# Quantum clebsch gordan coefficients and conformal field theory

Abhishek Hegde K R

Chennai Mathematical Institute, SIPCOT IT Park, Siruseri, Kelambakkam, Tamil Nadu 603103

Indian Institute of Technology Bombay, Powai, Mumbai, Maharashtra 400076

## Abstract

Group theory plays an important role in understanding the symmetries of a physical system. Tensor product representations SU(2) and SU(3) describe quantum mechanical spin systems and flavour symmetry respectively. Quantum groups are generalisations of classical Lie Algebras. The structure of Quantum groups SUq(2) and SUq(3) follow close analogies with classical Lie Algebras SU(2) and SU(3). After a brief review of Lie Algebra SU(3), tensor product representations of SUq(3) are studied and quantum Clebsch Gordan coefficients (qCGs) are tabulated for specific cases. Conformal symmetry in two dimensions imposes severe constraints on correlation functions of primary fields in a 2d conformal field theory. The conformal bootstrap program aims at exploiting these symmetries to solve the theory. Rational Conformal Field Theories (RCFTs) are characterised by finite number of primary fields. Additional Lie Algebra symmetries in the theory give rise to current algebras. Representation of SU(2) current algebras at level k and fusion rules of conformal families in RCFTs have close analogies with tensor product representations of quantum group SUq(2) where $q=e^{\frac{2\pi i}{k+2}}$. As part of the report, basics of conformal field theory are reviewed and the relationship between representation of SUq(2) and SU(2) current algebras in RCFTs are studied.

Keywords: quantum groups, conformal field theory, quantum clebsch gordan coefficients

## Structure Of Lie Group SU(3) and its Lie Algebra

Lie groups play an important role in modelling continuous symmetries of a system. Lie groups are equipped with a smooth manifold structure and product and inverse operations are smooth functions. The tangent space at identity of a Lie Group is called it's Lie Algebra. Lie algebra of a Lie group model infinitesimal transformations, these are transformations which differ slightly from the identity. Moreover, elements of (more precisely, connected components) Lie groups can be generated by exponentiation. Precise mathematical definitions are given below,

Definition 1.1.1: A smooth manifold $G$ is called a Lie group if the product operation $\mu :G\times G\rightarrow G$ is a smooth map

$\mu(x,y)=x^{-1}y$. Note that this already ensures that inverse operation is a smooth function.

Definition 1.1.2: The Lie Algebra $\mathfrak g$of a Lie group consists of its tangent space at identity.

$\mathfrak g$is also equipped with a Lie bracket structure,

$\displaystyle \left[ .,.\right]:\mathfrak g \times \mathfrak g\rightarrow \mathfrak g$

which an alternating billinear map and satisfies the jacobi identity,

$\displaystyle \left[ x,\left[ y,z\right]\right]+\left[ y,\left[ z,x\right]\right]+\left[ z,\left[ x,y\right]\right]=0$

this structure on the Lie algebra is inherited by the isomorphism between left invariant vector fields and Lie algebra $\mathfrak g$. See, (​Javier P Muniain, 1994​ ) for more details.​

Definition 1.1.3: Lie group is defined as the set of unitary 3 dimensional matrices whose determinant is one. $SU(3):=\left\{A \in U(3) , detA=1\right\}$ . It is an 8 dimensional Lie group. Lie algebra is denoted by $\mathfrak {su}{(3)}$.

## Representation Theory of $\mathfrak {su}{(3)}$​

​Here on,representations mean finite-dimensional representations.

We choose to work in a specific basis for $\mathfrak {su}{(3)}$. This basis in the defining representation consists of traceless hermitian matrices. They are called the Gell-Mann matrices.

