## Monday, September 19, 2016

### The 7-dimensional representation of $G_2$

The exceptional Lie algebra $G_2$ has an irreducible representation of dimension 7. In this post I calculate the matrices of this irrep.
Short overview of $G_2$
If $\alpha_1$ is the long simple root of $G_2$ and $\alpha_2$ is its short simple root, then all positive roots are \begin{equation*} \alpha_1, \alpha_2, \alpha_1 + \alpha_2, \alpha_1 + 2 \alpha_2, \alpha_1 + 3 \alpha_2, 2 \alpha_1 + 3\alpha_2 \end{equation*} In the Chevalley basis, the commutators are \begin{align*} [h_{\alpha} , h_{\beta} ] &= 0 &\\ [h_{\alpha} , e_{\beta} ] &= 2 \frac{\alpha \cdot \beta}{\alpha^2} e_{\beta} &\\ [e_{\alpha} , e_{-\alpha} ] &= h_{\alpha}\\ [e_{\alpha} , e_{\beta} ] &= n_{\alpha \beta}e_{\alpha+\beta}\quad&&\text{if $\alpha + \beta \neq 0$ and $\alpha + \beta$ is a root}\\ [e_{\alpha} , e_{\beta} ] &= 0 \quad&&\text{if $\alpha + \beta \neq 0$ and $\alpha + \beta$ is not a root} \end{align*} In a previous post, I have calculated the integers $n_{\alpha\beta}$.
Overview of the irreps of $G_2$
I use capital letters $H_{\alpha}$ and $E_{\alpha}$ for the representations of $h_{\alpha}$ and $e_{\alpha}$. These matrices act on vectors as \begin{align*} &H_{\alpha_1} | \mu_1 \mu_2 \rangle = \mu_1\\ &H_{\alpha_2} | \mu_1 \mu_2 \rangle = \mu_2\\ &E_{\alpha} |\mu\rangle = n_{\alpha,\mu} |\mu - \alpha \rangle \end{align*} $\mu_1$ and $\mu_2$ are called Dynkin labels, $|\mu\rangle = |\mu_1 \mu_2 \rangle$ is called a weight vector. If the $\alpha$-string through $|\mu\rangle$ goes from $|\mu - q \alpha\rangle$ up to $|\mu + p \alpha\rangle$ with $p,q \ge 0$ then [1] $$\label{eq:20160919} |n_{\alpha,\mu}|^2 = p (q+1)$$
The 7-dimensional irrep of $G_2$
The highest weight for the $\bf{7}$ is $| 0, 1 \rangle$. The list of all weights in $\bf{7}$ is [2] \begin{align*} |\Lambda\rangle & = | 0, 1 \rangle\\ |\Lambda-\alpha_2\rangle & = | 1,-1 \rangle\\ |\Lambda-\alpha_1-\alpha_2\rangle & = | -1,2\rangle\\ |\Lambda-\alpha_1-2 \alpha_2\rangle & = | 0,0\rangle\\ |\Lambda-\alpha_1-3 \alpha_2\rangle & = | 1,-2\rangle\\ |\Lambda-2 \alpha_1-3 \alpha_2\rangle & = | -1,1\rangle\\ |\Lambda-2 \alpha_1-4 \alpha_2\rangle & = | 0, -1\rangle \end{align*} The weights can also be plotted on the root diagram.
 7-dimensional irrep of $G_2$ The blue dots are the weights. The black arrows are the root vectors of $G_2$.
