The Polynomial Identities and Invariants of $n \times n$ Matrices
About this Title
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 1991: Volume 78
ISBNs: 978-0-8218-0730-9 (print); 978-1-4704-2438-1 (online)
MathSciNet review: MR1088481
MSC: Primary 16R10; Secondary 16Kxx, 16R20, 16R30, 16S50
The theory of polynomial identities, as a well-defined field of study, began with a well-known 1948 article of Kaplansky. The field since developed along two branches: the structural, which investigates the properties of rings that satisfy a polynomial identity; and the varietal, which investigates the set of polynomials in the free ring that vanish under all specializations in a given ring.
This book is based on lectures delivered during an NSF-CBMS Regional Conference, held at DePaul University in July 1990, at which the author was the principal lecturer. The first part of the book is concerned with polynomial identity rings. The emphasis is on those parts of the theory related to $n\times n$ matrices, including the major structure theorems and the construction of certain polynomial identities and central polynomials for $n\times n$ matrices. The ring of generic matrices and its center is described. The author then moves on to the invariants of $n\times n$ matrices, beginning with the first and second fundamental theorems, which are used to describe the polynomial identities satisfied by $n\times n$ matrices.
One of the exceptional features of this book is the way it emphasizes the connection between polynomial identities and invariants of $n\times n$ matrices. Accessible to those with background at the level of a first-year graduate course in algebra, this book gives readers an understanding of polynomial identity rings and invariant theory, as well as an indication of problems and research in these areas.
Table of Contents
- Polynomial Identity Rings
- The Standard Polynomial and the Amitsur-Levitzki Theorem
- Central Polynomials
- Posner’s Theorem and the Ring of Generic Matrices
- The Center of the Generic Division Ring
- The Capelli Polynomial and Artin’s Theorem
- Representation Theory of the Symmetric and General Linear Groups
- The First and Second Fundamental Theorems of Matrix Invariants
- Applications of the First and Second Fundamental Theorems
- The Nagata-Higman Theorem and Matrix Invariants
- Monographs and Survey Articles with Materials on Polynomial Identities