George Lusztig's Books

Introduction to Quantum Groups

By: George Lusztig

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Introduction to Quantum Groups

By: George Lusztig

The quantum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof. The theory of quantum groups has led to a new, extremely rigid structure, in which the objects of the theory are provided with canonical basis with rather remarkable properties. This book will be of interest to mathematicians working in the representation theory of Lie groups and Lie algebras, knot theorists and to theoretical physicists and graduate students. Since large parts of the book are independent of the theory of perverse sheaves, the book could also be used as a text book.

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361

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Springer Science & Business Media

Published Date

ISBN 10

0817647171

ISBN 13

9780817647179

Discrete Series of GLn Over a Finite Field. (AM-81), Volume 81

By: George Lusztig

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Discrete Series of GLn Over a Finite Field. (AM-81), Volume 81

By: George Lusztig

In this book Professor Lusztig solves an interesting problem by entirely new methods: specifically, the use of cohomology of buildings and related complexes. The book gives an explicit construction of one distinguished member, D(V), of the discrete series of GLn (Fq), where V is the n-dimensional F-vector space on which GLn(Fq) acts. This is a p-adic representation; more precisely D(V) is a free module of rank (q--1) (q2—1)...(qn-1—1) over the ring of Witt vectors WF of F. In Chapter 1 the author studies the homology of partially ordered sets, and proves some vanishing theorems for the homology of some partially ordered sets associated to geometric structures. Chapter 2 is a study of the representation △ of the affine group over a finite field. In Chapter 3 D(V) is defined, and its restriction to parabolic subgroups is determined. In Chapter 4 the author computes the character of D(V), and shows how to obtain other members of the discrete series by applying Galois automorphisms to D(V). Applications are in Chapter 5. As one of the main applications of his study the author gives a precise analysis of a Brauer lifting of the standard representation of GLn(Fq).

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Page Count

107

Publisher

Princeton University Press

Published Date

ISBN 10

1400881765

ISBN 13

9781400881765

Characters of Reductive Groups over a Finite Field. (AM-107), Volume 107

By: George Lusztig

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Characters of Reductive Groups over a Finite Field. (AM-107), Volume 107

By: George Lusztig

This book presents a classification of all (complex) irreducible representations of a reductive group with connected centre, over a finite field. To achieve this, the author uses etale intersection cohomology, and detailed information on representations of Weyl groups.

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408

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Princeton University Press

Published Date

ISBN 10

1400881773

ISBN 13

9781400881772

Iwahori-Hecke Algebras and their Representation Theory

Lectures given at the CIME Summer School held in Martina Franca, Italy, June 28 - July 6, 1999

By: Ivan Cherednik,Yavor Markov,Roger E. Howe,George Lusztig

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Iwahori-Hecke Algebras and their Representation Theory

Lectures given at the CIME Summer School held in Martina Franca, Italy, June 28 - July 6, 1999

By: Ivan Cherednik,Yavor Markov,Roger E. Howe,George Lusztig

Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. Algebras of Iwahori-Hecke type are one of the tools and were, probably, first considered in the context of representation theory of finite groups of Lie type. This volume consists of notes of the courses on Iwahori-Hecke algebras and their representation theory, given during the CIME summer school which took place in 1999 in Martina Franca, Italy.

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117

Publisher

Springer

Published Date

ISBN 10

3540362053

ISBN 13

9783540362050

Representations of Finite Chevalley Groups

By: George Lusztig

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Representations of Finite Chevalley Groups

By: George Lusztig

Features notes that arose from a series of lectures given by the author at a CBMS Regional Conference held at Madison, Wisconsin, in August 1977. The purpose of the notes was to show how $1$-adic cohomology of algebraic varieties over fields of characteristic $p>1$ can be used to get information on the representations of finite Chevalley groups.

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American Mathematical Soc.

Published Date

ISBN 10

0821816896

ISBN 13

9780821816899

Generalized Frobenius Partitions

By: George E. Andrews

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Generalized Frobenius Partitions

By: George E. Andrews

This paper is devoted to the study of equilength two-line arrays of non-negative integers. These are called generalized Frobenius partitions. It is shown that such objects have numerous interactions with modular forms, Kloosterman quadratic forms, the Lusztig-Macdonald-Wall conjectures as well as with classical theta functions and additive number theory.

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American Mathematical Soc.

