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 $G_2$ |

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 $G_2$ |

The dimensions of the weight spaces of the 1547-dimensional representation of $G_2$ |

**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

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