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.
Author(s): Corrado De Concini, Claudio Procesi
Series: University Lecture Series, 69
Publisher: American Mathematical Society
Year: 2017
Language: English
Pages: 153
City: Providence
Cover
Title page
Table of Contents
protect oindent Introduction and preliminaries
1. Sectionformat {Introduction}{1}
2. Sectionformat {Preliminaries}{1}
Part I . protect enspace protect oindent The classical theory
3. Sectionformat {Representation theory}{1}
4. Sectionformat {Algebras with trace}{1}
Part II . protect enspace protect oindent Quasi-hereditary algebras
5. Sectionformat {Modules}{1}
6. Sectionformat {Good filtrations and quasi-hereditary algebras}{1}
Part III . protect enspace protect oindent The Schur algebra
7. Sectionformat {The Schur algebra}{1}
8. Sectionformat {Double tableaux}{1}
9. Sectionformat {Modules for the Schur algebra}{1}
10. Sectionformat {Rational $GL(m)$-modules}{1}
11. Sectionformat {Tensor products}{1}
Part IV . protect enspace protect oindent Matrix functions and invariants
12. Sectionformat {A reduction for invariants of several matrices }{1}
13. Sectionformat {Polarization and specialization}{1}
14. Sectionformat {Exterior products}{1}
15. Sectionformat {Matrix functions and invariants}{1}
Part V . protect enspace protect oindent Relations
16. Sectionformat {Relations}{1}
17. Sectionformat {Describing $K_m$}{1}
18. Sectionformat {$K_m$ versus $ ilde K_m$}{1}
Part VI . protect enspace protect oindent The Schur algebra of~a~free~algebra
19. Sectionformat {Preliminary facts}{1}
20. Sectionformat {The Schur algebra of the free algebra}{1}
Bibliography
General Index
Symbol Index
Back Cover
