# Recurrence relation for the $6j$-symbol of $\mathrm{su}_{q}(2)$ as a
symmetric eigenvalue problem

Igor Khavkine1 1 Institute for Theoretical Physics, Utrecht, Leuvenlaan 4,
NL-3584 CE Utrecht, The Netherlands<EMAIL_ADDRESS>
###### Abstract

A well known recurrence relation for the $6j$-symbol of the quantum group
$\mathrm{su}_{q}(2)$ is realized as a tridiagonal, symmetric eigenvalue
problem. This formulation can be used to implement an efficient numerical
evaluation algorithm, taking advantage of existing specialized numerical
packages. For convenience, all formulas relevant for such an implementation
are collected in the appendix. This realization is a biproduct of an
alternative proof of the recurrence relation, which generalizes a classical
($q=1$) result of Schulten and Gordon and uses the diagrammatic spin network
formalism of Temperley-Lieb recoupling theory to simplify intermediate
calculations.

###### pacs:

02.20.Uw, 03.65.Fd

††: J. Phys. A: Math. Gen.

##  1 Introduction

Quantum groups first appeared in the study of quantum integrable systems.
Since then, they have proven useful in many applications, including among
others conformal field theory, statistical mechanics, representation theory
and the theory of hypergeometric functions, along with exhibiting a rich
internal structure. Quantum groups have appeared seminally in the mathematical
physics literature in connection with topological quantum field theory [1].
More recently, the quantum group $\mathrm{su}_{q}(2)$ was used to construct
“loop” [2] and “spin foam” [3] models of quantum gravity with a positive
cosmological constant.

The $6j$-symbol (or Racah-Wigner coefficients and henceforth the $6j$ or the
$q$-$6j$ for the classical and $q$-deformed versions respectively) has been
studied extensively. It first appeared in work on $q$-hypergeometric functions
[4]. Later, it was found to play a central role in the representation theory
of $\mathrm{su}_{q}(2)$ [5]. It is known to satisfy some recurrence relations
[4, 6], including a particular linear, three term, single argument one [5, 7,
8]. This recurrence has been used to analyze the asymptotics of the classical
[9, 10] and quantum [11] $6j$.

However, this $q$-$6j$ and the need to know its numerical values for a large
number of arguments also appear in other segments of the literature [3, 12],
less aware of these properties. An explicit formula for the $q$-$6j$ is well
known and involves a number of arithmetic operations that is linear in its
arguments (see Appendix). In applications where a large number of $6j$-symbols
is needed at once, e.g., for all values of one argument with others fixed as
is the case in [3], the total number of operations becomes quadratic in the
arguments. The above mentioned recurrence relation can be used to greatly
increase the efficiency of the calculation by reducing the total operation
count to be linear in the arguments. The main goal of this paper is to show
that, moreover, this recurrence can be realized as a tridiagonal, symmetric
eigenvalue problem, a property not shared by most recurrence relations,
especially since the equivalence is established using only rational
operations.

A significant advantage of the eigenvalue formulation is the ability to make
use of readily available, robust linear algebra packages, such as LAPACK [13],
which automatically take care of the important issues of numerical accuracy
and stability. When $q=1$ or when $q$ is a primitive root of unity, the
relevant inner product becomes either positive- or negative-definite and
standard, specialized numerical methods can be exploited to increase the
efficiency of the calculation even further. A secondary goal of this paper is
to concisely collect all the relevant information needed to readily implement
such an efficient $q$-$6j$ numerical evaluation algorithm without intimate
familiarity with the literature on quantum groups or $q$-hypergeometric
functions.

Sections 2 and 3, which can be skipped by those familiar with the mathematical
literature on $\mathrm{su}_{q}(2)$ recoupling theory, introduce the basic
notions of the spin network formalism [14, 15, 16], define the Kauffman-Lins
convention for the $6j$-symbol and summarize basic diagrammatic identities
needed for Section 4, where the recurrence relation is realized as an
eigenvalue problem. This is accomplished as a byproduct of an alternative
proof of the recurrence itself that generalizes the classical argument from
the Appendix of [10]. The diagrammatic spin network formalism makes all
intermediate calculations easy to check and reproduce. The Appendix
conveniently summarizes all formulas needed for a direct computer
implementation of the recurrence-based evaluation of the $q$-$6j$, including
the connection between the Kauffman-Lins and Racah-Wigner notational
convention, which is traditionally used in physics.

##  2 Spin networks

In a variety of physical and mathematical applications, one often encounters
tensor contraction expressions of the form

$T_{j\cdots}^{lm\cdots}=A_{ij\cdots}^{kl\cdots}B_{k\cdots}^{im\cdots}\cdots
Z_{\cdots}^{\cdots},$ (1)

where $T$, $A$, $B$, …, $Z$ are invariant tensors, with each index
transforming under a representation of a group or an algebra. The application
at hand usually calls for evaluating $T$, or at least simplifying it. An
extensive literature on this subject exists for the classical group $SU(2)$ or
its Lie algebra, a subject known as _angular momentum recoupling_ [17, 18]. It
is well known that such tensor contractions can be very efficiently expressed,
manipulated, and simplified using diagrams known as _spin networks_ [14].
Extensions of these techniques [15, 16] are also known for the _quantum_ (or
_$q$ -deformed_, since they depend on an arbitrary complex number $q\neq 0$)
analogs, the quantum group $\mathrm{su}_{q}(2)$ or $U_{q}(su(2))$. The basics
of this diagrammatic formalism, as needed for the derivation of the recurrence
relation, are given in this and next sections. All relevant formulas,
including explicit spin network evaluations in terms of _quantum integers_ are
listed in the Appendix.