$\displaystyle X_{ 1 }=\left( \begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \right) \quad X_{ 2 }=\left( \begin{matrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \right) \\ X_{ 3 }=\left( \begin{matrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{matrix} \right) \quad X_{ 4 }=\left( \begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} \right) \\ X_{ 5 }=\left( \begin{matrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{matrix} \right) \quad X_{ 6 }=\left( \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} \right) \\ X_{ 7 }=\left( \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{matrix} \right) \quad X_{ 8 }=\frac { 1 }{ \sqrt { 3 } } \left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{matrix} \right) \\$

Representations of Lie algebra on state space of a system helps in modelling operators of the theory. A maximal set of commuting lie subalgebra form the cartan subalgebra. They can be simultaneously diagonalised. We label states using eigenvalues and eigenvectors of these operators. Refer ​Howard Georgi, 2018​ for more details.

In Gell-Mann basis we see that $X_{3}$and $X_{8}$are diagonalised and we can also check that $[X_{3},X_{8}]=0$. The eigenvalues of these operators form weight vectors. For normalisation purposes, it is convinient to work with $H_{1}=\frac{X_{3}}{2}$and $H_{2}=\frac{X_{8}}{2}$. Simultaneous diagonalisation gives,

$H_{ 1 }{ e }_{ 1 }=\frac { 1 }{ 2 } { e }_{ 1 }\ ;{ H }_{ 2 }{ e }_{ 1 }=\frac { 1 }{ 2\sqrt { 3 } } e_{ 1 }\\H _{ 1 }{ e }_{ 2 }=\frac { -1 }{ 2 } { e }_{ 2 }\ ;{ H }_{ 2 }{ e }_{ 2 }=\frac { 1 }{ 2\sqrt { 3 } } { e }_{ 2 }\\H _{ 1 }{ e }_{ 3 }=0\ ; { H }_{ 2 }{ e }_{ 3 }=\frac { -1 }{ \sqrt { 3 } } { e }_{ 3 }$

​Hence we read off the weight vectors as,

$\mu _{1}= (\frac{1}{2},\frac{{1}}{2\sqrt{3}})\ ; \mu _{2}= (\frac{-1}{2},\frac{{1}}{2\sqrt{3}})\ ; \mu _{3}= (0,\frac{{-1}}{\sqrt{3}})$

$\mu_{1}$is a positive weight vector and $\mu_{2},\mu_{3}$are negative weight vectors.​

Commutation relations indicate that the operators,

$E_{\pm\alpha_1}=\frac1{2\sqrt2}\left(X_4\pm{iX}_5\right)\quad E_{\pm\alpha_2}=\frac1{2\sqrt2}\left(X_6\pm{iX}_7\right)\quad E_{\pm\alpha_3}=\frac1{2\sqrt2}\left(X_1\pm{iX}_2\right)$

act as raising and lowering operators on the basis $e_{1},e_{2},e_{3}$. It is convinient to label basis vectors by their weight labels. So we rename $e_{i}$to $\left| { \mu }_{ i } \right>$. Highest weight vector is the one which cannot be raised anymore by the action of raising operators. We conclude that $\left| { \mu }_{ 1 } \right>$is the highest weight vector.

Action of lowering operator give the roots of the algebra.

${ E }_{ -{ \alpha }_{ i } }\left| { \mu }_{ i } \right> \quad\propto \left| { \mu }_{ i }- \alpha_{i}\right >\\{ \alpha }_{ 1 }=\left( \frac { 1 }{ 2 } ,\frac { \sqrt { 3 } }{ 2 } \right) ; { \alpha }_{ 2 }=\left( \frac { 1 }{ 2 } ,\frac { -\sqrt { 3 } }{ 2 } \right) ;\ { \alpha }_{ 3 }={ \alpha }_{ 1 }+{ \alpha }_{ 2 }=\left( 1,0 \right)$

Here, are $\alpha_{1},\alpha_2$called simple roots (they cannot be written as the sum of other roots).

These are general features of finite dimensional irreducible representations of compact semi simple Lie algebras. Representations of $\mathfrak {su}(2)$ on state space of a quantum mechanical system models angular momentum and spin operators. From quantum mechanics, we know that eigenvalue of z-component of angular momentum runs from $m=j,j-1,..,0,..,-j$. And we move down from a higher eigenvalue to lower one by lowering operator $J_{-}$.Same analogy follows through. But,we need to replace eigenvalues by root systems and weight systems. ​The operator $J_{z}$ is replaced by operators $E_{\alpha_1},E_{\alpha_2}$ and raising and lowering operators are analogous with $E_{\pm \alpha_i}$. For more details see ​Howard Georgi, 2018​.

