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'''Wigner's''' 6-''j'' symbols were introduced by
[[Eugene Paul Wigner]] in 1940, and published in 1965.
They are defined by a sum over products of four [[Wigner 3-j symbols|3jm symbols]],
:<math>
  \begin{Bmatrix}
    j_1 & j_2 & j_3\\
    j_4 & j_5 & j_6
  \end{Bmatrix}
   = \sum_{m_i} (-1)^S
  \begin{pmatrix}
    j_1 & j_2 & j_3\\
    m_1 & m_2 & -m_3
  \end{pmatrix}
  \begin{pmatrix}
    j_1 & j_5 & j_6\\
    -m_1 & m_5 & m_6
  \end{pmatrix}
  \begin{pmatrix}
    j_4 & j_5 & j_3\\
    m_4 & -m_5 & m_3
  \end{pmatrix}
  \begin{pmatrix}
    j_4 & j_2 & j_6\\
    -m_4 & -m_2 & -m_6
  \end{pmatrix}
.
</math>
with phase <math>S=\sum_{k=1}^6 (j_k-m_k)</math>. The summation is over
all six {{math|''m''<sub>''i''</sub>}}, effectively confined by the selection rules of
the 3jm symbols.
They are related to [[Racah W-coefficient|Racah's W-coefficients]]
by
:<math>
  \begin{Bmatrix}
    j_1 & j_2 & j_3\\
    j_4 & j_5 & j_6
  \end{Bmatrix}
   = (-1)^{j_1+j_2+j_4+j_5}W(j_1j_2j_5j_4;j_3j_6).
</math>
They have higher symmetry than Racah's W-coefficients.

==Symmetry relations==
The 6-''j'' symbol is invariant under the permutation of any two columns:
:<math>
 \begin{Bmatrix}
    j_1 & j_2 & j_3\\
    j_4 & j_5 & j_6
 \end{Bmatrix}
 =
 \begin{Bmatrix}
    j_2 & j_1 & j_3\\
    j_5 & j_4 & j_6
 \end{Bmatrix}
=
 \begin{Bmatrix}
    j_1 & j_3 & j_2\\
    j_4 & j_6 & j_5
 \end{Bmatrix}
=
 \begin{Bmatrix}
    j_3 & j_2 & j_1\\
    j_6 & j_5 & j_4
 \end{Bmatrix}.
</math>
The 6-''j'' symbol is also invariant if upper and lower arguments
are interchanged in any two columns:
:<math>
 \begin{Bmatrix}
    j_1 & j_2 & j_3\\
    j_4 & j_5 & j_6
 \end{Bmatrix}
 =
 \begin{Bmatrix}
    j_4 & j_5 & j_3\\
    j_1 & j_2 & j_6
 \end{Bmatrix}
 =
 \begin{Bmatrix}
    j_1 & j_5 & j_6\\
    j_4 & j_2 & j_3
 \end{Bmatrix}
 =
 \begin{Bmatrix}
    j_4 & j_2 & j_6\\
    j_1 & j_5 & j_3
 \end{Bmatrix}.
</math>
These equations reflect the 24 symmetry operations of the [[Graph automorphism|automorphism group]] that leave the associated [[Table of simple cubic graphs#4 nodes|tetrahedral Yutsis graph]] with 6 edges invariant: mirror operations that exchange two vertices and a swap an adjacent pair of edges.

The 6-''j'' symbol
:<math>
 \begin{Bmatrix}
    j_1 & j_2 & j_3\\
    j_4 & j_5 & j_6
 \end{Bmatrix}
</math>
is zero unless ''j''<sub>1</sub>, ''j''<sub>2</sub>, and ''j''<sub>3</sub> satisfy triangle conditions,
i.e.,
:<math>
  j_1 = |j_2-j_3|, \ldots, j_2+j_3
</math>
In combination with the symmetry relation for interchanging upper and lower arguments this
shows that triangle conditions must also be satisfied for the triads (''j''<sub>1</sub>, ''j''<sub>5</sub>, ''j''<sub>6</sub>), (''j''<sub>4</sub>, ''j''<sub>2</sub>, ''j''<sub>6</sub>), and (''j''<sub>4</sub>, ''j''<sub>5</sub>, ''j''<sub>3</sub>).
Furthermore, the sum of each of the elements of a triad must be an integer. Therefore, the members of each triad are either all integers or contain one integer and two half-integers.

