A note on cocharacter sequence of Jordan upper triangular matrix algebra

Let UJ(n)(F) be the Jordan algebra of n x n upper triangular matrices over a field F of characteristic zero. This paper is devoted to the study of polynomial identities satisfied by UJ(2)(F) and UJ(3)(F). In particular, the goal is twofold. On one hand, we complete the description of G-graded polynomial identities of UJ(2)(F), where G is a finite abelian group. On the other hand, we compute the Gelfand-Kirillov dimension of the relatively free algebra of UJ(2)(F) and we give a bound for the Gelfand-Kirillov dimension of the relatively free algebra of UJ(3)(F) (AU)