E_6 unification model building III. Clebsch-Gordan coefficients in E_6 tensor products of the 27 with higher dimensional representations

TL;DR

This paper computes Clebsch-Gordan coefficients for tensor products of the fundamental 27 representation of E_6 with higher-dimensional irreducible representations, aiding unification model building.

Contribution

It provides explicit Clebsch-Gordan coefficients for products of the 27 with higher-dimensional E_6 representations, facilitating higher dimensional operator analysis in E_6 models.

Findings

Computed Clebsch-Gordan coefficients for 27 with 78, 351, 351', and their conjugates

Simplifies the application of higher dimensional operators in E_6 models

Enhances the toolkit for unification model building involving E_6

Abstract

$E_6$ is an attractive group for unification model building. However, the complexity of a rank 6 group makes it non-trivial to write down the structure of higher dimensional operators in an $E_6$ theory in terms of the states labeled by quantum numbers of the Standard Model gauge group. In this paper, we show the results of our computation of the Clebsch-Gordan coefficients for the products of the {\bf 27} with irreducible representations of higher dimensionality: ${\bf 78}$, ${\bf 351}$, ${\bf 351^\prime}$, ${\bf \ol{351}}$, and ${\bf \ol{351^\prime}}$. Application of these results to $E_6$ model building involving higher dimensional operators is straightforward.