Matrix algebras with involution and standard polynomial identities in symmetric variables

Abstract Let m be a positive integer and M 2 m ( F ) be the algebra of 2 m × 2 m 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 M 2 m ( 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 ( M 2 m ( F ) , s ) , where s is the symplectic involution.