
Journal of Lie Theory 33 (2023), No. 4, 11391176 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 9820230052@nankai.edu.cn Xiaoping Xu Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, P. R. China xiaoping@math.ac.cn [Abstractpdf] 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) 132154] 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 finitedimensional 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 firstorder differential operators in $n(n+1)/2$ variables. One can use the result to study tensor decompositions of finitedimensional simple modules by solving certain firstorder linear partial differential equations, and thereby obtain the corresponding physically interested ClebschGordan coefficients and exact solutions of KnizhnikZamolodchikov 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. [ Fulltextpdf (240 KB)] for subscribers only. 