Gradings on the algebra of upper triangular matrices of size three

Let UT3(F) be the algebra of 3×3 upper triangular matrices over a field F. On UT3(F), up to isomorphism, there are at most five non-trivial elementary gradings and we study the graded polynomial identities for such gradings. In case F is of characteristic zero we give a complete description of the space of multilinear graded identities in the language of Young diagrams through the representation theory of a Young subgroup of Sn. We finally compute the multiplicities in the graded cocharacter sequence for every elementary G-grading on UT3(F).