The Invariant Theory of Matrices

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. 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