= Solution
Write an element of the <symplectic Lie algebra> in $n\times n$ blocks. Direct multiplication in $AJ+JA^T=0$ gives
$$
A=\begin{pmatrix}P&Q\\R&-P^T\end{pmatrix},\qquad Q^T=Q,\quad R^T=R.
$$
The diagonal <Cartan subalgebra> consists of $H=\operatorname{diag}(h_1,\ldots,h_n,-h_1,\ldots,-h_n)$. Define $\varepsilon_i(H)=h_i$. Under the <Adjoint representation>, the off-diagonal <matrix units> in $P$ have <weights> $\varepsilon_i-\varepsilon_j$; the symmetric entries in $Q$ have <weights> $\varepsilon_i+\varepsilon_j$, including $2\varepsilon_i$ on the diagonal; those in $R$ have the negatives of these weights. All nonzero <root spaces> are one-dimensional, while the zero <weight space> is the diagonal <Cartan subalgebra>. Thus the <Cn root system> is
$$
\boxed{R=\{\pm\varepsilon_i\pm\varepsilon_j:i<j\}\cup\{\pm2\varepsilon_i:1\le i\le n\}.}
$$
Choose the <positive roots> and <simple roots> as
$$
\boxed{R^+=\{\varepsilon_i-\varepsilon_j,\varepsilon_i+\varepsilon_j:i<j\}\cup\{2\varepsilon_i\},\qquad
\Delta=\{\alpha_i=\varepsilon_i-\varepsilon_{i+1}:i<n\}\cup\{\alpha_n=2\varepsilon_n\}.}
$$
Their <highest root> and <Weyl vector> are
$$
\boxed{\theta=2\varepsilon_1=2\alpha_1+\cdots+2\alpha_{n-1}+\alpha_n,\qquad
\rho=\frac12\sum_{\alpha\in R^+}\alpha=\sum_{i=1}^n(n-i+1)\varepsilon_i.}
$$
For the last identity, each pair $i<j$ contributes $2\varepsilon_i$ before division by two, and the long positive root contributes another $\varepsilon_i$ afterward. Successive <simple roots> have equal length except for $\alpha_n$, whose squared length is twice that of $\alpha_{n-1}$. The <Cn Dynkin diagram> is a chain with a double last bond, whose arrow points toward $\alpha_{n-1}$, the short root:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-1-cn-dynkin.png]
{title=The Cn Dynkin diagram with its simple-root labels and arrow toward the short root}
The displayed schematic is for $n\ge4$; omit intermediate nodes for $n=2,3$. For $n=1$ the diagram has just the single node $\alpha_1=2\varepsilon_1$.
Back to article page
