
The Invariant Theory of Matrices
American Mathematical Society (Publisher)
Published on 30. December 2017
Book
Paperback/Softback
153 pages
978-1-4704-4187-6 (ISBN)
Description
This book gives a unified, complete, and self-contained exposition of the main algebraic theorems of invariant theory for matrices in a characteristic free approach. More precisely, it contains the description of polynomial functions in several variables on the set of $m\times m$ matrices with coefficients in an infinite field or even the ring of integers, invariant under simultaneous conjugation.
Following Hermann Weyl's classical approach, the ring of invariants is described by formulating and proving the first fundamental theorem that describes a set of generators in the ring of invariants, and the second fundamental theorem that describes relations between these generators. The authors study both the case of matrices over a field of characteristic 0 and the case of matrices over a field of positive characteristic. While the case of characteristic 0 can be treated following a classical approach, the case of positive characteristic (developed by Donkin and Zubkov) is much harder. A presentation of this case requires the development of a collection of tools. These tools and their application to the study of invariants are exlained in an elementary, self-contained way in the book.
Following Hermann Weyl's classical approach, the ring of invariants is described by formulating and proving the first fundamental theorem that describes a set of generators in the ring of invariants, and the second fundamental theorem that describes relations between these generators. The authors study both the case of matrices over a field of characteristic 0 and the case of matrices over a field of positive characteristic. While the case of characteristic 0 can be treated following a classical approach, the case of positive characteristic (developed by Donkin and Zubkov) is much harder. A presentation of this case requires the development of a collection of tools. These tools and their application to the study of invariants are exlained in an elementary, self-contained way in the book.
Reviews / Votes
The present book is a nice and introductory reference to graduate students or researchers who are working in the field of representation and invariant theory." - Yin Chen, Zentralblatt MATH"The choices made by the authors permit them to highlight the main results and also to keep the material within the reach of an interested reader. At the same time, the book remains open-ended with precise pointers to the literature on other approaches and the cases not treated here." - Felipe Zaldivar, MAA Reviews
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
Dimensions
Height: 254 mm
Width: 178 mm
Weight
295 gr
ISBN-13
978-1-4704-4187-6 (9781470441876)
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Schweitzer Classification
Persons
Corrado De Concini, Sapienza Universita di Roma, Rome, Italy.
Claudio Procesi, Sapienza Universita di Roma, Rome, Italy.
Claudio Procesi, Sapienza Universita di Roma, Rome, Italy.
Content
Introduction and preliminaries
The classical theory
Quasi-hereditary algebras
The Schur algebra
Matrix functions and invariants
Relations
The Schur algebra of a free algebra
Bibliography
General index
Symbol index.
The classical theory
Quasi-hereditary algebras
The Schur algebra
Matrix functions and invariants
Relations
The Schur algebra of a free algebra
Bibliography
General index
Symbol index.