Cocharacters of UT_n(E)

Let F be a field of characteristic 0 and let E be the infinite-dimensional Grassmann algebra over F. In the first part of this paper we give an algorithm calculating the generating function of the cocharacter sequence of the n x n upper triangular matrix algebra UTn(E) with entries in E, lying in a strip of a fixed size. In the second part we compute the double Hilbert series H(E; T_k, Y_l) of E, then we define the (k, l)-multiplicity series of any PI-algebra. As an application, we derive from H(E; T_k, Y_l) an easy algorithm determining the (k, l)-multiplicity series of UT_n(E).