Multiplicaton formulas and canonical basis for quantum affine gl_n

We will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra ${\frak H}_\Delta(n)$ of a cyclic quiver $\Delta(n)$ given in \cite[Thm~4.5]{DuFu2015quantum}. As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established in \cite{Guo1995hallpoly} and \cite{Ringel1993composition}, and derive a recursive formula to compute them. We will further use the formula and the construction of certain monomial base for ${\mathfrak H}_\Delta(n)$ given in \cite{DengDuXiao2007generic}, together with the double Ringel--Hall algebra realisation of the quantum loop algebra $U_v(\hat{gl}_n)$ in \cite{DengDuFu2012double}, to develop some algorithms and to compute the canonical basis for $U_v(\hat{gl}_n)^+$. 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 $U_v(\hat{gl}_n)$.