Clebsch–Gordan coefficients for E6 and SO(10) unification models

Abstract

We illustrate here a new method for computing Clebsch–Gordan coefficients (CGC) for E6 by computing CGC for the product 27⊗27 of the irreducible representation (100000) of E6 with itself. These CGC are calculated thrice: once in a weight vector basis independent of any semisimple subgroup, then in a basis which refers to SO(10)⊆E6, and finally in a basis referring to SU(5)⊆SO(10)⊆E6.

E 6 unification model building II. Clebsch–Gordan coefficients of 78⊗78

“…Despite the fact that it has received consideration for over twenty years [1], E 6 model building has not been extensively developed due to mathematical complexities associated with a rank 6 exceptional Lie group. The Clebsch-Gordan coefficients (CGCs), for instance, have only been known for the products of two fundamental irreducible representations (irreps) of the lowest dimensionality: 27 or 27 [2,3]. To our knowledge, the CGCs for higher dimensional irreps of E 6 have never been computed.…”

“…As discussed in [2], it is sufficient to present the tensor decomposition of the dominant weight states in the product. The CGCs of the other states can be obtained with the help of the charged conjugate operators introduced in [9] (or by direct lowerings).…”

E 6 unification model building. III. Clebsch–Gordan coefficients in E6 tensor products of the 27 with higher dimensional representations

“…|11 |12 |13 |14 |15 |16 |17 |18 |19 |20 |21 |22 |23 |24 |25 |26 |27 |28 |29 |30 |31 |32 |100000 |100000 1 |100000 |100000 2 |100000 |100000 3 |100000 |100000 4 1 |100000 |100000 6 |110000 |110000 1 1 |110000 |110000 2 |110000 |110000 3 |110000 |110000 4 1 1 |110000 |110000 6 |011000 |011000 1 1 |011000 |011000 2 |011000 |011000 3 |011000 |011000 4 1 1 |011000 |011000 6…”

“…The product 27 ⊗ 27 = 351 ′ ⊕ 351 ⊕ 27 is conjugated to the product studied in ref. [4]. We refer to this work to claim that all degenerate weight subspaces in the 351 ′ (or 351) are of the same dimensionality and that the degenerate weights follow the weight system of the 27.…”

“…[3] As for particular computations, to our knowledge only the Clebsch-Gordan decompositions of 27 ⊗ 27, 27 ⊗ 27, and 78 ⊗ 78 are presently available in the literature. [4,5,6]. (We note in passing that a separate paper [7] obtains a subset of the results needed for the operator 27 3 78 by studying a branching chain of E 6 .…”