Facts about irreducible representations of $\mathfrak {su}(3)$ are summarised below,

• Irreducible representations are characterised by a highest weight vector $\left| \Lambda \Lambda \right>$ . This vector cannot be raised to a higher state by applying raising operators.
• Fundamental weight vectors are given by $\mu _{1}= (\frac{1}{2},\frac{{1}}{2\sqrt{3}})\ ; \mu _{2}= (\frac{-1}{2},\frac{{1}}{2\sqrt{3}})$
• Simple roots are given by ${ \alpha }_{ 1 }=\left( \frac { 1 }{ 2 } ,\frac { \sqrt { 3 } }{ 2 } \right) ; { \alpha }_{ 2 }=\left( \frac { 1 }{ 2 } ,\frac { -\sqrt { 3 } }{ 2 } \right)$
• $\Lambda = a\mu_{1}+b\mu_{2} \quad ; a,b \in \mathbb Z$ and the irreducible representation is denoted by $D(a,b)$
• $dim(D(a,b))= \frac{1}{2}(a+1)(b+1)(a+b+2)$
• Action of raising and lower operators on states are given by,

$E_{ \alpha _{ i } }\left| \Lambda \Lambda \right> =\Lambda .\alpha _{ i }\left| \Lambda \Lambda \right> \\ E_{ \alpha _{ i } }\left| \Lambda \Lambda -m\alpha _{ 1 }-n\alpha _{ 2 } \right> =(\Lambda -m\alpha _{ 1 }-n\alpha _{ 2 }).\alpha _{ i }\left| \Lambda \Lambda -m\alpha _{ 1 }-n\alpha _{ 2 } \right> \\ \\ E_{ \pm \alpha _{ i } }\left| \Lambda \Lambda -m\alpha _{ 1 }-n\alpha _{ 2 } \right> =\sqrt { (\Lambda \mp (\Lambda -m\mu _{ 1 }-n\mu _{ 2 })).\alpha _{ i } } \sqrt { (\Lambda \pm (\Lambda -m\mu _{ 1 }-n\mu _{ 2 })).\alpha _{ i }+1 } \left| \Lambda \Lambda -m\alpha _{ 1 }-n\alpha _{ 2 }\pm { \alpha }_{ i } \right>$

## Tensor Representations, Young Tableaux and Clebsch Gordan Coefficients

Tensor product representations of two irreducible representations generally do not form irreducible representations. Tensor product of two irreducible representations can be decomposed as a Clebsch Gordan series.

$D(p_{1},q_{1}) \otimes D(p_{2},q_{2})=\sum_{P,Q}\oplus \sigma(P,Q)D(P,Q)$

Here are $\sigma(P,Q)$ integers which denote the multiplicities.

Two example of such a decomposition are,

• $D(1,0)\otimes D(0,1)=D(1,1)\oplus D(0,0)$
• $D(1,1)\otimes D(1,1)=D(2,2)\oplus D(3,0)\oplus D(1,1) \oplus D(1,1) \oplus D(0,0)$

Clebsch Gordan coefficients (CGs) are elements of change of basis matrix. They show up when we expand coupled state vector interms of uncoupled tensor product basis. The explicit tabulation of CGs is given in sec1.3.2.

A convenient method to study the structure of tensor product representations is using Young Tableaux. Refer ​Howard Georgi, 2018​ for more details.​

## Weight diagrams for $D(1,0)\otimes D(1,0)$​

Young Diagram

From Young Tableaux we see that, $D(1,0)\otimes D(1,0)= D(2,0)\oplus D(0,1)$.

