Matrix Algebras with Transpose or Symplectic Involution and their *-Polynomial Identities 1

Abstract: We look at the theory of *-polynomial identi-ties of the algebra of n X n matrices over a field. The represen-tation theory of the hyperoctahedral group and of the general linear group are applied for a quantitative study of the theory in characteristic zero. We examine the problem of determin-ing *-polynomial identities of minimal degree for symplectic and transpose involution and new *-polynomial identities of degree 2n- 1 are constructed