==Special case==
When ''j''<sub>6</sub> = 0 the expression for the 6-j symbol is:
:<math>
 \begin{Bmatrix}
    j_1 & j_2 & j_3\\
    j_4 & j_5 & 0
 \end{Bmatrix}
 = \frac{\delta_{j_2,j_4}\delta_{j_1,j_5}}{\sqrt{(2j_1+1)(2j_2+1)}} (-1)^{j_1+j_2+j_3}\{j_1,j_2,j_3\}.
</math>
The function {''j''<sub>1</sub>, ''j''<sub>2</sub>, ''j''<sub>3</sub>} is equal to 1 when the triad (''j''<sub>1</sub>, ''j''<sub>2</sub>, ''j''<sub>3</sub>) satisfies the triangle conditions, and zero otherwise. The symmetry relations can be used to find the expression when another ''j'' is equal to zero.

==Orthogonality relation==
The 6-j symbols satisfy this orthogonality relation:
:<math>
  \sum_{j_3} (2j_3+1)
 \begin{Bmatrix}
    j_1 & j_2 & j_3\\
    j_4 & j_5 & j_6
 \end{Bmatrix}
 \begin{Bmatrix}
    j_1 & j_2 & j_3\\
    j_4 & j_5 & j_6'
 \end{Bmatrix}
  = \frac{\delta_{j_6^{}j_6'}}{2j_6+1} \{j_1,j_5,j_6\} \{j_4,j_2,j_6\}.
</math>

==Asymptotics==
A remarkable formula for the asymptotic behavior of the 6-j symbol was first conjectured by Ponzano and Regge<ref>{{cite journal|last=Ponzano G and Regge T|title=Semiclassical Limit of Racah Coefficients|year=1968|pages=1–58|publisher=Amsterdam|location=in Spectroscopy and Group Theoretical Methods in Physics}}</ref> and later proven by Roberts.<ref>{{cite journal|last=Roberts J|title=Classical 6j-symbols and the tetrahedron|year=1998|journal=Geometry and Topology|pages=21–66|volume=3}}</ref> The asymptotic formula applies when all six quantum numbers ''j''<sub>1</sub>, ..., ''j''<sub>6</sub> are taken to be large and associates to the 6-j symbol the geometry of a tetrahedron. If the 6-j symbol is determined by the quantum numbers ''j''<sub>1</sub>, ..., ''j''<sub>6</sub> the associated tetrahedron has edge lengths ''J''<sub>i</sub> = ''j''<sub>i</sub>+1/2 (i=1,...,6) and the asymptotic formula is given by,
:<math>
\begin{Bmatrix}
j_1 & j_2 & j_3\\
j_4 & j_5 & j_6
\end{Bmatrix}
\sim \frac{1}{\sqrt{12 \pi |V|}} \cos{\left( \sum_{i=1}^{6} J_i \theta_i +\frac{\pi}{4}\right)}.
</math>
The notation is as follows: Each θ<sub>i</sub> is the external dihedral angle about the edge ''J''<sub>i</sub> of the associated tetrahedron and the amplitude factor is expressed in terms of the volume, ''V'', of this tetrahedron.

==Mathematical interpretation==

In [[representation theory]], 6j-symbols are matrix coefficients of the associator isomorphism in a [[tensor category]].<ref>{{cite book |last=Etingof |first=P. |coauthors=Gelaki S., Nikshych D., Ostrik V. |title=[http://www-math.mit.edu/~etingof/tenscat1.pdf Tensor Categories] |year=2009 }}</ref>  For example, if we are given three representations ''V''<sub>i</sub>, ''V''<sub>j</sub>, ''V''<sub>k</sub> of a group (or [[quantum group]]), one has a natural isomorphism
:<math>(V_i \otimes V_j) \otimes V_k \to V_i \otimes (V_j \otimes V_k)</math>
of tensor product representations, induced by coassociativity of the corresponding [[bialgebra]].  One of the axioms defining a monoidal category is that associators satisfy a pentagon identity, which is equivalent to the Biedenharn-Elliot identity for 6j-symbols.

When a monoidal category is semisimple, we can restrict our attention to irreducible objects, and define multiplicity spaces
:<math>H_{i,j}^\ell = \operatorname{Hom}(V_{\ell}, V_i \otimes V_j)</math>
so that tensor products are decomposed as:
:<math>V_i \otimes V_j = \bigoplus_\ell H_{i,j}^\ell V_\ell</math>
where the sum is over all isomorphism classes of irreducible objects.  Then:
:<math>(V_i \otimes V_j) \otimes V_k \cong \bigoplus_{\ell,m} H_{i,j}^\ell \otimes H_{\ell,k}^m \otimes V_m \qquad \text{while} \qquad V_i \otimes (V_j \otimes V_k) \cong \bigoplus_{m,n} H_{i,n}^m \otimes H_{j,k}^n \otimes V_m</math>
The associativity isomorphism induces a vector space isomorphism
:<math>\Phi_{i,j}^{k,m}: \bigoplus_{\ell} H_{i,j}^\ell \otimes H_{\ell,k}^m \to \bigoplus_n H_{i,n}^m \otimes H_{j,k}^n</math>
and the 6j symbols are defined as the component maps:
:<math>
  \begin{Bmatrix}
    i & j & \ell\\
    k & m & n
  \end{Bmatrix}
= (\Phi_{i,j}^{k,m})_{\ell,n}</math>
When the multiplicity spaces have canonical basis elements and dimension at most one (as in the case of ''SU''(2) in the traditional setting), these component maps can be interpreted as numbers, and the 6j-symbols become ordinary matrix coefficients.