The state vectors are represented in a weight diagram as follows,

Weight Diagram for  $\left| { \mu }_{ 1 } \right>$
Weight Diagram for  $\left| {2 \mu }_{ 1 } \right>$
Weight Diagram  $\left| { \mu }_{ 2} \right>$

## Clebsch Gordan Coefficients For $D(1,0)\otimes D(1,0)$​

 $\left| { 2\mu }_{ 1 } \right>$ $\left| { \mu }_{ 1 } \right>\left| { \mu }_{ 1 } \right>$ $1$
 $\left| { 2\mu }_{ 1 }-\alpha_1 \right>$ $\left| { \mu }_{ 2} \right>$ $\left| { \mu }_{ 1 }-\alpha_1 \right>\left| { \mu }_{ 1 } \right>$ $\frac1{\sqrt2}$ $\frac1{\sqrt2}$ $\left| { \mu }_{ 1 } \right>\left| { \mu }_{ 1 }-\alpha_1 \right>$ $\frac1{\sqrt2}$ $-\frac1{\sqrt2}$

## Note on reading CG's from the table:

• Table 1 is read as, $\left| { 2\mu }_{ 1 }{ 2\mu }_{ 1 } \right>=1\left| {\mu_1}\mu_1 \right>\otimes\left| {\mu_1}\mu_1 \right>$​.
• Table 2, 1st column is a state from $D(2,0)$ so $\left| { 2\mu }_{ 1 }-{ \alpha }_{ 1 } \right>$is read as $\left| { { 2\mu }_{ 1 }\quad 2\mu }_{ 1 }-{ \alpha }_{ 1 } \right>$.
• Table 2, 2nd column is a state from $D(0,1)$so $\left| { \mu }_{ 2} \right>$ is read as $\left| \mu _{ 2 } \mu _{ 2 } \right>$.
• Going down a column we get the CG coefficients, from Table 2, column1 is read as $\left| { 2\mu }_{ 1 } \quad { 2\mu }_{ 1 }-{ \alpha }_{ 1 } \right> = \frac{1}{\sqrt 2}\left| { \mu }_{ 1 } \quad { \mu }_{ 1 }-{ \alpha }_{ 1 } \right>\otimes\left| { \mu }_{ 1 } \quad { \mu }_{ 1 } \right> +\frac{1}{\sqrt 2}\left| { \mu }_{ 1 } \quad { \mu }_{ 1 } \right>\otimes\left| { \mu }_{ 1 } \quad { \mu }_{ 1 }-{ \alpha }_{ 1 } \right>$
• All the other tables are to be read as mentioned above.​
 $\left| { 2\mu }_{ 1 }-\alpha_1-\alpha_2 \right>$ $\left| { \mu }_{ 2 }-\alpha_2\right>$ $\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>\left| { \mu }_{ 1 } \right>$ $\frac1{\sqrt2}$ $\frac1{\sqrt2}$ $\left| { \mu }_{ 1 } \right>\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>$ $\frac1{\sqrt2}$ $-\frac1{\sqrt2}$
 $\left| { 2\mu }_{ 1 }-2\alpha_1-\alpha_2 \right>$ $\left| { \mu }_{ 2 }-\alpha_2-\alpha_1\right>$ $\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>\left| { \mu }_{ 1 } -\alpha_1\right>$ $\frac1{\sqrt2}$ $\frac1{\sqrt2}$ $\left| { \mu }_{ 1 }-\alpha_1 \right>\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>$ $\frac1{\sqrt2}$ $-\frac1{\sqrt2}$
 $\left| { 2\mu }_{ 1 }-2\alpha _{ 1 }-2\alpha _{ 2 } \right>$ $\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>$ $1$

Structure of quantum groups are studied in the following section.

## REPRESENTATIONS OF QUANTUM GROUPS

Quantum groups are deformations of Classical Lie Algebras. The mathematical structure of quantum groups is analysed in (​Christian Kassel, 1995​ ) . The structure of quantum group $\mathfrak {su}_q(2)$are mentioned below and connection representation theory of Classical Lie Algebra $\mathfrak{su}(2)$is outlined.

## Structure Of Quantum Group $\mathfrak {su}_q(2)$

Refer to (​A U Klimyk, 1990​) ,(​S. K. Suslov, 1994​) for more details.

