The Invariant Theory of Matrices (University Lecture Series) by Corrado De Concini (PDF)

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 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 (1) the first fundamental theorem that describes a set of generators in the ring of invariants, and (2) 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.

User’s Reviews

Editorial Reviews: Review 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 MATHThe 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

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