View source for Racah-Wigner coefficients

The mathematical entities known as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r1100105.png" /> coefficients enter mathematics and physics in describing the reduction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r1100106.png" />-fold Kronecker product of the unitary irreducible representations of the unitary unimodular group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r1100107.png" />, the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r1100108.png" /> unitary unimodular matrices, into unitary irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r1100109.png" />. This reduction has the form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001011.png" /></td> </tr></table>

where the symbols and quantities entering this relation have the following definitions:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001012.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001013.png" />-dimensional unitary irreducible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001014.png" /> given explicitly below, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001015.png" />, and this label is called an irreducible representation label or an angular momentum quantum number.

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001016.png" /> denotes the matrix direct product (Kronecker product), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001017.png" /> such products enter the left-hand side with arbitrary irreducible representation labels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001018.png" />.

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001019.png" /> denotes the matrix direct sum, and the direct sum on the right-hand side contains each irreducible representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001020.png" /> a number of times equal to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001022.png" /></td> </tr></table>

in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001023.png" /> is any irreducible representation label given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

(see below). For an arbitrary triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001025.png" /> of irreducible representation labels, the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001026.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001027.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001028.png" />, and it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001029.png" /> if and only if the so-called triangle conditions are satisfied:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001031.png" />.

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001032.png" /> denotes a real orthogonal matrix of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001033.png" />, with elements called Wigner–Clebsch–Gordan coefficients (WCG coefficients) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001034.png" /> and generalized Wigner–Clebsch–Gordan coefficients (generalized WCG coefficients) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001035.png" />.

5) The summation is over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001036.png" /> given by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001038.png" /> is given by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

and the minimum value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001040.png" /> is determined recursively from

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001041.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001042.png" /></td> </tr></table>

The content of (a1) is completed by giving the explicit form of the unitary irreducible representations entering the direct product. These may be described as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001043.png" /> denote an arbitrary non-negative integer and let the pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001045.png" /> consist of non-negative integers such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001046.png" /> (such pairs are called weights, or contents, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001047.png" />). Introduce the homogeneous polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001048.png" /> in the elements of an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001049.png" />-matrix of indeterminates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001050.png" /> defined by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>

where the summation is to be carried out over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001052.png" />-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001053.png" /> of non-negative integers with row and column sums given by the weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001055.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001056.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>

Significant properties of the functions (a8) are:

transposition:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001059.png" /> denotes [[matrix transposition]];

multiplication:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001060.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>

generation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001061.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a12)</td></tr></table>

orthogonality:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001062.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>

inner product:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001063.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a14)</td></tr></table>

where the bar denotes complex conjugation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001065.png" /> are arbitrary polynomials in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001066.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001067.png" /></td> </tr></table>

In the first two of these properties, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001068.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001069.png" />-dimensional matrix with rows and columns enumerated, respectively, by the weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001071.png" /> as they assume all possible values for the given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001072.png" />.

The unitary irreducible representation functions in (a1) are given in terms of the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001073.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001074.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a15)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001075.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001076.png" /></td> </tr></table>

that is, by choosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001081.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001082.png" /> is any integer or half odd integer, and for each such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001083.png" /> the so-called projection quantum numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001085.png" />, which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001086.png" /> subgroup irreducible representation labels, assume the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001087.png" />. By convention, these values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001089.png" /> label the rows and columns of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001090.png" />-dimensional unitary matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001091.png" />. (Conventionally, the rows are labelled from top to bottom by the greatest to the least value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001092.png" />, and the columns are similarly labelled from left to right by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001093.png" />).

Often, the irreducible representation functions (a15) are presented in a form in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001094.png" /> is parametrized. A general way of doing this is to use the property that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001095.png" /> unitary modular matrices are in one-to-one correspondence with the points on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001096.png" />-dimensional unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001097.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001098.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a16)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r11001099.png" /></td> </tr></table>

One can then parametrize the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010100.png" /> in any manner one chooses to obtain corresponding parametrizations of the representation functions.

The definitions and results given above are the basis relations underlying the origin of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010101.png" /> coefficients for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010102.png" />. The general problem leads to beautiful results and unsolved problems involving labelled binary trees, Cayley trivalent trees, and cubic graphs. This subject cannot be developed here, but its flavour can be indicated by the special results for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010103.png" />, which are presented in summary form below, focussing on these combinatorial aspects. The formulation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010104.png" /> problem can also be given in a purely Lie algebra setting, where the general procedure is known as addition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010106.png" /> angular momenta.

