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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.
An accessible text introducing algebraic geometries and algebraic
groups at advanced undergraduate and early graduate level, this
book develops the language of algebraic geometry from scratch and
uses it to set up the theory of affine algebraic groups from first
principles.
Building on the background material from algebraic geometry and
algebraic groups, the text provides an introduction to more
advanced and specialised material. An example is the representation
theory of finite groups of Lie type.
The text covers the conjugacy of Borel subgroups and maximal tori,
the theory of algebraic groups with a BN-pair, a thorough treatment
of Frobenius maps on affine varieties and algebraic groups, zeta
functions and Lefschetz numbers for varieties over finite fields.
Experts in the field will enjoy some of the new approaches to
classical results.
The text uses algebraic groups as the main examples, including
worked out examples, instructive exercises, as well as
bibliographical and historical remarks.
The representation theory of reductive algebraic groups and related
finite reductive groups is a subject of great topical interest and
has many applications. The articles in this volume provide
introductions to various aspects of the subject, including
algebraic groups and Lie algebras, reflection groups, abelian and
derived categories, the Deligne-Lusztig representation theory of
finite reductive groups, Harish-Chandra theory and its
generalisations, quantum groups, subgroup structure of algebraic
groups, intersection cohomology, and Lusztig's conjectured
character formula for irreducible representations in prime
characteristic. The articles are carefully designed to reinforce
one another, and are written by a team of distinguished authors: M.
Broue, R. W. Carter, S. Donkin, M. Geck, J. C. Jantzen, B. Keller,
M. W. Liebeck, G. Malle, J. C. Rickard and R. Rouquier. This volume
as a whole should provide a very accessible introduction to an
important, though technical, subject.
Finite Coxeter groups and related structures arise naturally in several branches of mathematics, for example, Lie algebras or theory of knots and links. This is the first book which develops the character theory of finite Coxeter groups and Iwahori-Hecke algebras in a systematic way, ranging from classical results to recent developments.
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