Full Conformal Oscillator Representations of Orthogonal Lie Algebras and Combinatorial Identities

TL;DR

This paper develops a universal differential operator framework for highest-weight representations of orthogonal Lie algebras, enabling explicit module construction, tensor decomposition analysis, and deriving new combinatorial identities.

Contribution

It introduces a functor-based method to realize orthogonal Lie algebra modules via differential operators, linking representation theory with combinatorial identities.

Findings

Explicit differential operator realizations for o(2n+3) and o(2n+2) modules.

New combinatorial identities for Steinberg modules.

Application to tensor decompositions and conformal field theory equations.

Abstract

Zhao and the second author (2013) constructed a functor from o(k)-Mod to o(k + 2)-Mod. In this paper, we use the functor successively to obtain an universal first-order differential operator realization for any highest-weight representation of o(2n + 3) in (n + 1)^2 variables and that of o(2n + 2) in n(n + 1) variables. When the highest weight is dominant integral, we determine the corresponding finite-dimensional irreducible module explicitly. 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. We also find an equation of counting the dimension of an irreducible o(k + 2)-module in terms of certain alternating sum of the dimensions of irreducible o(k)-modules. In the case of the Steinberg modules, we obtain new combinatorial identities of classical type.