Multiplicative Invariant Theory
Multiplicative Invariant Theory

Author: Martin Lorenz

Publisher: Springer Science & Business Media

Published: 2005-12-08

Total Pages: 179

ISBN-13: 3540273581

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Multiplicative invariant theory, as a research area in its own right within the wider spectrum of invariant theory, is of relatively recent vintage. The present text offers a coherent account of the basic results achieved thus far.. Multiplicative invariant theory is intimately tied to integral representations of finite groups. Therefore, the field has a predominantly discrete, algebraic flavor. Geometry, specifically the theory of algebraic groups, enters through Weyl groups and their root lattices as well as via character lattices of algebraic tori. Throughout the text, numerous explicit examples of multiplicative invariant algebras and fields are presented, including the complete list of all multiplicative invariant algebras for lattices of rank 2. The book is intended for graduate and postgraduate students as well as researchers in integral representation theory, commutative algebra and, mostly, invariant theory.

Introduction to Complex Reflection Groups and Their Braid Groups
Introduction to Complex Reflection Groups and Their Braid Groups

Author: Michel Broué

Publisher: Springer

Published: 2010-01-28

Total Pages: 150

ISBN-13: 3642111750

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This book covers basic properties of complex reflection groups, such as characterization, Steinberg theorem, Gutkin-Opdam matrices, Solomon theorem and applications, including the basic findings of Springer theory on eigenspaces.

Shift-invariant Uniform Algebras on Groups
Shift-invariant Uniform Algebras on Groups

Author: Suren A. Grigoryan

Publisher: Springer Science & Business Media

Published: 2006-08-08

Total Pages: 287

ISBN-13: 3764376058

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This book on the theory of shift-invariant algebras is the first monograph devoted entirely to an outgrowth of the established theory of generalized analytic functions on compact groups. Associated subalgebras of almost periodic functions of real variables and of bonded analytic functions on the unit disc are carried along within the general framework.

Unitary Reflection Groups
Unitary Reflection Groups

Author: Gustav I. Lehrer

Publisher: Cambridge University Press

Published: 2009-08-13

Total Pages: 303

ISBN-13: 0521749891

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A unitary reflection is a linear transformation of a complex vector space that fixes each point in a hyperplane. Intuitively, it resembles the transformation an image undergoes when it is viewed through a kaleidoscope, or an arrangement of mirrors. This book gives a complete classification of all finite groups which are generated by unitary reflections, using the method of line systems. Irreducible groups are studied in detail, and are identified with finite linear groups. The new invariant theoretic proof of Steinberg's fixed point theorem is treated fully. The same approach is used to develop the theory of eigenspaces of elements of reflection groups and their twisted analogues. This includes an extension of Springer's theory of regular elements to reflection cosets. An appendix outlines links to representation theory, topology and mathematical physics. Containing over 100 exercises, ranging in difficulty from elementary to research level, this book is ideal for honours and graduate students, or for researchers in algebra, topology and mathematical physics. Book jacket.

The Classical Groups
The Classical Groups

Author: Hermann Weyl

Publisher: Princeton University Press

Published: 1946

Total Pages: 335

ISBN-13: 0691057567

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The author discusses symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from algebra, he examines the various properties of the groups. The book also covers topics such as matrix algebras, semigroups, commutators, and spinors, which are important in understanding the group-theoretic structure of quantum mechanics.

Actions and Invariants of Algebraic Groups
Actions and Invariants of Algebraic Groups

Author: Walter Ricardo Ferrer Santos

Publisher: CRC Press

Published: 2017-09-19

Total Pages: 479

ISBN-13: 1482239167

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Actions and Invariants of Algebraic Groups, Second Edition presents a self-contained introduction to geometric invariant theory starting from the basic theory of affine algebraic groups and proceeding towards more sophisticated dimensions." Building on the first edition, this book provides an introduction to the theory by equipping the reader with the tools needed to read advanced research in the field. Beginning with commutative algebra, algebraic geometry and the theory of Lie algebras, the book develops the necessary background of affine algebraic groups over an algebraically closed field, and then moves toward the algebraic and geometric aspects of modern invariant theory and quotients.

Semigroups in Algebra, Geometry, and Analysis
Semigroups in Algebra, Geometry, and Analysis

Author: Karl Heinrich Hofmann

Publisher: Walter de Gruyter

Published: 1995

Total Pages: 394

ISBN-13: 3110143194

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The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)