## Tuesday, September 27, 2016

### Weight diagrams of $G_2$

This post contains pictures of weight diagrams of irreducible representations of the Lie algebra $G_2$.
The dimensions of the irreducible representations of $G_2$ are [1] \begin{equation*} 1, 7, 14, 27, 64, \ldots \end{equation*} Cahn [2] explains how to calculate the weights of representations and how to use Freudenthal's formula to calculate the dimensions of the weight spaces. I programmed these formulas in Mathematica to obtain the figures below.
The $7$-dimensional representation
The Dynkin labels of the highest weight are $| 0, 1 \rangle$. The list of all weights in $\bf{7}$ is $| 0, 1 \rangle$, $| 1,-1 \rangle$, $| -1,2\rangle$, $| 0,0\rangle$, $| 1,-2\rangle$, $| -1,1\rangle$ and $| 0, -1\rangle$. All weight spaces are 1-dimensional. If I plot them on the root diagram, I get the following picture
 The weights of the 7-dimensional representation of $G_2$. The black arrows are the root vectors.

The $14$-dimensional representation
The $14$-dimensional representation of $G_2$ is the adjoint representation, the weights thus coincide with the roots. The highest weight is $|1,0\rangle$. The weight space of $|0,0\rangle$ is $2$-dimensional, because the Cartan subalgebra is $2$-dimensional. All other weight spaces are 1-dimensional.
 The weights of the 14-dimensional representation of $G_2$

The $27$-dimensional representation
This is the representation with highest weight $|0,2\rangle$
 The weights of the 27-dimensional representation of $G_2$
In this case, the dimension of the weight spaces can be $1$, $2$ or $3$-dimensional. In the next figure, I have plotted the dimension of the weight space at the corresponding weight.
 The dimensions of the weight spaces of the 27-dimensional representation of $G_2$

The $64$-dimensional representation
This is the representation with highest weight $|1,1\rangle$. I only plot the dimensions of the weight spaces.
 The dimensions of the weight spaces of the 64-dimensional representation of $G_2$
All these pictures look quite similar, so I will only make one more. Here is the weight diagram of the $1547$-dimensional representation. This is the representation with highest weight $|2,3\rangle$
 The dimensions of the weight spaces of the 1547-dimensional representation of $G_2$
I was inspired to make these weight diagrams by a webpage from Bump [3].
References
[1] Wikipedia article
[2] Cahn, Semi-Simple Lie Algebras and Their Representations, Chapters X and XI
[3] Daniel Bump, Weight Diagrams of Weight Two