# Isomorphism symmetry group of 6j symbol

• Yoran91
In summary, the 6j symbol is a mathematical concept that is related to the symmetry group of a regular tetrahedron. To understand this relation, one can look at the permutations of the columns and upper/lower elements in a column, which correspond to the symmetries of the tetrahedron. However, it was later discovered that there are also other symmetries, known as Regge symmetries, that play a role in the 6j symbol. Ultimately, the 6j symbol is non-zero only when the spins of the representations correspond to the edges of a tetrahedron.
Yoran91
Hi everyone,

I read in 'Angular momentum in Quantum Mechanics' by A.R Edmonds that the symmetry group of the 6j symbol is isomorphic to the symmetry group of a regular tetahedron.

Is there an easy way of seeing this? I've tried working out what the symmetry relations of the 6j symbol do to the associated tetahedron, but I can't see the bigger picture. I have also tried constructing an isomorphism to S4 (which is isomorphic to the full symmetry group of a regular tetahedron), but no succes.

Can anyone help me out?

I had a look at wikipedia:
http://en.wikipedia.org/wiki/6-j_symbol
So the 6j's are invariant under a permutation of the colums and is S3. I start from (j1 j2 j3 / j4 j5 j6), then every column is described by the upper j alone, which I take as the label of a vertex in the tetrahedron. So apparently this group is isomorphic to the interchange of any of the three vertices in one face of the tetrahedron.
But is also invariant under the exchange of the elements in one column. You can see that this is equivalent to the choice of another triangular face where e.g. J1 is replaced by j4.
So this yields all the symmetry operations of the tetrahedron.

More mathematically, the permutations of the columns span the normal sub-group S3 (C3v) of S4 (Td) and the permutation of upper/ lower elements in a column from the corresponding coset.

Ah, I see! Thank you for your help.

Can you explain that last part a bit more? I see that the subgroup generated by the column permutations is isomorphic to [TEX]S_3[/TEX], but I don't see why the subgroup generated by the permutation of the upper and lower arguments is isomorphic to the corresponding coset.

Apparently the second class of symmetries specified in Wikipedia is not correct.
According to:

The 6j's are only invariant under a permutation of the upper and lower j's in two columns, which leaves only half of the permutations claimed in wikipedia. Then the isomorphism with the tetrahedral group becomes clear identifying the j's with the edges (not vertices) of a tetrahedron).

Interestingly, this are not all the symmetries of the 6j symbol. There are so-called Regge symmetries which were discovered later:

The easiest way is probably to recognize that a 6j is non-zero if and only if the value of the representations' spins correspond to the edges of a tetrahedron. This causes the four triangle inequalities (which are now the faces of the tetrahedron) to be automatically fulfilled.

## 1. What is the Isomorphism Symmetry Group of the 6j symbol?

The Isomorphism Symmetry Group of the 6j symbol is a mathematical concept used in the study of quantum mechanics and angular momentum. It refers to the group of symmetries that preserve the structure of the 6j symbol, which is a mathematical expression used to describe the coupling of three angular momenta. In simpler terms, it is a way to classify and understand the symmetries present in the 6j symbol.

## 2. How is the Isomorphism Symmetry Group of the 6j symbol calculated?

The Isomorphism Symmetry Group of the 6j symbol is calculated using group theory, a branch of mathematics that studies symmetry and its effects on objects. Specifically, it involves analyzing the properties of the 6j symbol and identifying the transformations that preserve these properties. These transformations then form the Isomorphism Symmetry Group.

## 3. What is the significance of the Isomorphism Symmetry Group of the 6j symbol?

The Isomorphism Symmetry Group of the 6j symbol has several important applications in physics and mathematics. It can be used to classify the different types of symmetries present in the 6j symbol, which can provide insights into the underlying physical systems. It also has implications in the study of quantum mechanics, where symmetries play a crucial role in understanding the behavior of particles.

## 4. How does the Isomorphism Symmetry Group of the 6j symbol relate to other symmetry groups?

The Isomorphism Symmetry Group of the 6j symbol is a specific type of symmetry group known as a Lie group. Lie groups are mathematical structures that describe continuous symmetries, and the Isomorphism Symmetry Group is a subgroup of a larger Lie group known as the Wigner-Eckart group. This group is important in the study of quantum mechanics and is closely related to other symmetry groups such as the rotation group and the permutation group.

## 5. Are there any real-world applications of the Isomorphism Symmetry Group of the 6j symbol?

While the Isomorphism Symmetry Group of the 6j symbol is primarily used in theoretical physics and mathematics, it has also found applications in other fields. For example, it has been used in the development of algorithms for computer simulations of quantum systems and in the analysis of molecular structures. It has also been applied in the study of crystallography and materials science, where symmetries play a crucial role in understanding the properties of materials.

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