Cocharacters for the weak polynomial identities of the Lie algebra of 3 × 3 skew-symmetric matrices

Abstract

Let so3(K) be the Lie algebra of 3×3 skew-symmetric matrices over a field K of characteristic 0. The ideal I(M3(K),so3(K)) of the weak polynomial identities of the pair (M3(K),so3(K)) consists of the elements f(x1,…,xn) of the free associative algebra K〈X〉 with the property that f(a1,…,an)=0 in the algebra M3(K) of all 3×3 matrices for all a1,…,an∈so3(K). The generators of I(M3(K),so3(K)) were found by Razmyslov in the 1980s. In this paper the cocharacter sequence of I(M3(K),so3(K)) is computed. In other words, the GLp(K)-module structure of the algebra generated by p generic skew-symmetric matrices is determined. Moreover, the same is done for the closely related algebra of SO3(K)-equivariant polynomial maps from the space of p-tuples of 3×3 skew-symmetric matrices into M3(K) (endowed with the conjugation action). In the special case p=3 the latter algebra is a module over a 6-variable polynomial subring in the algebra of SO3(K)-invariants of triples of 3×3 skew-symmetric matrices, and a free resolution of this module is found. The proofs involve methods and results of classical invariant theory, representation theory of the general linear group and explicit computations with matrices.