​The quantum algebra is generated by ${ J }_{ + },{ J }_{ - },{ J }_{ z }$and commutation relations are as follows,

$\displaystyle { \left[ J,{ J }_{ \pm } \right] }=J_{ \pm }$
$\displaystyle \left[ { J }_{ + },{ J }_{ - } \right] =\frac { { q }^{ J }-{ q }^{ -J } }{ { q }^{ \frac { 1 }{ 2 } }-{ q }^{ -\frac { 1 }{ 2 } } }$

q-numbers and factorials are defined as follows,

$\displaystyle \left[ n \right]\coloneqq \frac { { q }^{ \frac{n}{2}}-{ q }^{ -\frac{n}{2} } }{ { q }^{ \frac { 1 }{ 2 } }-{ q }^{ -\frac { 1 }{ 2 } } }$
$\displaystyle \left[ n \right]! \coloneqq \left[ n \right]\left[ n-1 \right]...[1]$

Square root operations are defined similarly. We see that $\lim_{q\to 1} [n] = n$.

Irreducible representations of $\mathfrak {su}_q(2)$follow close analogy with that of ​ $\mathfrak{su}(2)$. However tensor operators take different form.

$\displaystyle \triangle J=I\otimes J + J\otimes I$
$\displaystyle \triangle J_\pm=J_\pm\otimes q^\frac{J}2+\;q^{-\frac{J}2}\otimes J_\pm$

Facts about representations of $\mathfrak {su}_q(2)$are summarised below. Refer (5),(6) for details.

When $q$is not a root of unity:

• In analogy with the classical case, states of the representations are labelled $\left| l,m \right>$where $l$ takes integer and half integer values and $m=l,l-1,..,0,..-l$.
• The action of raising and lowering operators on $\left| l,m \right>$is given by,
$\displaystyle J\left| { l,m}\right>=m\left| { l,m}\right>$
$\displaystyle J_{ \pm }\left| { l,m } \right> =\sqrt { \left[ l\mp m \right] \left[ l\pm m+1 \right] } \left| { l,m } \right>$
• CG are defined as $\left| l,m \right> = \sum_{m_1,m_2} C^{lm}_{l_1m_1l_2m_2}\left| { l_{ 1 },m_{ 1 } } \right> \left| { l_{ 2 },m_{ 2 } } \right>$
• The general formula for CG is,
$\displaystyle C^{ jm }_{ j_{ 1 }m_{ 1 }j_{ 2 }m_{ 2 } }=\\ (-1)^{ j_{ 1 }-m_{ 1 } }\delta _{ m,m_{ 1 }+m_{ 2 } }q^{ \frac { 1 }{ 4 } (j_{ 2 }(j_{ 2 }+1)-j_{ 1 }(j_{ 1 }+1)-j(j+1))+\frac { (m+1)m_{ 1 } }{ 2 } }\\ \times { \left\{ \frac { \left[ 2j+1 \right] \left[ { j }_{ 1 }+{ j }_{ 2 }-j \right] !\left[ { j }_{ 1 }-{ m }_{ 1 } \right] !\left[ { j }_{ 2 }-{ m }_{ 2 } \right] !\left[ j+m \right] !\left[ j-m \right] ! }{ \left[ j+{ j }_{ 1 }+{ j }_{ 2 }+1 \right] !\left[ j+{ j }_{ 1 }-{ j }_{ 2 } \right] !\left[ j+{ j }_{ 2 }-{ j }_{ 1 } \right] !\left[ { j }_{ 1 }+{ m }_{ 1 } \right] !\left[ { j }_{ 2 }+{ m }_{ 2 } \right] ! } \right\} }^{ \frac { 1 }{ 2 } }\\ \times \sum _{ k=0 }^{ j-m }{ \frac { { (-1) }^{ k }{ q }^{ \frac { k(j+m+1) }{ 2 } }\left[ { j }_{ 2 }+j-{ m }_{ 1 }-k \right] !\left[ { j }_{ 1 }+{ m }_{ 1 }+k \right] ! }{ \left[ k \right] !\left[ j-m-k \right] !\left[ { j }_{ 2 }-j+{ m }_{ 1 }+k \right] \left[ { j }_{ 1 }-{ m }_{ 1 }-k \right] ! } }$

