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Representations of Hecke Algebras at Roots of Unity / by Meinolf Geck, Nicolas Jacon
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Algebra and Applications ; 15 | SpringerLink BücherPublisher: London : Springer-Verlag London Limited, 2011Description: Online-Ressource (XII, 401p. 6 illus, digital)ISBN:- 9780857297167
- 512 512.55 512/.55
- 512.2
- QA174-183
- QA174.2
Contents:
Summary: Generic Iwahori–Hecke algebras -- Kazhdan–Lusztig cells and cellular bases -- Specialisations and decomposition maps -- Hecke algebras and finite groups of Lie type -- Representation theory of Ariki–Koike algebras -- Canonical bases in affine type A and Ariki’s theorem -- Decomposition numbers for exceptional types.Summary: The modular representation theory of Iwahori-Hecke algebras and this theory's connection to groups of Lie type is an area of rapidly expanding interest; it is one that has also seen a number of breakthroughs in recent years. In classifying the irreducible representations of Iwahori-Hecke algebras at roots of unity, this book is a particularly valuable addition to current research in this field. Using the framework provided by the Kazhdan-Lusztig theory of cells, the authors develop an analogue of James' (1970) "characteristic-free'' approach to the representation theory of Iwahori-Hecke algebras in general. Presenting a systematic and unified treatment of representations of Hecke algebras at roots of unity, this book is unique in its approach and includes new results that have not yet been published in book form. It also serves as background reading to further active areas of current research such as the theory of affine Hecke algebras and Cherednik algebras. The main results of this book are obtained by an interaction of several branches of mathematics, namely the theory of Fock spaces for quantum affine Lie algebras and Ariki's theorem, the combinatorics of crystal bases, the theory of Kazhdan-Lusztig bases and cells, and computational methods. This book will be of use to researchers and graduate students in representation theory as well as any researchers outside of the field with an interest in Hecke algebras.PPN: PPN: 1650945752Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Preface; Contents; 1 Generic Iwahori-Hecke Algebras; 1.1 Coxeter Groups and Weight Functions; 1.2 Representations of H; 1.3 Lusztig's a-Invariants; 1.4 Balanced Representations; 1.5 The Asymptotic Algebra; 1.6 Introducing Cells; 1.7 Examples of Cells; 1.8 Cells and Leading Coefficients; 2 Kazhdan-Lusztig Cells and Cellular Bases; 2.1 The Kazhdan-Lusztig Basis; 2.2 A Pre-order Relation on Irr(W); 2.3 On Lusztig's Conjectures, I; 2.4 On Lusztig's Conjectures, II; 2.5 On Lusztig's Conjectures, III; 2.6 A Cellular Basis for H; 2.7 Further Properties of the Cellular Basis of H
2.8 The Case of the Symmetric Group3 Specialisations and Decomposition Maps; 3.1 Grothendieck Groups and Decomposition Maps; 3.2 Canonical Basic Sets; 3.3 Principal Specialisations and Blocks of Defect 1; 3.4 Towards Canonical Basic Sets for Classical Types; 3.5 A Canonical Basic Set for the Symmetric Group; 3.6 Factorisation of Decomposition Maps; 3.7 The General Version of James's Conjecture; 3.8 Blocks and Bad Specialisations; 4 Hecke Algebras and Finite Groups of Lie Type; 4.1 The Schur Functor and its Variations; 4.2 Hom Functors and Harish-Chandra Series
4.3 Unipotent Principal Series Representations, I4.4 Unipotent Principal Series Representations, II; 4.5 Examples and Conjectures; 4.6 A First Approach to Type Bn; 5 Representation Theory of Ariki-Koike Algebras; 5.1 The Complex Reflection Group of Type G(l,1,n); 5.2 Basic Properties of Ariki-Koike Algebras; 5.3 Ariki-Koike Algebras as Cellular Algebras; 5.4 Decomposition Maps for Ariki-Koike Algebras; 5.5 Cyclotomic Ariki-Koike Algebras; 5.6 A Fock Datum for Ariki-Koike Algebras; 5.7 FLOTW Multipartitions; 5.8 On Basic Sets for Ariki-Koike Algebras
6 Canonical Bases in Affine Type A and Ariki's Theorem6.1 The Quantum Affine Algebra Uq(sle); 6.2 The Fock Space and Ariki's Theorem; 6.3 Crystal and Canonical Bases for Highest Weight Modules; 6.4 Computing Decomposition Matrices of Ariki-Koike Algebras; 6.5 Uglov's Theory of Fock Spaces; 6.6 Canonical Bases for Fock Spaces; 6.7 Computation of Canonical Basic Sets for Iwahori-Hecke Algebras; 6.8 Recent Developments and Conjectures; 7 Decomposition Numbers for Exceptional Types; 7.1 Algorithmic Methods; 7.2 Decomposition Matrices for W of Dihedral Type
7.3 Decomposition Matrices for W of Type F47.4 Decomposition Matrices for Types H3, H4, E6, E7, E8; References; Index;
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