There is an extensive literature on this subject. Classical references include [[#References|[a1]]], [[#References|[a25]]], [[#References|[a26]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]], [[#References|[a15]]], [[#References|[a16]]], [[#References|[a17]]], [[#References|[a18]]], with more recent work in [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]], [[#References|[a22]]], [[#References|[a23]]], [[#References|[a24]]]. The notations in [[#References|[a19]]], [[#References|[a20]]] are followed.

==Vector space constructions.==
On assumes as given a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010107.png" /> with an inner product, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010109.png" /> is an arbitrary integer or half integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010110.png" />. This vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010111.png" /> also has an orthonormal basis given by the abstract ket vectors

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010112.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a17)</td></tr></table>

with an inner product

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010113.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a18)</td></tr></table>

Spaces corresponding to distinct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010114.png" /> are also to be perpendicular. Finally, there is an action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010115.png" />, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010116.png" />, of the unitary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010117.png" />, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010118.png" /> such that the action on the orthonormal basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010119.png" /> is given by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010120.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a19)</td></tr></table>

It is the tensor product space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010121.png" /> with orthonormal basis

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010122.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a20)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010123.png" /></td> </tr></table>

that underlies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010124.png" />-fold Kronecker product in the left-hand side of (a1). The action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010125.png" /> on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010126.png" /> is defined by its action on the basis (a17):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010127.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010128.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a21)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010129.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010130.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010131.png" /></td> </tr></table>

Concrete realizations of these spaces are provided by the vectors defined by (a8) and (a15) with inner product (a14):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010132.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a22)</td></tr></table>

which under left and right translations transform, in consequence of properties (a10) and (a11), as:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010133.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a23)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010134.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010135.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010136.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a24)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010137.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010138.png" /></td> </tr></table>

==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010139.png" /> coefficients.==


===Relation to Kronecker product.===
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010140.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a25)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010141.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010142.png" />-coefficient is a WCG coefficient, which is an element of the real orthogonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010143.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010144.png" />. The coefficient has the properties

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010145.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a26)</td></tr></table>

unless <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010146.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010147.png" />. These WCG coefficients satisfy the orthogonality relations

rows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010148.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a27)</td></tr></table>

columns:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010149.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a28)</td></tr></table>

===Vector space properties.===
Relation (a25) is valid when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010150.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010151.png" />. Using the inner product (a14) gives:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010152.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a29)</td></tr></table>

This relation (with a sign convention) may be used to calculate the WCG coefficients explicitly. One of the principal uses of the WCG coefficients in physical applications is the construction of a basis of the tensor product space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010153.png" /> that transforms irreducibly under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010154.png" />:

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010155.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a30)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010156.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a31)</td></tr></table>

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010157.png" /></td> </tr></table>

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

A binary tree type of notation has been used to indicate the angular momentum coupling scheme.

===Generating function.===
J. Schwinger [[#References|[a2]]] and T. Regge [[#References|[a11]]] discovered the generating function for the WCG coefficients to be the power of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010158.png" />-determinant:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010159.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a32)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010160.png" /></td> </tr></table>

where the summation is carried out over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010161.png" /> magic square arrays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010162.png" /> having row and column sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010163.png" />. The coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010164.png" /> is the restricted multinomial coefficient, given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010165.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a33)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010166.png" /></td> </tr></table>

where the summation is to be carried out over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010167.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010168.png" />, such that the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010169.png" /> is met:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010170.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a34)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010171.png" /></td> </tr></table>

Because of the restriction in the summation in (a33), the summation can be reduced (asymmetrically) to a sum with only one index.

The WCG coefficient is given by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010172.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a35)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010173.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010174.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010175.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010176.png" /></td> </tr></table>

===Symmetries.===
It follows directly from the generating function that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010177.png" /> coefficient, which is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010178.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a36)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010179.png" /></td> </tr></table>

has the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010180.png" /> determinantal symmetries associated with row interchange, column interchange, and transposition of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010181.png" />-determinant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010182.png" /> between the parentheses, which transfer directly to the array <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010183.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010184.png" /> coefficient is either invariant or changes its sign under the transformations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010185.png" /> induced by the following transformations of the array <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010186.png" />: invariance under even permutations of the rows or columns of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010187.png" />, and under transposition: multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010188.png" /> under odd permutations of the rows or columns.

