -- 5.3 The Irreducible Representations of An -- 5.4 Ambivalence of the Groups An -- 5.5 An Application to Isaacs' Going Down Theorem -- 5.6 Another Application: Analysis of p2-Extensions -- 5.7 Representation Theory of Finite Metacyclic Groups -- 5.8 Examples: Dihedral and Generalized Quaternion Groups -- 6 Central Extensions and the Orbit Method -- 6.1 Central Extensions. 327 $a6.2 2-Divisible Abelian Groups, Equalized Cocycles, and Schur Multipliers -- 6.3 Lie Rings -- 6.4 The Cocycle Decomposition -- 6.5 The Malcev Correspondence -- 6.6 The Orbit Method -- 6.7 More on the Orbit Method: Induced Representations -- 6.8 More on the Orbit Method: Restricting to a Subgroup -- 6.9 The Orbit Method for the Finite Heisenberg Group -- 6.10 Restricting from Hqt to Hq -- 6.11 The Little Group Method for the Heisenberg Group -- 7 Representations of Finite Group Extensions via Projective Representations -- 7.1 Mackey Obstruction -- 7.2 Unitary Projective Representations -- 7.3 The Dual of a Group Extension -- 7.4 Central Extensions and the Finite Heisenberg Group -- 7.5 Analysis of the Commutant -- 7.6 The Hecke Algebra -- 8 Induced Projective Representations -- 8.1 Basic Theory -- 8.2 Mackey's Theory for Induced Projective Representations -- 9 Clifford Theory for Projective Representations -- 9.1 Preliminaries and Notation -- 9.2 Basic Clifford Theory for Projective Representations -- 9.3 Projective Unitary Representations of a Group Extension -- 10 Projective Representations of Finite Abelian Groups with Applications -- 10.1 Bicharacters and 2-Cocycles on Finite Abelian Groups -- 10.2 The Irreducible Projective Representations of Finite Abelian Groups -- 10.3 Representation Theory of Finite Metabelian Groups -- 10.4 Representation Theory of Finite Step-2 Nilpotent Groups -- A Notes -- A.1 Group Extensions and Cohomology -- A.2 Clifford Theory -- A.3 The Little Group Method and Its Applications -- A.4 Lie Rings and the Orbit Method -- A.5 Projective Representations -- References -- Subject index -- Index of authors. 330 $aThis monograph adopts an operational and functional analytic approach to the following problem: given a short exact sequence (group extension) 1 N G H 1 of finite groups, describe the irreducible representations of G by means of the structure of the group extension. This problem has attracted many mathematicians, including I. Schur, A.H. Clifford, and G. Mackey and, more recently, M. Isaacs, B. Huppert, Y.G. Berkovich & E.M. Zhmud, and J.M.G. Fell & R.S. Doran. The main topics are, on the one hand, Clifford Theory and the Little Group Method (of Mackey and Wigner) for induced representations, and, on the other hand, Kirillov?s Orbit Method (for step-2 nilpotent groups of odd order) which establishes a natural and powerful correspondence between Lie rings and nilpotent groups. As an application, a detailed description is given of the representation theory of the alternating groups, of metacyclic, quaternionic, dihedral groups, and of the (finite) Heisenberg group. The Little Group Method may be applied if and only if a suitable unitary 2-cocycle (the Mackey obstruction) is trivial. To overcome this obstacle, (unitary) projective representations are introduced and corresponding Mackey and Clifford theories are developed. The commutant of an induced representation and the relative Hecke algebra is also examined. Finally, there is a comprehensive exposition of the theory of projective representations for finite Abelian groups which is applied to obtain a complete description of the irreducible representations of finite metabelian groups of odd order. 410 0$aSpringer Monographs in Mathematics,$x2196-9922 606 $aGroup theory 606 $aGroup Theory and Generalizations 606 $aGrups finits$2thub 606 $aTeoria de grups$2thub 608 $aLlibres electrònics$2thub 615 0$aGroup theory. 615 14$aGroup Theory and Generalizations. 615 7$aGrups finits 615 7$aTeoria de grups 676 $a512.2 676 $a512.2 700 $aCeccherini-Silberstein$b Tullio$0503338 702 $aScarabotti$b Fabio 702 $aTolli$b Filippo$f1968- 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910633923903321 996 $aRepresentation theory of finite group extensions$93088866 997 $aUNINA