  Using the projective oscillator representation of sl(n+1) and Shen's mixed
product for Witt algebras, Zhao and the second author (2011) constructed a new
functor from sl(n)-Mod to sl(n+1)-Mod. In this paper, we start from n = 2 and
use the functor successively to obtain a full projective oscillator realization
of any finite-dimensional irreducible representation of sl(n+1). The
representation formulas of all the root vectors of sl(n+1) are given in terms
of first-order differential operators in n(n+1)/2 variables. One can use the
result to study tensor decompositions of finite-dimensional irreducible modules
by solving certain first-order linear partial differential equations, and
thereby obtain the corresponding physically interested Clebsch-Gordan
coefficients and exact solutions of Knizhnik-Zamolodchikov equation in WZW
model of conformal field theory.