Single spin networks are edge-labeled graphs111Spin networks are actually
ribbon graphs, but since all diagrams in this paper are planar, the ribbon
structure can be added through blackboard framing., where each vertex has
valence either $1$ or $3$. General spin networks are formal linear
combinations of single spin networks. Edges attached to univalent vertices are
called _free_. Spin networks without free edges are called _closed_.
Conventionally, the labels are either integers (_spins_) of half-integers
(_twice-spins_), which correspond to irreducible representations of
$\mathrm{su}_{q}(2)$. Reference [15] labels all spin networks with twice-
spins. Unless otherwise indicated, all conventions in this paper follow [15].
Two spin networks may be equal even if not represented with identical labeled
graphs. A complete description of these identities are given in [15] and [16];
their study constitutes _spin network recoupling_ and is what allows us to
equate222A down-to-earth guide to this correspondence, for the classical $q=1$
case, can be found in Appendix A of [19]. Complete details with proofs can be
found in [16]. spin networks with $\mathrm{su}_{q}(2)$-invariant tensors and
their contractions.

In this correspondence, each index of a tensor, transforming under an
irreducible representation, corresponds to a spin network edge, labeled by the
same representation (free indices correspond to free edges). In particular, a
closed spin network corresponds to a complex number. Spin networks form a
graded algebra over $\mathbb{C}$ (as do tensors). The grading is given by the
number of free edges (free indices) and the product is diagrammatic
juxtaposition (tensor product).

##  3 Diagrammatic identities

The spin networks with $n$ free edges with fixed labels (_$n$ -valent_ spin
networks) form a linear space with a natural bilinear form (or inner product).
Suppose that the free edges are ordered in some canonical way, then, given two
spin networks, we can reflect one of them in a mirror and connect the free
edges in order. The value of the resulting closed spin network defines the
bilinear form, which is symmetric and non-degenerate [15]. We use the bra-ket
notation for this inner product $\langle s^{\prime}|s\rangle$, where $s$ and
$s^{\prime}$ are two spin networks. We also let $|s\rangle$ stand for $s$ and
$\langle s^{\prime}|$ for the reflection of $s^{\prime}$. The existence of an
inner product allows the following identities, whose proofs can be found in
[15]. For each identity, the corresponding well known fact of $SU(2)$
representation theory is given.

The space of $2$-valent spin networks, with ends labeled $a$ and $b$, is
$1$-dimensional if $a=b$, and $0$-dimensional otherwise. For non-trivial
dimension, the single edge gives a complete basis and therefore the _bubble
identity_ :

$\raisebox{-4.95801pt}{\leavevmode\hbox{\par\begin{picture}(654.0,293.0)(1249.0,-159.0)\put(1261.0,-16.0){\line(1,0){630.0}}
\put(1306.0,29.0){\makebox(0.0,0.0)[b]{\smash{{a}}}}
\put(1846.0,29.0){\makebox(0.0,0.0)[b]{\smash{{b}}}}
{\color[rgb]{.72,.72,.72}\put(1576.0,-16.0){\circle*{270.0}}
\color[rgb]{0,0,0}\put(1576.0,-16.0){\circle{240.0}}
}\end{picture}}}~{}=\delta_{ab}\frac{\raisebox{-7.70862pt}{\leavevmode\hbox{\par\begin{picture}(477.0,380.0)(1339.0,-253.0)\put(1456.0,-136.0){\oval(210.0,210.0)[bl]}
\put(1456.0,-121.0){\oval(210.0,210.0)[tl]}
\put(1696.0,-136.0){\oval(210.0,210.0)[br]}
\put(1696.0,-121.0){\oval(210.0,210.0)[tr]}
\put(1456.0,-241.0){\line(1,0){240.0}} \put(1456.0,-16.0){\line(1,0){240.0}}
\put(1351.0,-136.0){\line(0,1){15.0}} \put(1801.0,-136.0){\line(0,1){15.0}}
\put(1801.0,-16.0){\makebox(0.0,0.0)[lb]{\smash{{a}}}}
{\color[rgb]{.72,.72,.72}\put(1576.0,-16.0){\circle*{270.0}}
\color[rgb]{0,0,0}\put(1576.0,-16.0){\circle{240.0}}
}\end{picture}}}}{\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\end{picture}}}}~{}~{}\raisebox{-2.8081pt}{\leavevmode\hbox{\par\begin{picture}(474.0,225.0)(1339.0,-121.0)\put(1351.0,-16.0){\line(1,0){450.0}}
\put(1576.0,29.0){\makebox(0.0,0.0)[b]{\smash{{a}}}}
\put(1576.0,-106.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{1,1,1}a}}}}}
\end{picture}}}~{}.$ (2)

This identity the diagrammatic analog of Schur’s lemma for intertwiners
between irreducible representations.

The space of $3$-valent spin networks, with ends labeled $a$, $b$ and $c$, is
also $1$-dimensional if the triangle inequalities (31) and parity constraints
(32) are satisfied, and $0$-dimensional otherwise, if $q$ is generic. When $q$
is a primitive root of unity, the dimension also vanishes whenever the further
$r$-boundedness constraint (33) is violated. In the case of nontrivial
dimension, the canonical trivalent vertex gives a complete basis and therefore
the _vertex collapse identity_ :

cab
$\displaystyle=\frac{\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(605.0,608.0)(983.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
{\color[rgb]{.72,.72,.72}\put(1126.0,164.0){\circle*{270.0}}
\color[rgb]{0,0,0}\put(1126.0,164.0){\circle{240.0}}
}\end{picture}}}}{\theta(a,b,c)}~{}~{}\raisebox{-13.7157pt}{\leavevmode\hbox{\par\begin{picture}(342.0,570.0)(1834.0,-751.0){\color[rgb]{0,0,0}\put(2071.0,-466.0){\oval(450.0,450.0)[tl]}
\put(2071.0,-466.0){\oval(450.0,450.0)[bl]}
}{\color[rgb]{0,0,0}\put(1846.0,-466.0){\line(1,0){225.0}}
}\put(2161.0,-736.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(2161.0,-286.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(2161.0,-511.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\end{picture}}}~{}~{},$ (3) $\displaystyle\theta(a,b,c)$
$\displaystyle=\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\end{picture}}}.$ (4)

The normalization of the vertex, the value of the $\theta$-network, is
evaluated in Equation (30). This identity is the diagrammatic analog of the
uniqueness (up to normalization) of the Clebsch-Gordan intertwiner.