In abstract terms, the 6j-symbols are precisely the information that is lost when passing from a [[monoidal category]] to its [[Grothendieck group]], since one can reconstruct a monoidal structure using the associator.  For the case of representations of a finite group, the [[character table]], together with its 6j-symbols, uniquely determines the group up to isomorphism, while the character table alone does not.

==See also==

* [[Clebsch–Gordan coefficients]]
* [[3-jm symbol]]
* [[Racah W-coefficient]]
* [[9-j symbol]]

==Notes==
{{Reflist}}

==References==

* {{cite book |last= Biedenharn |first= L. C. |authorlink=Lawrence Biedenharn |coauthors= van Dam, H. 
    |title= Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers
    |year= 1965 |publisher= [[Academic Press]] |location= New York |isbn= 0-12-096056-7}}

* {{cite book |last= Edmonds |first= A. R. |title= Angular Momentum in Quantum Mechanics |year= 1957
    |publisher= [[Princeton University Press]] |location= Princeton, New Jersey |isbn= 0-691-07912-9}}

* {{cite book |last= Condon |first= Edward U. |coauthors= Shortley, G. H. |title= The Theory of Atomic Spectra |year= 1970
    |publisher= [[Cambridge University Press]] |location= Cambridge |isbn= 0-521-09209-4 |chapter= Chapter 3}}
*{{dlmf|id=34 |title=3j,6j,9j Symbols|first=Leonard C.|last= Maximon}}
* {{cite book |last= Messiah |first= Albert |authorlink=Albert Messiah |title= Quantum Mechanics (Volume II) |year= 1981 | edition= 12th
    |publisher= [[Elsevier|North Holland Publishing]] |location= New York |isbn= 0-7204-0045-7}}

* {{cite book |last= Brink |first= D. M. |coauthors= Satchler, G. R.  |title= Angular Momentum
    |year= 1993 |edition= 3rd |publisher= [[Clarendon Press]] |location= Oxford |isbn= 0-19-851759-9 |chapter= Chapter 2 }}

* {{cite book |last= Zare |first= Richard N. |title= Angular Momentum |year=1988
    |publisher= [[John Wiley & Sons|John Wiley]] |location= New York |isbn= 0-471-85892-7 |chapter= Chapter 2}}

* {{cite book |last= Biedenharn |first= L. C. |coauthors= Louck, J. D. |title= Angular Momentum in Quantum Physics
    |year= 1981 |publisher= [[Addison-Wesley]] |location= Reading, Massachusetts |isbn= 0-201-13507-8 }}

==External links==
<!-- that the spelling Simmetry is a part of the title and should not be corrected
-->
* {{cite journal
|journal=Nuovo Cimento
|first1=T.
|last1=Regge
|title=Simmetry Properties of Racah's Coefficients
|volume=11
|issue=1
|year=1959
|doi=10.1007/BF02724914
|pages=116&ndash;117
}}
* {{cite web
|first1=Anthony
|last1=Stone
|url=http://www-stone.ch.cam.ac.uk/wigner.shtml
|title=Wigner coefficient calculator
}} (Gives exact answer)
* {{cite web |url=http://geoweb.princeton.edu/people/simons/software.html |first1=Frederik J.|last1=Simons|title=Matlab software archive, the code SIXJ.M}}
* {{cite web
|first1=A
|last1=Volya
|url=http://www.volya.net/vc
|title=Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator
}}
* {{cite web
|url=http://plasma-gate.weizmann.ac.il/369j.html
|title=369j-symbol calculator
|author=Plasma Laboratory of Weizmann Institute of Science
}}
* {{ cite web
|url=http://www.gnu.org/software/gsl/manual/html_node/Coupling-Coefficients.html
|title=Coupling coefficients
|author=[[Gnu Scientific Library|GNU scientific library]]
}}
* {{cite web
|first1=Richard J.
|last1=Mathar
|title=(Python implementation)
|url=http://arxiv.org/src/1102.5125v2/anc 
}} [http://arxiv.org/src/0908.3030v2/anc Java implementation]

[[Category:Rotational symmetry]]
[[Category:Representation theory of Lie groups]]
[[Category:Quantum mechanics]]