摘要：In this talk, I will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra H?(n) of a cyclic quiver ?(n). As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, and derive a recursive formula to compute them. We will further use the formula and the construction of certain monomial base for H?(n), together with the double Ringel–Hall algebra realisation of the quantum loop algebra, to develop some algorithms and to compute the canonical basis for Uv(gln)^+. As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most 2 for the quantum group Uv(gl2). This talk is based on a joint work with Jie Du.