Inner Product in Highest-Weight Representation

Abstract: In this paper, we study the inner product of states corresponding to weights of finite-dimensional highest-weight representations of classical groups. We prove that the action of the raising operators would reduce a state of hight-weight representation to a linear combination of states of highest-weight representation, with the level decreased by one. Then we propose an iterative algorithm for calculating the inner products of sates efficiently, revealing the intricate structure of the representation. As applications, we discuss the unitarity of the highest-weight representation and propose a conjecture. We determine the norm of a special class of states. And we completely determine the inner products of states of the minuscule representations. The algorithm proposed is applicable to the highest-weight representation of affine Lie algebra without modifications. These findings can be used to study the construction of solutions to Kapustin-Witten equations which are based on the fundamental solutions of Toda systems.