Now, consider the space of $4$-valent networks with free edges labeled $a$,
$b$, $c$ and $d$. There are two natural bases, the vertical $\langle l|$ and
the horizontal $|\bar{j}\rangle$:

$\raisebox{-18.93237pt}{\leavevmode\hbox{\par\begin{picture}(300.0,735.0)(1426.0,-166.0){\color[rgb]{0,0,0}\put(1576.0,164.0){\oval(180.0,450.0)[tl]}
\put(1576.0,164.0){\oval(180.0,450.0)[bl]}
}{\color[rgb]{0,0,0}\put(1666.0,389.0){\oval(180.0,180.0)[tl]}
\put(1666.0,389.0){\oval(180.0,180.0)[bl]}
}{\color[rgb]{0,0,0}\put(1666.0,-61.0){\oval(180.0,180.0)[tl]}
\put(1666.0,-61.0){\oval(180.0,180.0)[bl]}
}\put(1711.0,299.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1711.0,29.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1711.0,479.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1711.0,-151.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1441.0,119.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\end{picture}}}\quad,\quad\raisebox{-18.93237pt}{\leavevmode\hbox{\par\begin{picture}(345.0,735.0)(1741.0,-166.0){\color[rgb]{0,0,0}\put(1801.0,164.0){\oval(180.0,270.0)[br]}
\put(1801.0,164.0){\oval(180.0,270.0)[tr]}
}{\color[rgb]{0,0,0}\put(1801.0,164.0){\oval(450.0,630.0)[br]}
\put(1801.0,164.0){\oval(450.0,630.0)[tr]}
}{\color[rgb]{0,0,0}\put(1891.0,164.0){\line(1,0){135.0}}
}\put(1756.0,-151.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1756.0,29.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1756.0,299.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1756.0,479.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(2071.0,119.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}~{}~{}.$ (5)

The admissible ranges for $j$ and $l$, the dimension $n$ of this space, and
the conditions on $(a,b,c,d)$ under which $n>0$ are given by Equations (41)
through (52). The transition matrix between the two bases is given by the so-
called $\operatorname{Tet}$-network:

$\operatorname{Tet}(a,b,c,d;j,l)=\raisebox{-18.23682pt}{\leavevmode\hbox{\par\begin{picture}(646.0,713.0)(1028.0,-1284.0){\color[rgb]{0,0,0}\put(1351.0,-961.0){\circle{630.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(0,1){315.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\end{picture}}}=\langle\bar{j}|l\rangle.$ (6)

The coefficients expressing the vertical basis in terms of the horizontal one
define the _$6j$ -symbol_, which can be expressed in terms of the
$\operatorname{Tet}$-network:

$\displaystyle|l\rangle$ $\displaystyle=\sum_{j}\begin{Bmatrix}{a}&{b}&{j}\\\
{c}&{d}&{l}\end{Bmatrix}_{KL}|\bar{j}\rangle,$ (7)
$\displaystyle\begin{Bmatrix}{a}&{b}&{j}\\\ {c}&{d}&{l}\end{Bmatrix}_{KL}$
$\displaystyle=\frac{\raisebox{-18.23682pt}{\leavevmode\hbox{\par\begin{picture}(646.0,713.0)(1028.0,-1284.0){\color[rgb]{0,0,0}\put(1351.0,-961.0){\circle{630.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(0,1){315.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\end{picture}}}}{\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\end{picture}}}\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\end{picture}}}}~{}\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}~{}.$ (8)

Note the subscript $KL$ for Kauffman-Lins, since this $6j$-symbol is defined
with respect to the conventions of [15]. The relation to the classical Racah-
Wigner $6j$-symbol used in the physics literature is given explicitly in
Equation 53.

##  4 Recurrence relation for the $\operatorname{Tet}$-network

The identities given in the previous section allow an alternative, elementary
derivation of the three-term recurrence relation for the
$\operatorname{Tet}$-network, distinct from the standard one. The standard
derivation is given in [7] and another is possible using the general theory of
recurrences for $q$-hypergeometric functions [6], but neither directly yields
the symmetric eigenvalue problem form.

It is easy to check, using the bubble identity, that both the vertical and
horizontal bases are orthogonal and that they are normalized as

$\displaystyle\langle\bar{j}|\bar{j}\rangle$
$\displaystyle=\raisebox{-19.78601pt}{\leavevmode\hbox{\par\begin{picture}(750.0,762.0)(1426.0,-163.0){\color[rgb]{0,0,0}\put(1981.0,164.0){\line(1,0){135.0}}
}{\color[rgb]{0,0,0}\put(1486.0,164.0){\line(1,0){135.0}}
}{\color[rgb]{0,0,0}\put(1711.0,74.0){\oval(450.0,450.0)[bl]}
\put(1711.0,254.0){\oval(450.0,450.0)[tl]}
\put(1891.0,74.0){\oval(450.0,450.0)[br]}
\put(1891.0,254.0){\oval(450.0,450.0)[tr]}
\put(1711.0,-151.0){\line(1,0){180.0}} \put(1711.0,479.0){\line(1,0){180.0}}
\put(1486.0,74.0){\line(0,1){180.0}} \put(2116.0,74.0){\line(0,1){180.0}}
}{\color[rgb]{0,0,0}\put(1756.0,164.0){\oval(270.0,270.0)[bl]}
\put(1756.0,164.0){\oval(270.0,270.0)[tl]}
\put(1846.0,164.0){\oval(270.0,270.0)[br]}
\put(1846.0,164.0){\oval(270.0,270.0)[tr]} \put(1756.0,29.0){\line(1,0){90.0}}
\put(1756.0,299.0){\line(1,0){90.0}} \put(1621.0,164.0){\line(0,1){0.0}}
\put(1981.0,164.0){\line(0,1){0.0}}
}\put(2161.0,119.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1801.0,524.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1801.0,344.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1801.0,74.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1801.0,-106.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1441.0,119.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}~{}=\frac{\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\end{picture}}}~{}\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\end{picture}}}}{\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}},$ (9) $\displaystyle\langle l|l\rangle$
$\displaystyle=\raisebox{-19.78601pt}{\leavevmode\hbox{\par\begin{picture}(660.0,762.0)(1201.0,-163.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\oval(180.0,450.0)[tl]}
\put(1351.0,164.0){\oval(180.0,450.0)[bl]}
}{\color[rgb]{0,0,0}\put(1711.0,164.0){\oval(180.0,450.0)[br]}
\put(1711.0,164.0){\oval(180.0,450.0)[tr]}
}{\color[rgb]{0,0,0}\put(1456.0,-46.0){\oval(210.0,210.0)[bl]}
\put(1456.0,-76.0){\oval(210.0,210.0)[tl]}
\put(1606.0,-46.0){\oval(210.0,210.0)[br]}
\put(1606.0,-76.0){\oval(210.0,210.0)[tr]}
\put(1456.0,-151.0){\line(1,0){150.0}} \put(1456.0,29.0){\line(1,0){150.0}}
}{\color[rgb]{0,0,0}\put(1456.0,404.0){\oval(210.0,210.0)[bl]}
\put(1456.0,374.0){\oval(210.0,210.0)[tl]}
\put(1606.0,404.0){\oval(210.0,210.0)[br]}
\put(1606.0,374.0){\oval(210.0,210.0)[tr]}
\put(1456.0,299.0){\line(1,0){150.0}} \put(1456.0,479.0){\line(1,0){150.0}}
}\put(1216.0,119.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1531.0,-106.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1531.0,74.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1846.0,119.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1531.0,344.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1531.0,524.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\end{picture}}}~{}=\frac{\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\end{picture}}}~{}\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\end{picture}}}}{\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\end{picture}}}}.$ (10)

