Explicit Calculations of Tensor Product Coefficients for E7

We propose a new method to calculate coupling coefficients of E7 tensor products. Our method is based on explicit use of E7 characters in the definition of a tensor product. When applying Weyl character formula for E7 Lie algebra, one needs to make sums over 2903040 elements of E7 Weyl group. To implement such enormous sums, we show we have a way which makes their calculations possible. This will be accomplished by decomposing an E7 character into 72 participating A7 characters.


Introduction
Let G 7 =E 7 ,A 7 and Λ, Λ′ be two dominant weights of G 7 where R(Λ) and R(Λ′) are corresponding irreducible representations. For general terms, we follow the book of Humphreys [1] as ever.
Tensor product of these two irreducible representations is defined by, where S(λ+λ′) is the set of Λ+Λ′ subdominants and t(λ<Λ+Λ′) s are tensor coupling coefficients. Though Steinberg formula is the best known way, a natural way to calculate tensor coupling coefficients is also to solve the equation The crucial fact here is that where S denotes order of set S. It is easy to see then to implement the sum in (I.4) would not be realizable explicitly. We, instead, propose 72 specifically chosen Weyl reflections which give us A 7 dominant weights participating within the same E 7 Weyl orbit W(Λ + ) for any E 7 dominant weight Λ + . As it is shown in the next section, this makes the evaluation of (I.4) realizable for E 7 but in terms of 72 A 7 characters and hence easily implementable.

A 7 Decomposition of E 7 Lie Algebra
For i=1,2…,7, let λ i 's and α i 's be respectively the fundamental dominant weigths and simple roots of A 7 Lie algebra with the following Dynkin diagram (Figure 1).  is A 7 Weyl vector. We suggest following relations allows us to embed A 7 subalgebra into E 7 algebra: This essentially means that  which tells us that there are at most 72 A 7 dominant weights inside a Weyl orbit W(Λ + ). Note here that it is exactly 72 when Λ + is a strictly dominant weight. From the now on, W(µ) will always denotes the Weyl orbit of a weight µ.
As the main point of view of this work, we present in appendix, 72 Weyl reflections to give 72 A 7 dominant weights participating in the same E 7 Weyl orbit W(Λ + ) when they are exerted on the dominant weight Λ + . To this end, the Weyl reflections with respect to simple roots α i will be called simple reflections σ i . We extend multiple products of simple reflections trivially by For s=1,…72, ∑(s)'s are 72 Weyl reflections mentioned above. As will also be seen by their definitions that, 1) ε(σ(s))=+1 s=1,2,…,36 2) ε(σ(s))=−1 s=37,38,…,72

Calculating Tensor Coupling Coefficients
Let us proceed in the instructive example To this end, we should care about specialization of formal exponentials [2]. Let us consider the so-called Fundamental Weights µ I which are defined for (I=1,…8) as in the following [3]: α i 's here are A 7 simple roots mentioned above and the best way to calculate A 7 and hence E 7 characters is to use the specialization in terms of parameters which are subjects of the condition µ 1 +µ 2 +… +µ 8 =0 or To exemplify (I.3) for E 7 , we would like to give detailed calculation of Ch(Λ 3 +Λ 4 ). By applying 72 specifically chosen Weyl reflections on strictly dominant weight is the permutation group of 8 objects.
To display our result here, we use the following specialization of formal exponentials with only one free parameter x:

Conclusion
One should note however that, the 1-parameter specialization (III.5) above is not enough to find all the tensor coupling coefficients completely so we saw that at least 3-parameters specializations will be sufficient, which we used the following one.