Invariant Theory and Superalgebras

Invariant Theory and Superalgebras PDF

Author: Frank D. Grosshans

Publisher: American Mathematical Soc.

Published: 1987-12-31

Total Pages: 106

ISBN-13: 0821807196

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This book brings the reader to the frontiers of research in some topics in superalgebras and symbolic method in invariant theory. Superalgebras are algebras containing positively-signed and negatively-signed variables. One of the book's major results is an extension of the standard basis theorem to superalgebras. This extension requires a rethinking of some basic concepts of linear algebra, such as matrices and coordinate systems, and may lead to an extension of the entire apparatus of linear algebra to ``signed'' modules. The authors also present the symbolic method for the invariant theory of symmetric and of skew-symmetric tensors. In both cases, the invariants are obtained from the symbolic representation by applying what the authors call the umbral operator. This operator can be used to systematically develop anticommutative analogs of concepts of algebraic geometry, and such results may ultimately turn out to be the main byproduct of this investigation. While it will be of special interest to mathematicians and physicists doing research in superalgebras, invariant theory, straightening algorithms, Young bitableaux, and Grassmann's calculus of extension, the book starts from basic principles and should therefore be accessible to those who have completed the standard graduate level courses in algebra and/or combinatorics.

Algebraic Homogeneous Spaces and Invariant Theory

Algebraic Homogeneous Spaces and Invariant Theory PDF

Author: Frank D. Grosshans

Publisher: Springer

Published: 2006-11-14

Total Pages: 158

ISBN-13: 3540696172

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The invariant theory of non-reductive groups has its roots in the 19th century but has seen some very interesting developments in the past twenty years. This book is an exposition of several related topics including observable subgroups, induced modules, maximal unipotent subgroups of reductive groups and the method of U-invariants, and the complexity of an action. Much of this material has not appeared previously in book form. The exposition assumes a basic knowledge of algebraic groups and then develops each topic systematically with applications to invariant theory. Exercises are included as well as many examples, some of which are related to geometry and physics.

Invariant Theory

Invariant Theory PDF

Author: Mara D. Neusel

Publisher: American Mathematical Soc.

Published: 2007

Total Pages: 326

ISBN-13: 0821841327

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This book presents the characteristic zero invariant theory of finite groups acting linearly on polynomial algebras. The author assumes basic knowledge of groups and rings, and introduces more advanced methods from commutative algebra along the way. The theory is illustrated by numerous examples and applications to physics, engineering, numerical analysis, combinatorics, coding theory, and graph theory. A wide selection of exercises and suggestions for further reading makes the book appropriate for an advanced undergraduate or first-year graduate level course.

Invariant Theory of Finite Groups

Invariant Theory of Finite Groups PDF

Author: Mara D. Neusel

Publisher: American Mathematical Soc.

Published: 2002

Total Pages: 388

ISBN-13: 9780821829165

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The questions that have been at the center of invariant theory since the 19th century have revolved around the following themes: finiteness, computation, and special classes of invariants. This book begins with a survey of many concrete examples chosen from these themes in the algebraic, homological, and combinatorial context. In further chapters, the authors pick one or the other of these questions as a departure point and present the known answers, open problems, and methods andtools needed to obtain these answers. Chapter 2 deals with algebraic finiteness. Chapter 3 deals with combinatorial finiteness. Chapter 4 presents Noetherian finiteness. Chapter 5 addresses homological finiteness. Chapter 6 presents special classes of invariants, which deal with modular invarianttheory and its particular problems and features. Chapter 7 collects results for special classes of invariants and coinvariants such as (pseudo) reflection groups and representations of low degree. If the ground field is finite, ad The book contains numerous examples to illustrate the theory, often of more than passing interest, and an appendix on commutative graded algebra, which provides some of the required basic background. There is an extensive reference list to provide the reader withorientation to the vast literature.

Lectures on Invariant Theory

Lectures on Invariant Theory PDF

Author: Igor Dolgachev

Publisher: Cambridge University Press

Published: 2003-08-07

Total Pages: 244

ISBN-13: 9780521525480

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The primary goal of this 2003 book is to give a brief introduction to the main ideas of algebraic and geometric invariant theory. It assumes only a minimal background in algebraic geometry, algebra and representation theory. Topics covered include the symbolic method for computation of invariants on the space of homogeneous forms, the problem of finite-generatedness of the algebra of invariants, the theory of covariants and constructions of categorical and geometric quotients. Throughout, the emphasis is on concrete examples which originate in classical algebraic geometry. Based on lectures given at University of Michigan, Harvard University and Seoul National University, the book is written in an accessible style and contains many examples and exercises. A novel feature of the book is a discussion of possible linearizations of actions and the variation of quotients under the change of linearization. Also includes the construction of toric varieties as torus quotients of affine spaces.

Actions and Invariants of Algebraic Groups

Actions and Invariants of Algebraic Groups PDF

Author: Walter Ricardo Ferrer Santos

Publisher: CRC Press

Published: 2017-09-19

Total Pages: 542

ISBN-13: 1351644777

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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.

Computational Invariant Theory

Computational Invariant Theory PDF

Author: Harm Derksen

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 272

ISBN-13: 3662049589

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This book, the first volume of a subseries on "Invariant Theory and Algebraic Transformation Groups", provides a comprehensive and up-to-date overview of the algorithmic aspects of invariant theory. Numerous illustrative examples and a careful selection of proofs make the book accessible to non-specialists.

Algebraic Geometry IV

Algebraic Geometry IV PDF

Author: A.N. Parshin

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 291

ISBN-13: 366203073X

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Two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory, by well-known experts in the fields. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.

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