Journal of Lie Theory

Using the projective oscillator representation of $\mathfrak{sl}(n+1)$ and Shen's mixed product for Witt algebras, Y. Zhao and the second author [{\it Generalized projective representations for $\mathfrak{sl}(n+1)$}, J. Algebra 328 (2011) 132--154] constructed a new functor from $\mathfrak{sl}(n)$-{\bf Mod} to $\mathfrak{sl}(n+1)$-{\bf 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 $\mathfrak{sl}(n+1)$. The representation formulas of all the root vectors of $\mathfrak{sl}(n+1)$ 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 simple 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.