Matrix algebras with involution and standard polynomial identities in symmetric variables

Abstract

Let m be a positive integer and M2m(F) be the algebra of 2m×2m matrices over an algebraically closed field of characteristic zero F endowed with the transpose or the symplectic involution. In this paper, we construct a basis B of M2m(F) over F such that ±B is a group whose elements are symmetric or skew with respect to the given involution. Moreover all elements of this basis commute or anti commute among themselves. The construction is based on a specific irreducible representation of ±B, an extra-special 2-group. As an application, this basis solves the problem on finding the minimal degree of a standard polynomial identity in symmetric variables of (M2m(F),s), where s is the symplectic involution.