Calculation of $E_{-\alpha_1}$
I calculate the matrix $E_{-\alpha_1}$ by calculating $E_{-\alpha_1}|\mu\rangle$ for all weights $|\mu\rangle$ in the irrep $\bf{7}$. Because $|\Lambda - \alpha_1\rangle$ is not a weight of $\bf{7}$, it follows that \begin{equation*} E_{-\alpha_1}|\Lambda\rangle = 0 \end{equation*} The calculation of $E_{-\alpha_1}|\Lambda-\alpha_2\rangle$ proceeds as follows. Because the $\alpha_1$-string through $|\Lambda-\alpha_2\rangle$ goes from $|\Lambda-\alpha_2\rangle$ to $|\Lambda-\alpha_2-\alpha_1\rangle$, $q=0$ and $p=1$ in formula \eqref{eq:20160919}. Thus $|n_{-\alpha_1,\Lambda - \alpha_2}|^2=1$, I choose the phase of $|\Lambda - \alpha_2 -\alpha_1 \rangle$ such that \begin{equation*} E_{-\alpha_1}|\Lambda-\alpha_2\rangle = |\Lambda-\alpha_2-\alpha_1\rangle \end{equation*} The calculation of $E_{-\alpha_1}|\mu\rangle$ for the other weights is similar:
• either $|\mu - \alpha_1\rangle$ is not a weight and then $E_{-\alpha_1}|\mu\rangle=0$
• or $|n_{-\alpha_1,\mu}|^2 = 1$ and then I can choose the phase such that $E_{-\alpha_1}|\mu\rangle = |\mu - \alpha_1 \rangle$
Calculation of $E_{-\alpha_2}$
This proceeds as above. The most interesting case is $E_{-\alpha_2}|\Lambda-\alpha_1-\alpha_2\rangle$. In this case the $\alpha_2$-string is \begin{equation*} |\Lambda-\alpha_1-\alpha_2\rangle , |\Lambda-\alpha_1-2\alpha_2\rangle, |\Lambda-\alpha_1-3\alpha_2\rangle \end{equation*} thus $p=2$ and $q=0$, hence $|n_{-\alpha_2,\Lambda - \alpha_1 - \alpha_2}|^2 = 2$. I choose the phase of $| \Lambda - \alpha_1 - 2 \alpha_2 \rangle$ such that \begin{equation*} E_{-\alpha_2}|\Lambda-\alpha_1 - \alpha_2\rangle = \sqrt{2} |\Lambda-\alpha_1-2 \alpha_2\rangle \end{equation*} The rest of the calculation of $E_{-\alpha_2}$ proceeds in the same way. Luckily, I encounter each weight only once, so I can always choose the positive square root in formula \eqref{eq:20160919}.
Calculation of $E_{-\alpha}$
If $\alpha < 0$ and $\alpha$ is not simple, I can calculate $E_{-\alpha}$ from the commutators. For example, because $[ e_{-\alpha_1}, e_{-\alpha_2} ] = - e_{-\alpha_1-\alpha_2}$ I know that $E_{-\alpha_1-\alpha_2} = -[ E_{-\alpha_1}, E_{-\alpha_2} ]$. For the positive roots I use $E_{-\alpha}^{\dagger} = E_{\alpha}$.
Result
Here is the list of the explicit matrices \begin{align*} H_{\alpha_1}&= \left( \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)& H_{\alpha_2}&= \left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ \end{array} \right)\\[3mm] E_{-\alpha_2} &= \left( \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \sqrt{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ \end{array} \right)& E_{-\alpha_1} &= \left( \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\\ E_{-\alpha_1-\alpha_2} &= \left( \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -\sqrt{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ \end{array} \right)& E_{-\alpha_1-2\alpha_2} &= \left( \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -\sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -\sqrt{2} & 0 & 0 & 0 \\ \end{array} \right)\\[3mm] E_{-\alpha_1-3\alpha_2} &= \left( \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ \end{array} \right)& E_{-2\alpha_1-3\alpha_2} &= \left( \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\\[3mm] E_{\alpha_2} &= \left( \begin{array}{ccccccc} 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \sqrt{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)& E_{\alpha_1} &= \left( \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\\[3mm] E_{\alpha_1+\alpha_2} &= \left( \begin{array}{ccccccc} 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -\sqrt{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \sqrt{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)& E_{\alpha_1+2\alpha_2} &= \left( \begin{array}{ccccccc} 0 & 0 & 0 & -\sqrt{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\\[3mm] E_{\alpha_1+3\alpha_2} &= \left( \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)& E_{2\alpha_1+3\alpha_2} &= \left( \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right) \end{align*} It can be checked that these matrices satisfy the commutation relations of $G_2$.
References
[1] Jones, Groups, Representations and Physics, page 196
[2] Cahn, Semi-Simple Lie Algebras and Their Representations, Chapter X