Department of Physics, Faculty of Sciences and Letters, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey
Received Date: October 28, 2016; Accepted Date: February 17, 2017; Published Date: February 27, 2017
Citation: Gungormez M, Karadayi HR (2017) Explicit Calculations of Tensor Product Coefficients for E7. J Generalized Lie Theory Appl 11: 254. doi: 10.4172/1736- 4337.1000254
Copyright: © 2017 Gungormez M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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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.
Coupling coefficients; Lie algebra; Irreducible representations; Subdominants; Tensor coupling coefficients
Let G7=E7,A7 and Λ, Λ′ be two dominant weights of G7 where R(Λ) and R(Λ′) are corresponding irreducible representations. For general terms, we follow the book of Humphreys  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
for tensor coupling coefficients. Ch(λ) here is the character of an irreducible representation R(λ) which corresponds to a dominant weight λ and it is defined by the famous Weyl Character formula:
where for a weight μ in general
W(G7) is the Weyl Group of G7 and each and every element σ is the so-called Weyl reflection while ε(σ) denotes its sign and eσ (λ ++ ) ’s here are known as formal exponentials.Thoroughout this work, we assume λ++ λ ++ denotes a strictly dominant weight defined for a dominant λ+ by
where is the Weyl vector of G7.
The crucial fact here is that
where 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 A7 dominant weights participating within the same E7 Weyl orbit W(Λ+) for any E7 dominant weight Λ+. As it is shown in the next section, this makes the evaluation of (I.4) realizable for E7 but in terms of 72 A7 characters and hence easily implementable.
For i=1,2…,7, let λi’s and αi’s be respectively the fundamental dominant weigths and simple roots of A7 Lie algebra with the following Dynkin diagram (Figure 1).
where is A7 Weyl vector and Λ i’ s be fundamental dominant weights of E7 Lie algebra in according with the following Dynkin diagram, (Figure 2).
where is A7 Weyl vector. We suggest following relations allows us to embed A7 subalgebra into E7 algebra:
This essentially means that
which tells us that there are at most 72 A7 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 A7 dominant weights participating in the same E7 Weyl orbit W(Λ+) when they are exerted on the dominant weight Λ+. To this end, the Weyl reflections with respect to simple roots αi will be called simple reflections σi. We extend multiple products of simple reflections trivially by
For s=1,…72, Σ(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
Let us proceed in the instructive example
of (I.1). One can see that there are 39 sub-dominant weights θj of Λ3+Λ4 :
αi’s here are A7 simple roots mentioned above and the best way to calculate A7 and hence E7 characters is to use the specialization in terms of parameters which are subjects of the condition μ1+μ2+… +μ8=0 or
To exemplify (I.3) for E7, we would like to give detailed calculation of Ch(Λ3+Λ4). By applying 72 specifically chosen Weyl reflections on strictly dominant weight , one can see we have the following decompositions:
A7 characters Ch(vk) ’s are defined by
Note here that W(A7) 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:
In this specialization, one obtains the following one-parameter characters:
Now, one can see that the characters above fulfill the following equation:
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.
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