On the *-cocharacter sequence of 3×3 matrices Original Research Article

Let M3(F) be the algebra of 3×3 matrices with involution * over a field F of characteristic zero. We study the *-polynomial identities of M3(F), where *=t is the transpose involution, through the representation theory of the hyperoctahedral group Bn. After decomposing the space of multilinear *-polynomial identities of degree n under the Bn-action, we determine which irreducible Bn-modules appear with non-zero multiplicity. In symbols, we write the nth *-cocharacterimagewhere λ and μ are partitions of r and n−r, respectively, χλ,μ is the irreducible Bn-character associated to the pair (λ,μ) and mλ,μgreater-or-equal, slanted0 is the corresponding multiplicity. We prove that, for any n, the multiplicities mλ,μ are always non-zero except the trivial case λ=(16) and μ=empty set︀.