Curiously, when these normalizations are fully expanded using formulas from
the Appendix, they take the form $(-)^{\sigma}P/Q$, where $P$ and $Q$ are
products of positive quantum integers. In both cases, $\sigma=(a+b+c+d)/2$, is
an integer independent of $j$ or $l$. When $q=1$ or when $q$ is a primitive
root of unity, positive quantum integers are positive real numbers. Hence the
above inner product is real and either positive- or negative-definite. On the
other hand, for arbitrary complex $q$, the normalizations (9) and (10) can be
essentially arbitrary complex numbers.

If we can find a linear operator $L$ that is diagonal in one basis, but not in
the other, then we can obtain $\langle\bar{j}|l\rangle$ as matrix elements of
the diagonalizing transformation. Furthermore, if the non-diagonal form of $L$
is tridiagonal, then the linear equations defining $\langle\bar{j}|l\rangle$
reduce to a three-term recurrence relation.

We can construct such an operator by generalizing the argument for the
classical case, found in the Appendix of [10]. For brevity of notation, we
introduce a special modified version of the trivalent vertex:

$\raisebox{-12.00842pt}{\leavevmode\hbox{\par\begin{picture}(1380.0,516.0)(1516.0,-697.0){\color[rgb]{0,0,0}\put(1801.0,-511.0){\circle*{46.0}}
}{\color[rgb]{0,0,0}\put(1531.0,-511.0){\line(1,0){540.0}}
}{\color[rgb]{0,0,0}\put(1801.0,-511.0){\line(0,1){270.0}}
}{\color[rgb]{0,0,0}\put(2341.0,-511.0){\line(1,0){540.0}}
}{\color[rgb]{0,0,0}\put(2611.0,-511.0){\line(0,1){270.0}}
}\put(2071.0,-466.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1531.0,-466.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(2881.0,-466.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(2341.0,-466.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(2656.0,-286.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}2}}}}}
\put(2611.0,-646.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$[a]$}}}}}
\put(2206.0,-421.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$=$}}}}}
\end{picture}}}.$ (11)

The unlabeled edge implicitly carries twice-spin $2$ and the bold dot
indicates the multiplicative factor of $[a]$. Using it, we can define a
symmetric operator $L$. Its diagrammatic representation and its matrix
elements are given below.

The operator $L$ is diagonal in the $|l\rangle$-basis and its matrix elements
$L_{ll^{\prime}}=\langle l|L|l^{\prime}\rangle$ are

$\displaystyle L_{ll^{\prime}}$
$\displaystyle=\raisebox{-18.36328pt}{\leavevmode\hbox{\par\begin{picture}(660.0,717.0)(1201.0,-163.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\oval(180.0,450.0)[tl]}
\put(1351.0,164.0){\oval(180.0,450.0)[bl]}
}{\color[rgb]{0,0,0}\put(1711.0,164.0){\oval(180.0,450.0)[br]}
\put(1711.0,164.0){\oval(180.0,450.0)[tr]}
}{\color[rgb]{0,0,0}\put(1531.0,479.0){\circle*{46.0}}
}{\color[rgb]{0,0,0}\put(1531.0,299.0){\circle*{46.0}}
}{\color[rgb]{0,0,0}\put(1456.0,-46.0){\oval(210.0,210.0)[bl]}
\put(1456.0,-76.0){\oval(210.0,210.0)[tl]}
\put(1606.0,-46.0){\oval(210.0,210.0)[br]}
\put(1606.0,-76.0){\oval(210.0,210.0)[tr]}
\put(1456.0,-151.0){\line(1,0){150.0}} \put(1456.0,29.0){\line(1,0){150.0}}
}{\color[rgb]{0,0,0}\put(1456.0,404.0){\oval(210.0,210.0)[bl]}
\put(1456.0,374.0){\oval(210.0,210.0)[tl]}
\put(1606.0,404.0){\oval(210.0,210.0)[br]}
\put(1606.0,374.0){\oval(210.0,210.0)[tr]}
\put(1456.0,299.0){\line(1,0){150.0}} \put(1456.0,479.0){\line(1,0){150.0}}
}{\color[rgb]{0,0,0}\put(1531.0,479.0){\line(0,-1){180.0}}
}\put(1216.0,119.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1531.0,-106.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1531.0,74.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1846.0,119.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}l'}}}}}
\put(1396.0,209.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1666.0,209.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1666.0,479.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1396.0,479.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\end{picture}}}~{}=\frac{[a][b]}{\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\end{picture}}}}~{}\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\end{picture}}}~{}\raisebox{-18.23682pt}{\leavevmode\hbox{\par\begin{picture}(646.0,713.0)(1028.0,-1284.0){\color[rgb]{0,0,0}\put(1351.0,-961.0){\circle{630.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(0,1){315.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\end{picture}}}~{}\delta_{ll^{\prime}}$ (12)
$\displaystyle=\lambda(a,b,l)\langle l|l^{\prime}\rangle,$ (13)

