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An accessible text introducing algebraic geometry 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 geometries from basic principles.

Geometry, Algebraic. --- Linear algebraic groups. --- Algebraic groups, Linear --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- Algebraic geometry --- Geometry

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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.

Algebra. --- Electronic books. --- Hecke algebras. --- Orthogonal polynomials. --- Hecke algebras --- Representations of algebras --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Algebras, Hecke --- Mathematics. --- Associative rings. --- Rings (Algebra). --- Group theory. --- Group Theory and Generalizations. --- Associative Rings and Algebras. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Math --- Science --- Group algebras --- Mathematical analysis

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Coxeter groups. --- Finite groups. --- Hecke algebras.

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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. Broué, 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.

Representations of groups. --- Linear algebraic groups. --- Lie algebras.

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Representations of groups.