*-Polynomial Identities of Matrices

Abstract

Let m > 1 be a positive integer, F be a field, K2m( F, t) be the subspace of M2 m(F) of matrices skew-symmetric with respect to the transpose involution, and H2 m(F, s) be the subspace of matrices symmetric with respect to the symplectic involution. We show that K 2m(F, t) and H 2m(F, s) both satisfy qm, a multilinear identity of degree 4m-3. As corollaries we obtain both new proofs and refinements of theorems of Kostant and Rowen concerning s 4m-2, a so-called "standard" polynomial identity for K2m( F, t) and H2m( F, s), and s4m -4, a "standard" polynomial identity for K2 m-1(F, t).