with

$\lambda(a,b,l)=\frac{\left[\frac{a-b+l}{2}\right]\left[\frac{-a+b+l}{2}\right]-\left[\frac{a+b-l}{2}\right]\left[\frac{a+b+l}{2}+2\right]}{[2]},$
(14)

where we have evaluated $\operatorname{Tet}(a,a,b,b;l,2)$ as

$\raisebox{-18.23682pt}{\leavevmode\hbox{\par\begin{picture}(646.0,713.0)(1028.0,-1284.0){\color[rgb]{0,0,0}\put(1351.0,-961.0){\circle{630.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(0,1){315.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\end{picture}}}=\frac{\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}l}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\end{picture}}}}{[a][b]}\lambda(a,b,l).$ (15)

This result may be obtained directly from Equation (34), where the sum reduces
to two terms, or from more fundamental considerations [20]. In the limit,
$q\to 1$, the eigenvalues simplify to
$\lambda(a,b,l)=\frac{1}{4}[l(l+2)-a(a+2)-b(b+2)]$, which shows that the
operator $L$ is closely related to the “square of angular momentum” in quantum
mechanics, which was used to obtain the classical version of this recurrence
relation [10].

On the other hand, in the $|\bar{j}\rangle$ basis, the operator $L$ is not
diagonal and the matrix elements
$\bar{L}_{jj^{\prime}}=\langle\bar{j}|L|\bar{j}^{\prime}\rangle$, making use
of the vertex collapse identity, are

$\bar{L}_{jj^{\prime}}=\raisebox{-18.36328pt}{\leavevmode\hbox{\par\begin{picture}(750.0,717.0)(1426.0,-163.0){\color[rgb]{0,0,0}\put(1801.0,479.0){\circle*{46.0}}
}{\color[rgb]{0,0,0}\put(1801.0,299.0){\circle*{46.0}}
}{\color[rgb]{0,0,0}\put(1981.0,164.0){\line(1,0){135.0}}
}{\color[rgb]{0,0,0}\put(1486.0,164.0){\line(1,0){135.0}}
}{\color[rgb]{0,0,0}\put(1711.0,74.0){\oval(450.0,450.0)[bl]}
\put(1711.0,254.0){\oval(450.0,450.0)[tl]}
\put(1891.0,74.0){\oval(450.0,450.0)[br]}
\put(1891.0,254.0){\oval(450.0,450.0)[tr]}
\put(1711.0,-151.0){\line(1,0){180.0}} \put(1711.0,479.0){\line(1,0){180.0}}
\put(1486.0,74.0){\line(0,1){180.0}} \put(2116.0,74.0){\line(0,1){180.0}}
}{\color[rgb]{0,0,0}\put(1756.0,164.0){\oval(270.0,270.0)[bl]}
\put(1756.0,164.0){\oval(270.0,270.0)[tl]}
\put(1846.0,164.0){\oval(270.0,270.0)[br]}
\put(1846.0,164.0){\oval(270.0,270.0)[tr]} \put(1756.0,29.0){\line(1,0){90.0}}
\put(1756.0,299.0){\line(1,0){90.0}} \put(1621.0,164.0){\line(0,1){0.0}}
\put(1981.0,164.0){\line(0,1){0.0}}
}{\color[rgb]{0,0,0}\put(1801.0,479.0){\line(0,-1){180.0}}
}\put(1801.0,74.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1801.0,-106.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1441.0,119.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1936.0,299.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1981.0,479.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1621.0,479.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1666.0,299.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(2161.0,119.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$j^{\prime}$}}}}}
\end{picture}}}~{}=\frac{[a][b]}{\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$j^{\prime}$}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}}~{}\raisebox{-18.23682pt}{\leavevmode\hbox{\par\begin{picture}(646.0,713.0)(1028.0,-1284.0){\color[rgb]{0,0,0}\put(1351.0,-961.0){\circle{630.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(0,1){315.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$j^{\prime}$}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\end{picture}}}~{}\raisebox{-18.23682pt}{\leavevmode\hbox{\par\begin{picture}(646.0,713.0)(1028.0,-1284.0){\color[rgb]{0,0,0}\put(1351.0,-961.0){\circle{630.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(0,1){315.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}c}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$j^{\prime}$}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}b}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\end{picture}}},$ (16)

with the special case $\bar{L}_{00}=0$. Fortunately, though
$\bar{L}_{jj^{\prime}}$ is not diagonal, it is tridiagonal. This property is a
consequence of the conditions enforced at the central vertex in both
$\operatorname{Tet}$-networks above: the triangle inequality,
$|j-j^{\prime}|\leq 2$, and the parity constraint, which forces admissible
values of $j$ to change by $2$. If these conditions are violated, the matrix
element $\bar{L}_{jj^{\prime}}$ vanishes.