## Structure of Quantum Group $\mathfrak{su}_q(3)$​

Analogy carries on with $\mathfrak{su}_q(3)$, $q$is not a root of untiy. Facts about representations of $\mathfrak{su}_q(3)$is summarised below.​

• Irreducible representations are characterised by a highest weight vector $\left| \Lambda \Lambda \right>$ . This vector cannot be raised to a higher state by applying raising operators.
• The fundamental weights and simple roots are the same as that of $\mathfrak {su}(3)$
• Tensor operators are replaced by,
$\displaystyle \triangle J_{\alpha_i}=I\otimes J_{\alpha_i}+J_{\alpha_i}\otimes I$
$\displaystyle \triangle J_{\pm\alpha_i}= J_{\pm\alpha_i}\otimes q^\frac{ J_{\alpha_i}}{2}+\;q^{-\frac{ J_{\alpha_i}}{2}}\otimes J_{\pm\alpha_i}$
• Raising and lowering operators act on states as,
$\displaystyle J_{ \alpha _{ i } }\left| \Lambda \Lambda \right> =\Lambda .\alpha _{ i }\left| \Lambda \Lambda \right>$
$\displaystyle J_{ \alpha _{ i } }\left| \Lambda \Lambda -m\alpha _{ 1 }-n\alpha _{ 2 } \right> =(\Lambda -m\alpha _{ 1 }-n\alpha _{ 2 }).\alpha _{ i }\left| \Lambda \Lambda -m\alpha _{ 1 }-n\alpha _{ 2 } \right>$
$\displaystyle J_{ \pm \alpha _{ i } }\left| \Lambda \Lambda -m\alpha _{ 1 }-n\alpha _{ 2 } \right> =$

$J_{ \pm \alpha _{ i } }\left| \Lambda \Lambda -m\alpha _{ 1 }-n\alpha _{ 2 } \right> =\\\sqrt { \left[ \left( \Lambda \mp (\Lambda -m\mu _{ 1 }-n\mu _{ 2 }) \right) .{ \alpha }_{ i } \right] } \sqrt { \left[ \left( \Lambda \pm (\Lambda -m\mu _{ 1 }-n\mu _{ 2 }) \right) .{ \alpha }_{ i }+1 \right] } \\\left| \Lambda \Lambda -m\alpha _{ 1 }-n\alpha _{ 2 }\pm { \alpha }_{ i } \right>$

## qCG for $D(1,0)\otimes D(1,0)$​

$D(1,0)\otimes D(1,0)=D(2,0)\oplus D(0,1)$

Young Tableaux and Weight Diagrams are given in section 2.3.2

qCGs are tabulated below:

