**University Lecture Series**

Volume: 69;
2017;
153 pp;
Softcover

MSC: Primary 15; 14; 20;

**Print ISBN: 978-1-4704-4187-6
Product Code: ULECT/69**

List Price: $44.00

AMS Member Price: $35.20

MAA Member Price: $39.60

**Electronic ISBN: 978-1-4704-4346-7
Product Code: ULECT/69.E**

List Price: $44.00

AMS Member Price: $35.20

MAA Member Price: $39.60

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#### Supplemental Materials

# The Invariant Theory of Matrices

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*Corrado De Concini; Claudio Procesi*

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

(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.

The print version of this book is only available in black and
white. The eBook is in full color.

#### Readership

Undergraduate and graduate students and researchers interested in linear algebra, representation theory, and invariant theory.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## The Invariant Theory of Matrices

- Cover Cover11
- Title page iii4
- Copyright iv5
- Contents v6
- Introduction and preliminaries 18
- The classical theory 1926
- Quasi-hereditary algebras 3946
- The Schur algebra 4956
- Matrix functions and invariants 8794
- Relations 107114
- The Schur algebra of a free algebra 131138
- Bibliography 145152
- General index 149156
- Symbol index 151158
- Back Cover Back Cover1162