The diagonal elements $\bar{L}_{jj}$ can be evaluated using (15). For the off-
diagonal elements $\bar{L}_{j+2,j}=\bar{L}_{j,{j+2}}$ we also need

$\raisebox{-18.23682pt}{\leavevmode\hbox{\par\begin{picture}(646.0,713.0)(1028.0,-1284.0){\color[rgb]{0,0,0}\put(1351.0,-961.0){\circle{630.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(0,1){315.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(-1,-1){225.0}}
}{\color[rgb]{0,0,0}\put(1351.0,-961.0){\line(1,-1){225.0}}
}\put(1351.0,-1231.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1216.0,-1051.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1486.0,-1051.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$j{+}2$}}}}}
\put(1126.0,-691.0){\makebox(0.0,0.0)[rb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1576.0,-691.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\put(1396.0,-871.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\end{picture}}}=\frac{1}{[a]}\left[\frac{a+d-j}{2}\right]~{}\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}d}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$j+2$}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}a}}}}}
\end{picture}}},$ (17)

which can be obtained in the same way as (15). Finally, we need the identities

$\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}=-\frac{[j+2]}{[2][j]}~{}\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}\quad\text{and}\quad\raisebox{-14.91711pt}{\leavevmode\hbox{\par\begin{picture}(474.0,608.0)(1114.0,-69.0){\color[rgb]{0,0,0}\put(1351.0,164.0){\circle{450.0}}
}{\color[rgb]{0,0,0}\put(1126.0,164.0){\line(1,0){450.0}}
}\put(1351.0,-16.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\put(1351.0,434.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$2$}}}}}
\put(1351.0,209.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}$j{+}2$}}}}}
\end{picture}}}=\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}$j{+}2$}}}}}
\end{picture}}}\phantom{j+{}}.$ (18)

The $|j\rangle$-basis matrix elements can now be expressed as (again, recall
the special case $\bar{L}_{00}=0$)

$\displaystyle\bar{L}_{jj}$ $\displaystyle=-\langle
j|j\rangle\frac{[2]\lambda(a,j,d)\lambda(b,j,c)}{[j][j+2]},$ (19)
$\displaystyle\bar{L}_{j,j+2}$ $\displaystyle=\langle
j{+}2|j{+}2\rangle\left[\frac{a+d-j}{2}\right]\left[\frac{b+c-j}{2}\right].$
(20)

The transition matrix elements $\langle\bar{j}|l\rangle$ can now be obtained
by solving an eigenvalue problem in the $|\bar{j}\rangle$-basis:

$\displaystyle\langle\bar{j}|L-\lambda_{l}|l\rangle$
$\displaystyle=\sum_{j^{\prime}}\frac{\langle\bar{j}|L-\lambda_{l}|\bar{j}^{\prime}\rangle}{\langle\bar{j}^{\prime}|\bar{j}^{\prime}\rangle}\langle\bar{j}^{\prime}|l\rangle,$
(21) $\displaystyle 0$
$\displaystyle=\sum_{j^{\prime}}\left(\frac{\bar{L}_{jj^{\prime}}}{\langle\bar{j}^{\prime}|\bar{j}^{\prime}\rangle}-\lambda_{l}\delta_{jj^{\prime}}\right)\langle\bar{j}^{\prime}|l\rangle,$
(22) $\displaystyle 0$
$\displaystyle=\sum_{j^{\prime}}\left(\bar{L}_{jj^{\prime}}-\lambda_{l}\langle\bar{j}^{\prime}|\bar{j}^{\prime}\rangle\delta_{jj^{\prime}}\right)\frac{\langle\bar{j}^{\prime}|l\rangle}{\langle\bar{j}^{\prime}|j^{\prime}\rangle},$
(23)

where $\lambda_{l}=\lambda(a,b,l)$. Since $\bar{L}_{jj^{\prime}}$ is
tridiagonal, we obtain a three-term recurrence relation for the
$\langle\bar{j}|l\rangle$ transition coefficients. Expanding the expression
for $\bar{L}_{jj^{\prime}}$, we find the following general form of the
recurrence relation:

$\frac{\bar{L}_{j,j-2}}{\langle\overline{j-2}|\overline{j-2}\rangle}\langle\overline{j-2}|l\rangle+\left(\frac{\bar{L}_{jj}}{\langle\bar{j}|\bar{j}\rangle}-\lambda_{l}\right)\langle\bar{j}|l\rangle+\frac{\bar{L}_{j,j+2}}{\langle\overline{j+2}|\overline{j+2}\rangle}\langle\overline{j+2}|l\rangle=0,$
(24)

with the provision that $\bar{L}_{jj^{\prime}}$ vanishes whenever either of
the indices fall outside the admissible range or $j=j^{\prime}=0$. Finally,
the transition coefficients are uniquely determined (up to sign) by requiring
the normalization condition

$\sum_{j}\frac{\langle
l|\bar{j}\rangle\langle\bar{j}|l\rangle}{\langle\bar{j}|\bar{j}\rangle}=\langle
l|l\rangle.$ (25)

Practically, it is more convenient to recover the correct normalization for
all $j$ and fixed $l$, or vice versa, by requiring $\langle\bar{j}|l\rangle$
to agree with (34) for $j=\underline{j}$, cf. (41), where the sum reduces to a
single term.

Once the $\operatorname{Tet}$-network has been evaluated recursively, the
$6j$-symbol can be obtained from Equation (7). Alternatively, a linear, three-
term recurrence relation directly for the $6j$-symbol follows from (24) and
the linear, two-term recurrence relations for the bubble and
$\theta$-networks, obvious from (29) and (30). However, because of the
additional normalization factors in Equation (7), this direct recurrence
relation cannot be cast in the form of a symmetric eigenvalue problem like
(23) using rational operations alone.

The author thanks Dan Christensen for many helpful discussions as well as
Nicolai Reshetikhin and an anonymous referee for pointing out previous work on
the above recurrence relation. In the course of this work, the author was
supported by Postgraduate (PGS) and Postdoctoral (PDF) Fellowships from the
Natural Science and Engineering Research Council (NSERC) of Canada.

