The graded identities of upper triangular matrices of size two

Let UT2 be the algebra of 2 x 2 upper triangular matrices over a field F. We first classify all possible gradings on UT2 by a group G. It turns out that, up to isomorphism. there is only one non-trivial grading and we study all the graded polynomial identities for such algebra. 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 the hyperoctahedral group. We finally establish a result concerning the rate of growth of the identities for such algebra by proving that its sequence of graded codimensions has almost polynomial growth. (C) 2001 Elsevier Science B.V. All rights reserved.