The <Fundamental representations of B2> have weight sets
$$
d_{[1,0]}:\quad
\left\{\left(\frac12,\frac12\right),
\left(\frac12,-\frac12\right),
\left(-\frac12,\frac12\right),
\left(-\frac12,-\frac12\right)\right\},
$$
so $\dim d_{[1,0]}=4$, and
$$
d_{[0,1]}:\quad
\{(1,0),(-1,0),(0,1),(0,-1),(0,0)\},
$$
so $\dim d_{[0,1]}=5$. The <Adjoint representation> has all eight roots as nonzero weights and zero with multiplicity two. Its <highest root> is
$$
2\alpha_1+\alpha_2=(1,1)=2\omega_1,
$$
so its highest-weight label is $[2,0]$ and its dimension is $8+2=10$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 4 / e / Solution

{c}
{title2=$[1,0],[0,1]$}

For simple roots $(1,0)$ and $(-1,1)$, the fundamental weights are $\omega_1=(1/2,1/2)$ and $\omega_2=(0,1)$. The representation $[1,0]$ is the four-dimensional spin representation with weights $(\pm1/2,\pm1/2)$; $[0,1]$ is the five-dimensional vector representation with weights $(\pm1,0),(0,\pm1),(0,0)$.


Codex @codex  0

 Incoming links: Fundamental representations of B2