##  Appendix A Formulas

For a complex number $q\neq 0$ and an integer $n$ the corresponding _quantum
integer_ is defined as

$[n]=\frac{q^{n}-q^{-n}}{q-q^{-1}}.$ (26)

In the limit $q\to 1$, we recover the regular integers, $[n]\to n$. When
$q=\exp(i\pi/r)$, for some integer $r>1$, it is a _primitive root of unity_
and the definition reduces to

$[n]=\frac{\sin(n\pi/r)}{\sin(\pi/r)},$ (27)

This expression is clearly real and positive in the range $0<n<r$. _Quantum
factorials_ are direct analogs of classical factorials:

$[0]!=1,\quad[n]!=[1][2]\cdots[n].$ (28)

Next, we give the evaluations of some spin networks needed in the paper. They
are reproduced from Ch. 9 of [15]. The _bubble diagram_ evaluates to

$\raisebox{-4.23083pt}{\leavevmode\hbox{\par\begin{picture}(330.0,270.0)(1396.0,-466.0){\color[rgb]{0,0,0}\put(1531.0,-331.0){\circle{254.0}}
}\put(1711.0,-376.0){\makebox(0.0,0.0)[lb]{\smash{{{\color[rgb]{0,0,0}j}}}}}
\end{picture}}}=(-)^{j}[j+1]$ (29)

whenever it is non-vanishing. For generic $q$, it vanishes if $j<0$ and if $q$
is a primitive root of unity then it also vanishes when $j>r-2$. The _$\theta$
-network_ evaluates to

$\theta(a,b,c)=\frac{(-)^{s}[s+1]![s-a]![s-b]![s-c]!}{[a]![b]![c]!},$ (30)

with $s=(a+b+c)/2$, whenever the twice-spins $(a,b,c)$ are admissible and
vanishes otherwise. Admissibility consists of the following criteria (besides
the obvious $a,b,c\geq 0$):

triangle inequalities $\displaystyle\left\\{\begin{aligned} a&\leq b+c\\\
b&\leq c+a\\\ c&\leq a+b\end{aligned}\right.,$ (31) parity $\displaystyle
a+b+c\equiv 0\pmod{2}.$ (32) When $q$ is a primitive root of unity, further
constraints needs to be satisfied: $r$-boundedness
$\displaystyle\left\\{\begin{aligned} a,b,c&\leq r-2\\\ a+b+c&\leq
2r-4\end{aligned}\right..$ (33)

The _tetrahedral-_ or _$\operatorname{Tet}$ -network_ evaluates to

$\operatorname{Tet}(a,b,c,d;j,l)=\frac{\mathcal{I}!}{\mathcal{E}!}\sum_{S}\frac{(-)^{S}[S+1]!}{\prod_{\imath}[S-a_{\imath}]!\prod_{\jmath}[b_{\jmath}-S]!},$
(34)

where the summation is over the range $m\leq S\leq M$ and

$\displaystyle\mathcal{I}!$
$\displaystyle=\prod_{\imath,\jmath}[b_{\jmath}-a_{\imath}]!,$
$\displaystyle\mathcal{E}!$ $\displaystyle=[a]![b]![c]![d]![j]![l]!,$ (35)
$\displaystyle a_{1}$ $\displaystyle=(a+d+j)/2,$ $\displaystyle b_{1}$
$\displaystyle=(b+d+j+l)/2,$ (36) $\displaystyle a_{2}$
$\displaystyle=(b+c+j)/2,$ $\displaystyle b_{2}$ $\displaystyle=(a+c+j+l)/2,$
(37) $\displaystyle a_{3}$ $\displaystyle=(a+b+l)/2,$ $\displaystyle b_{3}$
$\displaystyle=(a+b+c+d)/2,$ (38) $\displaystyle a_{4}$
$\displaystyle=(c+d+l)/2,$ $\displaystyle m$
$\displaystyle=\max\\{a_{\imath}\\},$ (39) $\displaystyle M$
$\displaystyle=\min\\{b_{\jmath}\\}.$ (40)

The indices $\imath$ and $\jmath$ fully span the defined ranges. Each of the
triples of twice-spins $(a,b,l)$, $(c,d,l)$, $(a,d,j)$ and $(c,b,j)$ must be
admissible, otherwise the $\operatorname{Tet}$-network vanishes. Then, due to
parity constraints, $a_{\imath}$, $b_{\jmath}$, $m$, $M$, and $S$ are always
integers. If the twice-spins $(a,b,c,d)$ are fixed, the admissibility
conditions for generic $q$ enforce the ranges of $\underline{j}\leq
j\leq\overline{j}$ and $\underline{l}\leq l\leq\overline{l}$ to

$\displaystyle\underline{j}$ $\displaystyle=\max\\{|a-d|,|b-c|\\},$
$\displaystyle\overline{j}$ $\displaystyle=\min\\{a+d,b+c\\},$ (41)
$\displaystyle\underline{l}$ $\displaystyle=\max\\{|a-b|,|c-d|\\},$
$\displaystyle\overline{l}$ $\displaystyle=\min\\{a+b,c+d\\},$ (42)

with

$\displaystyle j$ $\displaystyle\equiv a+b\equiv c+d\pmod{2},$ (43)
$\displaystyle l$ $\displaystyle\equiv a+d\equiv b+c\pmod{2}.$ (44)

The number of admissible values is the same for $j$ and $l$ and is equal to
$n=\max\\{0,\bar{n}\\}$, where

$\displaystyle\bar{n}$ $\displaystyle=\min\\{m,s-M\\}+1,$ $\displaystyle m$
$\displaystyle=\min\\{a,b,c,d\\},$ (45) $\displaystyle s$
$\displaystyle=(a+b+c+d)/2,$ $\displaystyle M$
$\displaystyle=\max\\{a,b,c,d\\}.$ (46)

This number $n$ is also the dimension of the space of $4$-valent spin networks
with fixed twice-spins $(a,b,c,d)$ labeling the free edges. This dimension is
non-vanishing, $n>0$, precisely when the twice-spins satisfy the conditions

$\displaystyle a+b+c+d$ $\displaystyle\leq 2\max\\{a,b,c,d\\},$ (47)
$\displaystyle a+b+c+d$ $\displaystyle\equiv 0\pmod{2}.$ (48)

When $q$ is a primitive root of unity, the admissible ranges shrink to
$\underline{l}\leq j\leq\overline{j}_{r}$ and $\underline{l}\leq
l\leq\overline{l}_{r}$, where

$\displaystyle\overline{j}_{r}$
$\displaystyle=\min\left\\{\overline{j},r-2,2r-4-\max\\{a+d,b+c\\}\right\\},$
(49) $\displaystyle\overline{l}_{r}$
$\displaystyle=\min\left\\{\overline{l},r-2,2r-4-\max\\{a+b,c+d\\}\right\\}.$
(50)

The number of admissible values in each range is thus restricted to
$n=\max\\{0,\bar{n}_{r}\\}$, where

$\bar{n}_{r}=\min\left\\{\bar{n},r-1-\max\\{M,s-m\\}\right\\}.$ (51)

The condition $n>0$ requires (47), (48) and

$a+b+c+d\leq 2\min\\{a,b,c,d\\}+2r-4.$ (52)

The above admissibility criteria are well known. However, the consequent
explicit expressions for the constraints on $(a,b,c,d)$, the bounds on $j$ and
$l$, and the dimension $n$ are not easily found in the literature.

