Blocks and Families for Cyclotomic Hecke Algebras (Lecture Notes in Mathematics, 1981)

The definition of Rouquier for the families of characters introduced by Lusztig for Weyl groups in terms of blocks of the Hecke algebras has made possible the generalization of this notion to the case of complex reflection groups. The aim of this book is to study the blocks and to determine the families of characters for all cyclotomic Hecke algebras associated to complex reflection groups. This volume offers a thorough study of symmetric algebras, covering topics such as block theory, representation theory and Clifford theory, and can also serve as an introduction to the Hecke algebras of complex reflection groups. Uploaded by zhang. Preface Chapter 1. On Commutative Algebra 1.1 Localizations 1.2 Integrally Closed Rings 1.2.1 Lifting Prime Ideals 1.2.2 Valuations 1.2.3 Discrete Valuation Rings and Krull Rings 1.3 Completions 1.4 Morphisms Associated with Monomials and Adapted Morphisms 1.5 Irreducibility Chapter 2. On Blocks 2.1 General Results 2.1.1 Blocks and Integral Closure 2.1.2 Blocks and Prime Ideals 2.1.3 Blocks and Quotient Blocks 2.1.4 Blocks and Central Characters 2.2 Symmetric Algebras 2.3 Twisted Symmetric Algebras of Finite Groups 2.3.1 Action of G on Z \over{A} 2.3.2 Multiplication of an A-Module by an OG-Module 2.3.3 Induction and Restriction of KA-Modules and K \over{A}-Modules 2.3.4 Blocks of A and Blocks of \over{A} 2.4 Representation Theory of Symmetric Algebras 2.4.1 Grothendieck Groups 2.4.2 Integrality 2.4.3 The Decomposition Map 2.4.4 A Variation for Tits’ Deformation Theorem 2.4.5 Symmetric Algebras over Discrete Valuation Rings Chapter 3. On Essential Algebras 3.1 General Facts 3.2 Specialization via Morphisms Associated with Monomials 3.3 Specialization via Adapted Morphisms 3.4 The Map In Chapter 4. On Hecke 4.1 Complex Reflection Groups and Associated Braid Groups 4.1.1 Complex Reflection Groups 4.1.2 Braid Groups Associated to Complex Reflection Groups 4.2 Generic Hecke Algebras 4.3 Cyclotomic Hecke Algebras 4.3.1 Essential Hyperplanes 4.3.2 Group Algebra 4.4 Rouquier Blocks of the Cyclotomic Hecke Algebras 4.4.1 Rouquier Blocks and Central Morphisms 4.4.2 Rouquier Blocks and Functions a and A Chapter 5. On the Determination of the Rouquier Blocks 5.1 General Principles 5.2 The Exceptional Irreducible Complex Reflection Groups 5.2.1 Essential Hyperplanes 5.2.2 Algorithm 5.2.3 Results 5.3 The Groups G(d, 1, r) 5.3.1 Combinatorics 5.3.2 Ariki-Koike Algebras 5.3.3 Rouquier Blocks, Charged Content and Residues 5.3.4 Essential Hyperplanes 5.3.5 Results 5.4 The Groups G(2d, 2, 2) 5.4.1 Essential Hyperplanes 5.4.2 Results 5.5 The Groups G(de, e, r) 5.5.1 The groups G(de, e, r), r > 2 5.5.2 The Groups G(de, e, 2) Appendix A. Clifford Theory and Schur Elements for Generic Hecke Algebras A.1 The Groups G4, G5, G6, G7 A.2 The Groups G8, G9, G10, G11, G12, G13, G14, G15 A.3 The Groups G16, G17, G18, G19, G20, G21, G22 A.4 The Groups G25, G26 A.5 The Group G28 (“F4”) A.6 The Group G32 A.7 The Groups G(de, e, r) A.7.1 The Groups G(de, e, r), r > 2 A.7.2 The Groups G(de, e, 2), e Odd A.7.3 The Groups G(de, e, 2), e Even References 20 47 Index abcdefghijk lmnoprstvw Series "The definition of Rouquier for the families of characters introduced by Lusztig for Weyl groups in terms of blocks of the Hecke algebras has made possible the generalization of this notion to the case of complex reflection groups. The aim of this book is to study the blocks and to determine the families of characters for all cyclotomic Hecke algebras associated to complex reflection groups. This volume offers a thorough study of symmetric algebras, covering topics such as block theory, representation theory and Clifford theory, and can also serve as an introduction to the Hecke algebras of complex reflection groups."--Publisher's website

The definition of Rouquier for the families of characters introduced by Lusztig for Weyl groups in terms of blocks of the Hecke algebras has made possible the generalization of this notion to the case of complex reflection groups. The aim of this book is to study the blocks and to determine the families of characters for all cyclotomic Hecke algebras associated to complex reflection groups.

This volume offers a thorough study of symmetric algebras, covering topics such as block theory, representation theory and Clifford theory, and can also serve as an introduction to the Hecke algebras of complex reflection groups.

This volume offers a thorough study of symmetric algebras, covering topics such as block theory, representation theory and Clifford theory. It can also serve as an introduction to the Hecke algebras of complex reflection groups.