On the minimal degree of the *-polynomial identities for the matrix algebra of order 6 with symplectic involution

Abstract

We consider the *-polynomial identities in symmetric variables for the 6×6 matrix algebraM6(K,*) with symplectic involution over a fieldK of characteristic O. We prove that their minimal degree is greater than 8. It is known that in the general case forM2n(K,*), ifn>1, this degree is greater than 2n+1. Our result is based on the fact that the *-identities in symmetric variables are also ordinary polynomial identities forM3(K) and the description of the identities of degree 8 forM3(K) is known. The proof uses the representation theory of the general linear group and involves computer calculations.