$E_6$ unification model building I: Clebsch-Gordan coefficients of $27\otimes \ol{27}$

TL;DR

This paper computes the Clebsch-Gordan coefficients for the fundamental and conjugate representations of E6, providing essential tools for grand unified model building involving higher representations.

Contribution

It presents the first detailed calculation of Clebsch-Gordan coefficients for key E6 representations, aiding the analysis of operators in grand unified theories.

Findings

Computed Clebsch-Gordan coefficients for (27) ⊗ (27*) of E6.

Presented results in terms of dominant weight states.

Applied coefficients to construct the operator 27^3.

Abstract

In an effort to develop tools for grand unified model building for the Lie group $E_6$, in this paper we present the computation of the Clebsch-Gordan coefficients for the product (100000) $\otimes$ (000010), where (100000) is the fundamental 27-dimensional representation of $E_6$ and (000010) is its charged conjugate. The results are presented in terms of the dominant weight states of the irreducible representations in this product. These results are necessary for the group analysis of $E_6$ operators involving also higher representations, which is the next step in this project. In this paper we apply the results to the construction of the operator ${\bf 27}^3$.