Gradings on the algebra of upper triangular matrices and their graded identities

Let K be an infinite field and let UTn(K) denote the algebra of n×n upper triangular matrices over K. We describe all elementary gradings on this algebra. Further we describe the generators of the ideals of graded polynomial identities of UTn(K) and we produce linear bases of the corresponding relatively free graded algebras. We prove that one can distinguish the elementary gradings by their graded identities. We describe bases of the graded polynomial identities in several “typical” cases. Although in these cases we consider elementary gradings by cyclic groups, the same methods serve for elementary gradings by any finite group.