The Invariant Theory of Matrices
About this Title
Corrado De Concini, Sapienza Università di Roma, Rome, Italy and Claudio Procesi, Sapienza Università di Roma, Rome, Italy
Publication: University Lecture Series
Publication Year: 2017; Volume 69
ISBNs: 978-1-4704-4187-6 (print); 978-1-4704-4346-7 (online)
MathSciNet review: MR3726879
MSC: Primary 15A72; Secondary 13A50, 20G20
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.
Undergraduate and graduate students and researchers interested in linear algebra, representation theory, and invariant theory.
Table of Contents
- Introduction and preliminaries
- The classical theory
- Quasi-hereditary algebras
- The Schur algebra
- Matrix functions and invariants
- The Schur algebra of a free algebra