Benanti, Francesca; Demmel, James; Drensky, Vesselin; Koev, Plamen 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]. Reviewer: Matyas Domokos (Budapest) Cited in 3 Documents 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 Keywords:polynomial identities of matrices; consequences of standard identities; highest weight vectors; weak polynomial identities; central polynomials; identities with involution × Cite Format Result Cite Review PDF