Inner Product in Highest-Weight Representation

TL;DR

This paper introduces an iterative algorithm for efficiently computing inner products in highest-weight representations of classical groups, with applications to unitarity, norms, and affine Lie algebras.

Contribution

It presents a novel iterative method for calculating inner products in highest-weight representations, applicable to affine Lie algebras and related to solutions of Toda systems.

Findings

Algorithm efficiently computes inner products.

Determined norms of specific states.

Fully characterized inner products in minuscule representations.

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.