 $\left| { 2\mu }_{ 1 } \right>$ $\left| { \mu }_{ 1 } \right>\left| { \mu }_{ 1 } \right>$ $1$
 $\left| { 2\mu }_{ 1 }-\alpha_1 \right>$ $\left| { \mu }_{ 2} \right>$ $\left| { \mu }_{ 1 }-\alpha_1 \right>\left| { \mu }_{ 1 } \right>$ $\frac { { q }^{ \frac { 1 }{ 4 } } }{ \sqrt { \left[ 2 \right] } }$ $\frac{q^{-\frac14}}{\sqrt{\left[2\right]}}$ $\left| { \mu }_{ 1 } \right>\left| { \mu }_{ 1 }-\alpha_1 \right>$ $\frac{q^{-\frac14}}{\sqrt{\left[2\right]}}$ $-\frac{q^{\frac{1}{4}}}{\sqrt{\left[2\right]}}$
 $\left| { 2\mu }_{ 1 }-\alpha_1-\alpha_2 \right>$ $\left| { \mu }_{ 2 }-\alpha_2\right>$ $\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>\left| { \mu }_{ 1 } \right>$ $\frac { { q }^{ \frac { 1 }{ 4 } } }{ \sqrt { \left[ 2 \right] } }$ $\frac{q^{-\frac14}}{\sqrt{\left[2\right]}}$ $\left| { \mu }_{ 1 } \right>\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>$ $\frac{q^{-\frac14}}{\sqrt{\left[2\right]}}$ $-\frac{q^{\frac{1}{4}}}{\sqrt{\left[2\right]}}$
 $\left| { 2\mu }_{ 1 }-2\alpha_1-\alpha_2 \right>$ $\left| { \mu }_{ 2 }-\alpha_2-\alpha_1\right>$ $\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>\left| { \mu }_{ 1 } -\alpha_1\right>$ $\frac { { q }^{ \frac { 1 }{ 4 } } }{ \sqrt { \left[ 2 \right] } }$ $\frac{q^{-\frac14}}{\sqrt{\left[2\right]}}$ $\left| { \mu }_{ 1 }-\alpha_1 \right>\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>$ $\frac{q^{-\frac14}}{\sqrt{\left[2\right]}}$ $-\frac{q^{\frac{1}{4}}}{\sqrt{\left[2\right]}}$
 $\left| { 2\mu }_{ 1 }-2\alpha _{ 1 }-2\alpha _{ 2 } \right>$ $\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>\left| { \mu }_{ 1 }-\alpha _{ 1 }-\alpha _{ 2 } \right>$ $1$

## qCG for $D(2,0)\otimes D(2,0)$​

$D(2,0)\otimes D(2,0)=D(4,0)\oplus D(2,1)\oplus D(0,2)$

## Weight diagrams

Young Diagram
Weight Diagram for  $\left| { 2\mu }_{ 1} \right>$
Weight Diagram for  $\left| { 4\mu }_{ 1 } \right>$
Weight Diagram for  $\left| { 2\mu }_{ 1 }+\mu _{ 2 } \right>$
Weight Diagram for  $\left| \mu _{ 2 }\right>$

