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Using the projective oscillator representation of sl(n+1) and Shen's mixedproduct for Witt algebras, Zhao and the second author (2011) constructed a newfunctor from sl(n)-Mod to sl(n+1)-Mod. In this paper, we start from n = 2 anduse the functor successively to obtain a full projective oscillator realizationof any finite-dimensional irreducible representation of sl(n+1). Therepresentation formulas of all the root vectors of sl(n+1) are given in termsof first-order differential operators in n(n+1)/2 variables. One can use theresult to study tensor decompositions of finite-dimensional irreducible modulesby solving certain first-order linear partial differential equations, andthereby obtain the corresponding physically interested Clebsch-Gordancoefficients and exact solutions of Knizhnik-Zamolodchikov equation in WZWmodel of conformal field theory.

Full Projective Oscillator Representations of Special Linear Lie Algebras and Combinatorial Identities