Irreducible characters and Clebsch-Gordan series for the exceptional algebra $E_6$: an approach through the quantum Calogero-Sutherland model

TL;DR

This paper develops a method to express the quantum Calogero-Sutherland model for the exceptional Lie algebra E6 using irreducible characters, enabling computation of characters and Clebsch-Gordan series.

Contribution

It introduces a systematic procedure to derive Clebsch-Gordan series and re-express the model in terms of fundamental irreducible characters for E6.

Findings

Systematic method for Clebsch-Gordan series derivation

Re-expression of the quantum Hamiltonian in character variables

Enhanced ability to compute characters and series for E6

Abstract

We re-express the quantum Calogero-Sutherland model for the Lie algebra $E_6$ and the particular value of the coupling constant $\kappa=1$ by using the fundamental irreducible characters of the algebra as dynamical variables. For that, we need to develop a systematic procedure to obtain all the Clebsch-Gordan series required to perform the change of variables. We describe how the resulting quantum Hamiltonian operator can be used to compute more characters and Clebsch-Gordan series for this exceptional algebra.