## qCG Tables

 $\left| { 4\mu }_{ 1 } \right>$ $\left| { 2\mu }_{ 1 } \right>\left| { 2\mu }_{ 1 } \right>$ $1$
 $\left| { 4\mu }_{ 1 }-{ \alpha }_{ 1 } \right>$ $\left| { 2\mu }_{ 1 }+\mu _{ 2 } \right>$ $\left| { 2\mu }_{ 1 }-{ \alpha }_{ 1 } \right> \left| { 2\mu }_{ 1 } \right>$ $\frac { \sqrt { \left[ 2 \right] } }{ \sqrt { \left[ 4 \right] } } { q }^{ \frac { 1 }{ 2 } }$ $\frac { \sqrt { \left[ 2 \right] } }{ \sqrt { \left[ 4 \right] } } { q }^{ -\frac { 1 }{ 2 } }$ $\left| { 2\mu }_{ 1 } \right>\left| { 2\mu }_{ 1 }-{ \alpha }_{ 1 } \right>$ $\frac { \sqrt { \left[ 2 \right] } }{ \sqrt { \left[ 4 \right] } } { q }^{ -\frac { 1 }{ 2 } }$ $-\frac { \sqrt { \left[ 2 \right] } }{ \sqrt { \left[ 4 \right] } } { q }^{ \frac { 1 }{ 2 } }$
 $\left| { 4\mu }_{ 1 }-{ 2\alpha }_{ 1 } \right>$ $\left| { 2\mu }_{ 1 }+\mu _{ 2 } -\alpha_1\right>$ $\left| { 2\mu }_{ 2 } \right>$ $\left| { 2\mu }_{ 1 }-{ 2\alpha }_{ 1 } \right>\left| { 2\mu }_{ 1 } \right>$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 3 \right] \left[ 4 \right] } } q$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } }$ $\frac { { q }^{ -\frac { 1 }{ 2 } } }{ \sqrt { \left[ 3 \right] } }$ $\left| { 2\mu }_{ 1 } \right>\left| { 2\mu }_{ 1 }-{ 2\alpha }_{ 1 } \right>$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 3 \right] \left[ 4 \right] } } { q }^{ -1 }$ $-\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } }$ $\frac { { q }^{ \frac { 1 }{ 2 } } }{ \sqrt { \left[ 3 \right] } }$ $\left| { 2\mu }_{ 1 }-{ \alpha }_{ 1 } \right>\left| { 2\mu }_{ 1 }-{ \alpha }_{ 1 } \right>$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 3 \right] \left[ 4 \right] } } \left[ 2 \right]$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } } \left( { q }^{ -\frac { 1 }{ 2 } }-{ q }^{ \frac { 1 }{ 2 } } \right)$ $-\frac1{\sqrt{\left[3\right]}}$
 $\left| { 4\mu }_{ 1 }-{ 3\alpha }_{ 1 } \right>$ $\left| { 2\mu }_{ 1 }+\mu _{ 2 }-2\alpha _{ 1 } \right>$ $\left| { 2\mu }_{ 1 }-{ 2\alpha }_{ 1 } \right> \left| { 2\mu }_{ 1 }-{ \alpha }_{ 1 } \right>$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } } { q }^{ \frac { 1 }{ 2 } }$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } } { q }^{ -\frac { 1 }{ 2 } }$ $\left| { 2\mu }_{ 1 }-{ \alpha }_{ 1 } \right>\left| { 2\mu }_{ 1 }-{ 2\alpha }_{ 1 } \right>$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } } { q }^{ -\frac { 1 }{ 2 } }$ $-\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } } { q }^{ \frac { 1 }{ 2 } }$
 $\left| { 4\mu }_{ 1 }-{ 4\alpha }_{ 1 } \right>$ $\left| { 2\mu }_{ 1 }-{ 2\alpha }_{ 1 } \right>\left| { 2\mu }_{ 1 }-{ 2\alpha }_{ 1 } \right>$ $1$
 $\left| { 4\mu }_{ 1 }-{ \alpha }_{ 1 }-{ \alpha }_{ 2 } \right>$ $\left| { 2\mu }_{ 1 }+\mu _{ 2 }-\alpha _{ 2 } \right>$ $\left| { 2\mu }_{ 1 }-{ \alpha }_{ 1 }-{ \alpha }_{ 2 } \right> \left| { 2\mu }_{ 1 } \right>$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } } { q }^{ \frac { 1 }{ 2 } }$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } } { q }^{ -\frac { 1 }{ 2 } }$ $\left| { 2\mu }_{ 1 } \right>\left| { 2\mu }_{ 1 }-{ \alpha }_{ 1 }-{ \alpha }_{ 2 } \right>$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } } { q }^{ -\frac { 1 }{ 2 } }$ $-\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } } { q }^{ \frac { 1 }{ 2 } }$
 $\left| { 4\mu }_{ 1 }-{ 2\alpha }_{ 1 }-{ \alpha }_{ 2 } \right>$ ${ \left| { 2\mu }_{ 1 }+\mu _{ 2 }-{ \alpha }_{ 1 }-\alpha _{ 2 } \right> }_{ 1 }$ ${ \left| { 2\mu }_{ 1 }+\mu _{ 2 }-{ \alpha }_{ 1 }-\alpha _{ 2 } \right> }_{ 2}$ $\left| { 2\mu }_{ 2 }-{ \alpha }_{ 2 } \right>$ $\left| { 2\mu }_{ 1 }-{ 2\alpha }_{ 1 }-{ \alpha }_{ 2 } \right> \left| { 2\mu }_{ 1 } \right>$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 3 \right] \left[ 4 \right] } } q\quad$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } }$ $0$ $\frac { { q }^{ -\frac { 1 }{ 2 } } }{ \sqrt { \left[ 3 \right] } }$ $\left| { 2\mu }_{ 1 } \right>\left| { 2\mu }_{ 1 }-{ 2\alpha }_{ 1 }-{ \alpha }_{ 2 } \right>$ $\sqrt { \frac { \left[ 2 \right] }{ \left[ 3 \right] \left[ 4 \right] } } { q }^{ -1 }$ $-\sqrt { \frac { \left[ 2 \right] }{ \left[ 4 \right] } }$ $0$