==Binary coupling theory.==
In the physics literature, the phrase  "binary coupling theory"  addresses the problem of reducing the multiple Kronecker product (a1) by repeatedly using the reduction result (a25) for two angular momenta. One encounters immediately the problem of how to associate pairs in multiple Kronecker products. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010189.png" />, there are two such associations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010190.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a37)</td></tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010191.png" />, there are five such associations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010192.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a38)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010193.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010194.png" /></td> </tr></table>

In addition, all permutations of the irreducible representation labels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010195.png" /> must be considered. Entire textbooks ([[#References|[a14]]], [[#References|[a17]]], [[#References|[a22]]]) have been devoted to this subject, which leads to the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010196.png" /> coefficients by pairs of labelled binary trees and cubic graphs ([[#References|[a20]]], p. 435). Some limited aspects of this subject are presented here in the context of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010197.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010198.png" /> coefficients.

An important topic, but one that is only partially developed, is that of generating functions for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010199.png" /> coefficients. The property of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010200.png" /> coefficient that characterizes its presentation in terms of a pair of labelled binary trees or in terms of a cubic graph is the set of angular momentum triangles associated with it. This number is always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010201.png" />, the number of vertices in the associated cubic graph. A key concept for the generating function of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010202.png" /> coefficient is that of a triangle monomial. These monomials are defined as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010203.png" /> denote three distinct indeterminates and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010204.png" /> three angular momentum quantum numbers constituting a triangle. The triangle monomial associated with these symbols is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a39)</td></tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010206.png" /> denote the set of triangles associated with a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010207.png" /> coefficient. The triangle monomial for this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010208.png" /> coefficient is defined as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a40)</td></tr></table>

The indeterminates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010210.png" /> are coordinates associated with each of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010211.png" /> points of the cubic graph corresponding to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010212.png" /> coefficient. This is illustrated concretely below for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010213.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010214.png" /> coefficients.

It must be pointed out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010215.png" /> coefficients for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010216.png" /> are qualitatively different objects than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010217.png" /> coefficients. The vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010218.png" /> with basis (a20) is the carrier space of the reducible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010219.png" /> given by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010220.png" />-fold Kronecker product in (a1). This space, which is of dimension

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010221.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a41)</td></tr></table>

may be written as a direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010222.png" /> perpendicular vector subspaces:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010223.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a42)</td></tr></table>

Each subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010224.png" /> is of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010225.png" /> and has a basis of the form (a17) that transforms under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010226.png" /> in the standard way given by (a19). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010227.png" /> coefficients (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010228.png" />) are elements of real orthogonal matrices that transform between certain pairs of these subspaces that arise in binary coupling theory, and these coefficients, taken collectively, provide the transformation between all such pairs of subspaces. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010229.png" /> coefficients provide a single such space, as a first step in the construction of the pairs in the binary coupling method.

One of the consequences of this qualitative difference is that no generating function for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010230.png" /> coefficient (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010231.png" />) is a power of a determinant. It is known for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010232.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010233.png" /> coefficients ([[#References|[a2]]], [[#References|[a15]]], [[#References|[a16]]], [[#References|[a21]]]) that these coefficients have generating functions that can be presented in the following uniform manner [[#References|[a22]]]: There exists a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010234.png" /> with integer coefficients associated with the cubic graph corresponding to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010235.png" /> coefficient such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010236.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a43)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010237.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a44)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010238.png" /></td> </tr></table>

where the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010239.png" /> denotes the so-called triangle coefficient, defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010240.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a45)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010241.png" /></td> </tr></table>

==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010242.png" /> coefficients.==


===Relation to the Kronecker product.===
The coefficients defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010243.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a46)</td></tr></table>

are elements of the orthogonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010244.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010245.png" /> in (a1) that completely reduces the Kronecker product in the first coupling scheme given in (a37).

