Identities and cocharacters of the algebra of 3 × 3 matrices with non-trivial grading and transpose graded involution

Abstract

Let F be a field of characteristic zero and M2,1(F) the algebra of 3×3 matrices over F endowed with non-trivial Z2-grading. The transpose involution t on M2,1(F) preserves the homogeneous components of the grading and so, we consider (M2,1(F),t) as a superalgebra with graded involution. We study the (Z2,⁎)-identities of this algebra and make explicit the decomposition of the space of multilinear (Z2,⁎)-identities into the sum of irreducibles under the action of the group (Z2×Z2)≀Sn in order to determine all the irreducible characters appearing with non-zero multiplicity in the decomposition of the ⁎-graded cocharacter of (M2,1(F),t). Along the way, using the representation theory of the general linear group, we determine all the (Z2,⁎)-identities of (M2,1(F),t) up to degree 3.