Published Date

ISBN 10

0821823027

ISBN 13

9780821823026

Category Theory 1991: Proceedings of the 1991 Summer Category Theory Meeting, Montreal, Canada

Proceedings of an International Summer Category Theory Meeting, Held June 23-30, 1991

By: Robert Andrew George Seely

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Category Theory 1991: Proceedings of the 1991 Summer Category Theory Meeting, Montreal, Canada

Proceedings of an International Summer Category Theory Meeting, Held June 23-30, 1991

By: Robert Andrew George Seely

Representing this diversity of the field, this book contains the proceedings of an international conference on category theory. The subjects covered here range from topology and geometry to logic and theoretical computer science, from homotopy to braids and conformal field theory. Although generally aimed at experts in the various fields represented, the book will also provide an excellent opportunity for nonexperts to get a feel for the diversity of current applications of category theory.

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462

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821860186

ISBN 13

9780821860182

$q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra

Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra

By: George E. Andrews

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$q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra

Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra

By: George E. Andrews

Integrates developments and related applications in $q$-series with a historical development of the field. This book develops important analytic topics (Bailey chains, integrals, and constant terms) and applications to additive number theory.

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Page Count

144

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821807161

ISBN 13

9780821807163

Geometric Analysis and Function Spaces

By: Steven George Krantz

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Geometric Analysis and Function Spaces

By: Steven George Krantz

This book brings into focus the synergistic interaction between analysis and geometry by examining a variety of topics in function theory, real analysis, harmonic analysis, several complex variables, and group actions. Krantz's approach is motivated by examples, both classical and modern, which highlight the symbiotic relationship between analysis and geometry. Creating a synthesis among a host of different topics, this book is useful to researchers in geometry and analysis and may be of interest to physicists, astronomers, and engineers in certain areas. The book is based on lectures presented at an NSF-CBMS Regional Conference held in May 1992.

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224

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821889257

ISBN 13

9780821889251

Dynamical Systems, Graphs, and Algorithms

By: George Osipenko

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Dynamical Systems, Graphs, and Algorithms

By: George Osipenko

This book describes a family of algorithms for studying the global structure of systems. By a finite covering of the phase space we construct a directed graph with vertices corresponding to cells of the covering and edges corresponding to admissible transitions. The method is used, among other things, to locate the periodic orbits and the chain recurrent set, to construct the attractors and their basins, to estimate the entropy, and more.

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Page Count

286

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Springer

Published Date

ISBN 10

3540355952

ISBN 13

9783540355953

Constructions of Lie Algebras and their Modules

By: George B. Seligman

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Constructions of Lie Algebras and their Modules

By: George B. Seligman

This book deals with central simple Lie algebras over arbitrary fields of characteristic zero. It aims to give constructions of the algebras and their finite-dimensional modules in terms that are rational with respect to the given ground field. All isotropic algebras with non-reduced relative root systems are treated, along with classical anisotropic algebras. The latter are treated by what seems to be a novel device, namely by studying certain modules for isotropic classical algebras in which they are embedded. In this development, symmetric powers of central simple associative algebras, along with generalized even Clifford algebras of involutorial algebras, play central roles. Considerable attention is given to exceptional algebras. The pace is that of a rather expansive research monograph. The reader who has at hand a standard introductory text on Lie algebras, such as Jacobson or Humphreys, should be in a position to understand the results. More technical matters arise in some of the detailed arguments. The book is intended for researchers and students of algebraic Lie theory, as well as for other researchers who are seeking explicit realizations of algebras or modules. It will probably be more useful as a resource to be dipped into, than as a text to be worked straight through.

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Page Count

203

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Springer

Published Date

ISBN 10

3540388648

ISBN 13

9783540388647

Introduction to Quantum Groups

By: George Lusztig

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Introduction to Quantum Groups

By: George Lusztig

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361

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Springer Science & Business Media

Published Date

ISBN 10

0817647171

ISBN 13

9780817647179

Representations of Finite Chevalley Groups

By: George Lusztig

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Representations of Finite Chevalley Groups

By: George Lusztig

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American Mathematical Soc.

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ISBN 10

0821816896

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9780821816899

Hecke Algebras with Unequal Parameters

By: George Lusztig

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Hecke Algebras with Unequal Parameters

By: George Lusztig

Hecke algebras arise in representation theory as endomorphism algebras of induced representations. One of the most important classes of Hecke algebras is related to representations of reductive algebraic groups over $p$-adic or finite fields. In 1979, in the simplest (equal parameter) case of such Hecke algebras, Kazhdan and Lusztig discovered a particular basis (the KL-basis) in a Hecke algebra, which is very important in studying relations between representation theory and geometry of the corresponding flag varieties. It turned out that the elements of the KL-basis also possess very interesting combinatorial properties. In the present book, the author extends the theory of the KL-basis to a more general class of Hecke algebras, the so-called algebras with unequal parameters. In particular, he formulates conjectures describing the properties of Hecke algebras with unequal parameters and presents examples verifying these conjectures in particular cases. Written in the author's precise style, the book gives researchers and graduate students working in the theory of algebraic groups and their representations an invaluable insight and a wealth of new and useful information.

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