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