The polynomial identities and invariants of n\(\times n\) matrices. (English) Zbl 0714.16001

Formanek, Edward

Regional Conference Series in Mathematics, 78. Providence, RI: American Mathematical Society (AMS). v, 55 p. $ 22.00 (1991).

This survey represents the most significant topics in the theory of polynomial identities in the matrix algebra over a field of characteristic zero. Classical results as well as recent ones can be found in the survey. It is divided into ten parts (lectures). The author considers first the famous Amitsur and Levitzki theorem. Then the constructions of central polynomials are described. Generic matrices are considered in parts 4 and 5. Some applications of central polynomials and the Capelli identities are given in part 6. Representations of the symmetric and general linear groups have proved their usefulness in the investigations of PI-algebras and part 7 is devoted to these methods. The last three parts concern the matrix invariants. One of the significant advantages of the survey is that both structural and combinatorial approaches in that theory are given. Many open problems and conjectures are mentioned with some remarks. The survey is written in an appropriate way and it will be very useful and interesting for all specialists in the theory of rings.

Reviewer: P.Koshlukov

MSC:

16-02	Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16R10	\(T\)-ideals, identities, varieties of associative rings and algebras
16R30	Trace rings and invariant theory (associative rings and algebras)
16R50	Other kinds of identities (generalized polynomial, rational, involution)

Keywords:

polynomial identities; matrix algebra; central polynomials; Generic matrices; Capelli identities; PI-algebras; matrix invariants