The <root lattice> is $Q=\sum_i\mathbb Z\alpha^{(i)}$. The <weight lattice> is
$$
P=\{\lambda:\langle\lambda,(\alpha^{(i)})^\vee\rangle\in\mathbb Z\text{ for every }i\}.
$$
Because every <Cartan integer> $\langle\alpha^{(j)},(\alpha^{(i)})^\vee\rangle$ is integral, $Q\subseteq P$. The <Dynkin labels> of $\lambda$ are
$$
[m_1,\ldots,m_r],
\qquad m_i=\langle\lambda,(\alpha^{(i)})^\vee\rangle.
$$

There are three isomorphism classes of complex simple rank-three Lie algebras: types <A3 root system>$A_3$, <B3 root system>$B_3$, and <C3 root system>$C_3$. Use the convention $A_{ij}=\langle\alpha^{(i)},(\alpha^{(j)})^\vee\rangle$, and order the chain as $1$--$2$--$3$ with $|\alpha^{(1)}|=|\alpha^{(2)}|$.

* For $A_3$, all roots have the same length and the diagram has two single edges. Its <Cartan matrix> and angles are
$$
A_{A_3}=\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{pmatrix},
\qquad
\theta_{12}=\theta_{23}=120^\circ,quad\theta_{13}=90^\circ,quad
\frac{|\alpha^{(2)}|}{|\alpha^{(3)}|}=1.
$$
* For $B_3$, take $\alpha^{(1)}=e_1-e_2$, $\alpha^{(2)}=e_2-e_3$, and $\alpha^{(3)}=e_3$. The double-edge arrow points to the short third root, and
$$
A_{B_3}=\begin{pmatrix}2&-1&0\\-1&2&-2\\0&-1&2\end{pmatrix},
\qquad
\theta_{12}=120^\circ,quad\theta_{23}=135^\circ,quad\theta_{13}=90^\circ,quad
\frac{|\alpha^{(2)}|}{|\alpha^{(3)}|}=\sqrt2.
$$
* For $C_3$, take $\alpha^{(1)}=e_1-e_2$, $\alpha^{(2)}=e_2-e_3$, and $\alpha^{(3)}=2e_3$. The double-edge arrow points to the short second root, and
$$
A_{C_3}=\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-2&2\end{pmatrix},
\qquad
\theta_{12}=120^\circ,quad\theta_{23}=135^\circ,quad\theta_{13}=90^\circ,quad
\frac{|\alpha^{(2)}|}{|\alpha^{(3)}|}=\frac1{\sqrt2}.
$$

The Dynkin labels of a finite-dimensional <irreducible representation> are those of its <highest weight>. Thus $[1,0,0]$ means the <fundamental representation> $V(\omega_1)$. The weights, written in Dynkin labels and in a lowering order, are
$$
\begin{array}{c|l|c}
\text{type}&\text{weight labels}&\dim V(\omega_1)\\ \hline
A_3&[1,0,0],[-1,1,0],[0,-1,1],[0,0,-1]&4\\
B_3&[1,0,0],[-1,1,0],[0,-1,2],[0,0,0],[0,1,-2],[1,-1,0],[-1,0,0]&7\\
C_3&[1,0,0],[-1,1,0],[0,-1,1],[0,1,-1],[1,-1,0],[-1,0,0]&6
\end{array}.
$$
For $A_3$ these are the weights of the defining representation of $\mathfrak{sl}_4$; for $B_3$ they are $\{\pm e_1,\pm e_2,\pm e_3,0\}$ in the vector representation of $\mathfrak{so}_7$; for $C_3$ they are $\{\pm e_1,\pm e_2,\pm e_3\}$ in the defining representation of $\mathfrak{sp}_6$. Hence the requested dimensions are \b[$4$, $7$, and $6$], respectively.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 302 / 3 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-302/3/solution