 Solution

For $x\alpha +y\beta$, the simple-coroot coordinates are $2x-y$ and $-3x+2y$. Solving for the fundamental weights givesHence $C_1=2$, $C_2=3$, and $C_3=2$. The weight $\omega$ is the fundamental weight of the short root, and its highest-weight representation is the seven-dimensional fundamental representation of $G_2$. Its weights are zero and the six short roots.