Full conformal oscillator representations of odd orthogonal Lie algebras and combinatorial identities

Abstract

Zhao and the second author (2013) constructed a functor from $o\left(k\right)$-Mod to $o(k+2)$-Mod. In this paper, we use the functor successively to obtain an inhomogeneous first-order differential operator realization for any highest-weight representation of $o(2n+3)$ in ${(n+1)}^2$ variables. When the highest weight is dominant integral, we find a span set for the corresponding finite-dimensional irreducible module. One can use the result to study tensor decomposition 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. We also find an equation of counting the dimension of an irreducible $o(2n+3)$-module in terms of certain alternating sum of the dimensions of irreducible $o(2n+1)$-modules. In the case of the Steinberg modules, we obtain new combinatorial identities of classical type.