In the classical $q=1$ case, the Kauffman-Lins version of the $6j$-symbol (7)
differs from the Racah-Wigner convention used in the physics literature, which
preserves the symmetries of the underlying $\operatorname{Tet}$-network. The
two $6j$-symbols are related through the formula

$\begin{Bmatrix}{j_{1}/2}&{j_{2}/2}&{j_{3}/2}\\\
{J_{1}/2}&{J_{2}/2}&{J_{3}/2}\end{Bmatrix}_{RW}=\frac{\operatorname{Tet}(J_{1},J_{2},j_{1},j_{2};J_{3},j_{3})}{\sqrt{|\theta(J_{1},J_{2},j_{3})\theta(j_{1},j_{2},j_{3})\theta(J_{1},j_{2},J_{3})\theta(J_{2},j_{1},J_{3})}|},$
(53)

which can be obtained by comparing the explicit expressions (34) and (6.3.7)
of [17]. Note that the argument of the absolute value under the square root
has sign $(-)^{j_{3}-J_{3}}$.

## References

## References

  * [1] Edward Witten.  Quantum field theory and the Jones polynomial.  Communications in Mathematical Physics, 121(3):351–399, 1989.
  * [2] Lee Smolin.  Quantum gravity with a positive cosmological constant.  2002\.
  * [3] Igor Khavkine and J. Daniel Christensen.  $q$-deformed spin foam models of quantum gravity.  Classical and Quantum Gravity, 24(13):3271–3290, 2007.
  * [4] Richard Askey and James Wilson.  A Set of Orthogonal Polynomials That Generalize the Racah Coefficients or $6-j$ Symbols.  SIAM Journal on Mathematical Analysis, 10(5):1008, 1979.
  * [5] A. N. Kirillov and N. Yu Reshetikhin.  Representations of the algebra $U_{q}(sl(2))$, $q$-orthogonal polynomials and invariants of links.  In Victor G. Kac, editor, Infinite-dimensional Lie Algebras and Groups, volume 7 of Advanced Series in Mathematical Physics, pages 285–339, Singapore, 1989. World Scientific.
  * [6] I. I. Kachurik and A. U. Klimyk.  General recurrence relations for clebsch-gordan coefficients of the quantum algebra $U_{q}(\mathrm{su}_{2})$.  Journal of Physics A: Mathematical and General, 24(17):4009–4015, 1991.
  * [7] Shun’ya Mizoguchi and Tsukasa Tada.  3-Dimensional Gravity and the Turaev-Viro Invariant.  Progress of Theoretical Physics Supplement, 110:207–227, 1992.
  * [8] Shun’ya Mizoguchi and Tsukasa Tada.  Three-dimensional gravity from the Turaev-Viro invariant.  Physical Review Letters, 68(12):1795–1798, 1992.
  * [9] G. Ponzano and T. Regge.  Semiclassical limit of racah coefficients.  In F. Bloch, S. G. Cohen, A. De-Shalit, S. Sambursky, and I. Talmi, editors, Spectroscopic and group theoretical methods in physics, pages 1–98, Amsterdam, 1968. North-Holland.
  * [10] Klaus Schulten and Roy G. Gordon.  Exact recursive evaluation of 3j- and 6j-coefficients for quantum-mechanical coupling of angular momenta.  Journal of Mathematical Physics, 16(10):1961–1970, 1975.
  * [11] Yuka Taylor and Christopher Woodward.  6j symbols for u_q(sl_2) and non-Euclidean tetrahedra.  Selecta Mathematica (New Series), 11(3):539–571, 2006.
  * [12] P. P. Raychev, R. P. Roussev, and Yu F. Smirnov.  The quantum algebra SUq(2) and rotational spectra of deformed nuclei.  Journal of Physics G: Nuclear and Particle Physics, 16(8):L137, 1990\.
  * [13] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen.  LAPACK Users’ Guide (Software, Environments and Tools).  Society for Industrial Mathematics, 3 edition, 1987.
  * [14] R. Penrose.  Applications of negative dimension tensors.  In D. J. A. Welsh, editor, Combinatorial Mathematics and its Applications, pages 221–244. Academic Press, 1971.
  * [15] Louis H. Kauffman and Sóstenes Lins.  Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, volume 134 of Annals of Mathematics Studies.  Princeton University Press, 1994.
  * [16] J. Scott Carter, Daniel E. Flath, and Masahico Saito.  The Classical and Quantum 6j-symbols, volume 43 of Mathematical Notes.  Princeton University Press, 1995.
  * [17] A. R. Edmonds.  Angular Momentum in Quantum Mechanics.  Princeton Landmarks in Physics. Princeton University Press, 1996.
  * [18] A. P. Yutsis, J. B. Levinson, and V. V. Vanagas.  Mathematical Apparatus of the Theory of Angular Momentum.  Israel Program for Scientific Translation, 1962.
  * [19] J. Wade Cherrington, J. Daniel Christensen, and Igor Khavkine.  Dual computations of non-abelian yang-mills theories on the lattice.  Physical Review D, 76(9):094503, 2007.
  * [20] G. Massbaum and P. Vogel.  3-valent graphs and the kauffman bracket.  Pacific Journal of Mathematics, 164(2):361–381, 1994.