Computational Approach to Polynomial Identities of Matrices — a Survey

Abstract

We present a survey on polynomial identities of matrices over a field of characteristic 0 from computational point of view. We describe several computational methods for calculation with polynomial identities of matrices and related objects. Among the other applications, these methods have been successfully used: The work of this author was partially supported by MURST of Italy. The work of this author was partially supported by Grant MM-1106/2001 of the Bulgarian Foundation for Scientific Research. (i) to show that all polynomial identities of degree 2n + 2 for the n× n matrix algebra for n = 3, 4, 5 are consequences of the standard identity s2n; (ii) to obtain upper bounds for the multiplicities of the irreducible S9-characters of the multilinear identities of degree 9 in 3×3 matrices; (iii) in the discovery of new central polynomials of low degree for matrices of order 3 and 4; (iv) to obtain upper bounds for the multiplicities of the irreducible S9-characters of ∗-polynomial identities of degree 9 in symmetric variables only for 6× 6 matrices with symplectic involution.