===Vector space interpretation.===


<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010246.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a47)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010247.png" /></td> </tr></table>

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010248.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a48)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010249.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010250.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010251.png" /> coefficient is called a Racah coefficient. In fact, one could take (a48) as the definition of the Racah coefficient. It is independent of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010252.png" /> quantum number; that is, it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010253.png" /> invariant. All other non-trivial binary couplings of three angular momenta lead again to Racah coefficients, up to sign. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010254.png" /> there is only one invariant, the Racah coefficient.

It was a great feat of G. Racah [[#References|[a25]]], [[#References|[a26]]] (see also [[#References|[a9]]], p. 162) to reduce the summation over four WCG coefficients in (a48), each of which is itself a summation over multinomial coefficients, to an expression involving but a single summation index. In the Wigner <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010255.png" /> notation, Racah's result is given by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010256.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a49)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010257.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010258.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010259.png" /></td> </tr></table>

The triangles in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010260.png" /> (Racah) coefficient are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010261.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010262.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010263.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010264.png" />, as inherited from the WCG coefficients in (a48), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010265.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010266.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010267.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010268.png" /> denote the triangle coefficients defined in (a45). The quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010269.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010270.png" /> are the discrete vertex and edge functions associated with the tetrahedron:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010271.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a50)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010272.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010273.png" /></td> </tr></table>

See (a55), in which the following identifications are to be made:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010274.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a51)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010275.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010276.png" /></td> </tr></table>

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010277.png" /> coefficient (now using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010278.png" /> labels throughout) labelled as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010279.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a52)</td></tr></table>

has four triangles, which are conveniently presented as the columns of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010280.png" />-array:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010281.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a53)</td></tr></table>

===Generating function.===
For the description of the generating function, the real coordinates

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010282.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a54)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010283.png" /></td> </tr></table>

are introduced, and these are displayed, analogously to (a53) in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010284.png" />-array, the columns of which are interpreted as the vertices, in Cartesian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010285.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010286.png" />, of a tetrahedron:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010287.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a55)</td></tr></table>

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

The array of triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010288.png" /> in (a53) is obtained from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010289.png" /> by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010290.png" />.

The generating function of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010291.png" /> coefficients is now obtained from the labelled tetrahedron in (a55) by the following rules. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010292.png" /> on the left-hand side of (a43) is given as follows. Interchange the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010293.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010294.png" /> symbols. Then form

the vertex function:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010295.png" /></td> </tr></table>

the edge function:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010296.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a56)</td></tr></table>

the tetrahedral function:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010297.png" /></td> </tr></table>

Using the tetrahedral function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010298.png" /> in the general form (a43) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010299.png" /> gives the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010300.png" /> with the labels (a52), where the variables in a given triangle monomial (a40) with exponents given by the columns of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010301.png" /> in (a53) are those in the corresponding columns of the array <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010302.png" /> on the left hand side of (a55).

===Symmetries.===
The symmetries of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010303.png" /> coefficients may be verified directly from the explicit form (a49), or from the generating function. For the coefficient written in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010304.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a57)</td></tr></table>

the symmetries are best expressed in terms of the Bargmann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010306.png" />-array

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010307.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a58)</td></tr></table>

as follows: The coefficient (a57) is invariant under all transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010308.png" /> induced by row or column permutations of the Bargmann array. This gives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010309.png" /> symmetry relations, some simple substitutions, and other simple linear transformations.

==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010310.png" /> coefficients.==


===Relation to the Kronecker product.===
The coefficients defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010311.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a59)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010312.png" /></td> </tr></table>

are the elements of the orthogonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010313.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010314.png" /> in (a1) that completely reduces the Kronecker product in the bottom binary coupling scheme given in (a38).

===Vector space interpretation.===


<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010315.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a60)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010316.png" /></td> </tr></table>

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010317.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a61)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010318.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010319.png" /></td> </tr></table>

The bracket object is Wigner's notation for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010320.png" /> coefficient, and this relation (a61) may be taken as its definition. The summation over six WCG coefficients coming from the inner product in (a61), using (a60), may be reduced to the following summation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010321.png" /> coefficients (a switch to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010322.png" /> notation has been made):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010323.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a62)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010324.png" /></td> </tr></table>

The six triangles in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010325.png" /> coefficient are inherited from its expression in terms of WCG coefficients, or, equivalently, from the triangles in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010326.png" /> coefficients in (a62) not involving the summation index. It is convenient to display these six triangles as the columns in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010327.png" />-array:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010328.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a63)</td></tr></table>

