# The Polynomial Identities and Invariants of \(n ×n\) Matrices

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*Edward Formanek*

A co-publication of the AMS and CBMS

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.

#### Readership

#### Reviews & Endorsements

This monograph provides an excellent overview of the subject and can serve nonexperts as an introduction to the field and serve experts as a handy reference.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## The Polynomial Identities and Invariants of $n xn$ Matrices

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents iii4 free
- Introduction v6 free
- 1. Polynomial Identity Rings 18 free
- 2. The Standard Polynomial and the Amitsur-Levitzki Theorem 512
- 3. Central Polynomials 916
- 4. Posner's Theorem and the Ring of Generic Matrices 1522
- 5. The Center of the Generic Division Ring 2128
- 6. The Capelli Polynomial and Artin's Theorem 2734
- 7. Representation Theory of the Symmetric and General Linear Groups 3340
- 8. The First and Second Fundamental Theorems of Matrix Invariants 3946
- 9. Applications of the First and Second Fundamental Theorems 4552
- 10. The Nagata-Higman Theorem and Matrix Invariants 5158
- Monographs and Survey Articles with Materials on Polynomial Identities 5764
- Back Cover Back Cover165