On the [formula omitted]-cocharacter sequence of [formula omitted] matrices

Abstract

Let M 3(F) be the algebra of 3×3 matrices with involution * over a field F of characteristic zero. We study the ∗-polynomial identities of M 3(F) , where ∗=t is the transpose involution, through the representation theory of the hyperoctahedral group B n . After decomposing the space of multilinear ∗-polynomial identities of degree n under the B n -action, we determine which irreducible B n -modules appear with non-zero multiplicity. In symbols, we write the nth ∗-cocharacter χ n(M 3(F),*)=∑ r=0 n ∑ λ⊢r,h(λ)⩽6 μ⊢n−r,h(μ)⩽3 m λ,μχ λ,μ, where λ and μ are partitions of r and n−r, respectively, χ λ,μ is the irreducible B n -character associated to the pair (λ,μ) and m λ,μ⩾0 is the corresponding multiplicity. We prove that, for any n, the multiplicities m λ,μ are always non-zero except the trivial case λ=(1 6) and μ=∅.