===Generating function.===
For the description of the generating function, the real coordinates

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010329.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a64)</td></tr></table>

are introduced, and these are displayed, analogously to (a63), in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010330.png" />-array, the columns of which are interpreted as the vertices, in Cartesian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010331.png" />-dimensional space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010332.png" />, of a cubic graph on six vertices:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010333.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a65)</td></tr></table>

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />

The array of triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010334.png" /> in (a63) is obtained from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010335.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010336.png" /> by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010337.png" />. The generating function of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010338.png" /> coefficients is now obtained from the labelled cubic graph by the following rules. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010339.png" /> in the left hand side of (a43) is obtained as follows. Interchange the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010340.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010341.png" /> symbols. Then form

the vertex function:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010342.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a66)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010343.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010344.png" /></td> </tr></table>

the edge function:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010345.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a67)</td></tr></table>

the six point cubic graph function:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010346.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a68)</td></tr></table>

Using the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010347.png" /> in the general form (a43) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010348.png" /> gives the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010349.png" /> coefficient with the labels (a62), where the variables in a given triangle monomial (a40) with exponents given by the columns of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010350.png" /> in (a63) are those in the corresponding columns of the array <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010351.png" /> on the left hand side of (a65).

===Symmetries.===
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010352.png" /> coefficient has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010353.png" /> (known) symmetries, and these are just those originating from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010354.png" /> symmetries of the six <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010355.png" /> coefficients implicit on the left hand side of (a61). In terms of the coefficient written in the form (a62), these symmetries are expressed by: The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010356.png" /> coefficient (a62) is invariant under even permutations of its rows or columns, under transposition of the array, and is multiplied by the factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010357.png" /> under odd permutations of its rows or columns.

==Concluding remarks.==
A great deal of work has been done on the relationship between cubic graphs and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010358.png" /> coefficients, and the results can not even be surveyed here. It is important to point out, however, that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010359.png" /> there is a second non-isomorphic cubic graph on six vertices, the prism, or wedge, which leads to a simple product (with dimensional factors) of two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010360.png" /> coefficients. While this is important for transforming between the relevant coupling schemes, it does not lead to a new <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010361.png" /> coefficient. Indeed, it can be proved that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010362.png" /> coefficient is a sum over products of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010363.png" /> coefficients, and sometimes this sum may degenerate to a product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010364.png" /> coefficients (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010365.png" />), in which case it is not counted as new. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010366.png" /> there are always fewer  "new"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010367.png" /> coefficients than non-isomorphic cubic graphs. The general problem of the relation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010368.png" /> coefficients and non-isomorphic cubic graphs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010369.png" /> vertices is unsolved.

Relevant to the classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010370.png" /> coefficients, indeed, for the definition of what the term  "classification"  means, are three basic relations:

triangle transformation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010371.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a69)</td></tr></table>

Racah sum rule:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010372.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a70)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010373.png" /></td> </tr></table>

Biedenharn–Elliot identity, [[#References|[a4]]], [[#References|[a5]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010374.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a71)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010375.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010376.png" /></td> </tr></table>

