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

TL;DR

This paper develops a comprehensive projective oscillator framework for representing all finite-dimensional irreducible modules of sl(n+1), enabling explicit formulas for root vectors and applications in tensor decompositions and conformal field theory.

Contribution

It introduces a full realization of sl(n+1) representations using differential operators, extending previous functor constructions to all finite-dimensional irreducible modules.

Findings

Explicit differential operator formulas for all root vectors of sl(n+1)

Application to tensor product decompositions and Clebsch-Gordan coefficients

Solutions to Knizhnik-Zamolodchikov equations in conformal field theory

Abstract

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.