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Journal of Lie Theory 33 (2023), No. 4, 1139--1176
Copyright Heldermann Verlag 2023

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

Zhenyu Zhou
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, P. R. China

Xiaoping Xu
Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, P. R. China


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.

Keywords: Special linear Lie algebra, projective oscillator representation, simple module, singular vectors, combinatorial identities.

MSC: 17B10; 05A19.

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