Each of these relations has an interpretation in terms of fundamental operations in mathematics: The triangle transformation rule is a consequence of the associativity of three symplecton polynomials, which are polynomials of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010377.png" /> over the four non-commuting variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010378.png" /> (or, equivalently, over two bosons and their conjugates) which transform irreducibly under a well-defined commutation action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010379.png" />. Racah's sum rule has an interpretation in terms of the commutativity of a diagram associated with the coupling of three angular momenta. The Biedenharn–Elliot identity is a consequence of the associative law for three unit tensor operators. For a discussion of each of these results, see [[#References|[a20]]], pp. 243; 453; 30.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.P. Wigner,   "On the matrices which reduce the Kronecker products of representation of S.R. groups"  L.C. Biedenharn (ed.)  H. van Dam (ed.) , ''Quantum Theory of Angular Momentum'' , Acad. Press  (1965)  pp. 87–133  (Original: 1940 (unpublished))</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Schwinger,   "On angular momentum"  L.C. Bidenharn (ed.)  H. van Dam (ed.) , ''Quantum Theory of Angular Momentum'' , Acad. Press  (1965)  pp. 229–279  (Original: U.S. Atomic Energy Commission Reports NYO–3071, 1952 (unpublished))</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.C. Biedenharn,   J.M. Blatt,   M.E. Rose,   "Some properties of the Racah and associated coefficients"  ''Rev. Mod. Phys.'' , '''24'''  (1952)  pp. 249–257</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L.C. Biedenharn,   "An identity satisfied by Racah coefficients"  ''J. Math. Phys.'' , '''31'''  (1953)  pp. 287–293</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.P. Elliot,   "Theoretical studies in nuclear spectroscopy. V: The matrix elements of non-central forces with an application to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010380.png" />-shell"  ''Proc. Roy. Soc. A'' , '''218'''  (1953)  pp. 345–370</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H.A. Jahn,   J. Hope,   "Symmetry properties of the Wigner <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010381.png" /> symbol"  ''Phys. Rev.'' , '''93'''  (1954)  pp. 318–321</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Arima,   H. Horie,   Y. Tanabe,   "Generalized Racah coefficient and its application"  ''Progress Theor. Phys. (Kyoto)'' , '''11'''  (1954)  pp. 143–154</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.R. Edmonds,   "Angular momentum in quantum mechanics" , Princeton Univ. Press  (1957)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M.E. Rose,   "Elementary theory of angular momentum" , Wiley  (1957)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  U. Fano,   G. Racah,   "Irreducible tensorial sets" , Acad. Press  (1959)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  T. Regge,   "Symmetry properties of Clebsch–Gordan coefficients"  ''Nuovo Cim.'' , '''10'''  (1958)  pp. 544–545</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  T. Regge,   "Symmetry properties of Racah's coefficients"  ''Nuovo Cim.'' , '''11'''  (1959)  pp. 116–117</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  W.T. Sharp,   "Racah algebra and the contraction of groups" , Thesis Princeton Univ. Press  (1960)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  A.P. Yutsis (Jucys),   I.B. Levinson,   V.V. Vanagas,   "Mathematical apparatus of the theory of angular momentum" , Gordon&amp;Breach  (1964)  (In Russian)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  V. Bargmann,   "On the representations of the rotation group"  ''Rev. Mod. Phys.'' , '''345'''  (1962)  pp. 829–845</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  A.C.T. Wu,   "Structure of the Wigner <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010382.png" /> coefficients in the Bargmann approach"  ''J. Math. Phys.'' , '''13'''  (1972)  pp. 84–90</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  D.A. Varshalovich,   A.N. Moskalev,   V.K. Khersonskii,   "Quantum theory of angular momentum" , Nauka  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  A.P. Yutsis (Jucys),   A.A. Bandzaitis,   "Angular momentum theory in quantum mechanics" , Molslas , Vilnius  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  L.C. Biedenharn,   J.D. Louck,   "Angular momentum in quantum physics" , ''Encycl. Math. Sci.'' , '''8''' , Cambridge Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  L.C. Biedenharn,   J.D. Louck,   "The Racah–Wigner algebra in quantum theory" , ''Encycl. Math. Sci.'' , '''9''' , Cambridge Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  B.R. Judd,   G.M.S. Lister,   "A class of generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010383.png" /> coefficients for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010384.png" />"  ''J. Phys. A: Math. Gen.'' , '''20'''  (1986)  pp. 3159–3169</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  L.C. Biedenharn,   W.Y.C. Chen,   M.A. Lohe,   J.D. Louck,   "The role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010385.png" /> coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110010/r110010386.png" />"  T. Lulek (ed.)  W. Florek (ed.)  S. Walcerz (ed.) , ''Proc. 3rd SSCPM'' , World Sci.  (1995)  pp. 150–182</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  M. Danos,   "??" , ??  (199?)</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top">  J.D. Louck,   "Unitary symmetry, combinatorics and generating functions"  ''J. Discrete Math.'' , '''??'''  (199?)</TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top">  G. Racah,   "Theory of complex spectra. II"  ''Phys. Rev.'' , '''62'''  (1942)  pp. 438–462</TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top">  G. Racah,   "Theory of complex spectra. III"  ''Phys. Rev.'' , '''63'''  (1943)  pp. 367–382</TD></TR></table>