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Computational approach to polynomial identities of matrices – a survey. (English) Zbl 1067.16041

Giambruno, Antonio (ed.) et al., Polynomial identities and combinatorial methods, Pantelleria, Italy. New York, NY: Marcel Dekker (ISBN 0-8247-4051-3/pbk). Lect. Notes Pure Appl. Math. 235, 141-178 (2003).
This is a survey on computational techniques for working with polynomial identities of \(n\times n\) matrices, mainly over a field of characteristic zero.
Recall that the identities of \(2\times 2\) matrices are well understood, but we are very far from having a complete picture about the identities of matrices of size \(3\times 3\) or greater.
The authors outline computational methods that have been applied to attack the following open problems: the minimal degree of a polynomial identity that does not follow from the standard identity of degree \(2n\); the minimal degree of a central polynomial; the minimal degree of a \(*\)-polynomial identity; the cocharacter series of the algebra of \(n\times n\) matrices.
The paper contains a rather complete account on what is known about low degree explicit identities and central polynomials of \(n\times n\) matrices.
For the entire collection see [Zbl 1027.00013].

MSC:

16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16R50 Other kinds of identities (generalized polynomial, rational, involution)
20C30 Representations of finite symmetric groups