= Solution
Let $\mathfrak g$ be a complex <semisimple Lie algebra>, choose a <Cartan subalgebra> and positive <root system> $R^+$, and put $\rho=\frac12\sum_{\alpha\in R^+}\alpha$. For a <dominant integral weight> $\lambda$, let $L(\lambda)$ be the finite-dimensional <highest-weight representation> with <highest weight> $\lambda$. The <Weyl dimension formula> is
$$
\boxed{\dim L(\lambda)=\prod_{\alpha\in R^+}\frac{\langle\lambda+\rho,\alpha^\vee\rangle}{\langle\rho,\alpha^\vee\rangle}.}
$$
Here $\alpha^\vee=2\alpha/(\alpha,\alpha)$ is the <coroot>, the pairing is between a <weight> and a <coroot>, and $\rho$ is the <Weyl vector>. The <inner product> can be any <Weyl group> invariant normalization on each simple factor: its scaling cancels from the ratios. The dominance condition means that $\langle\lambda,\alpha_i^\vee\rangle$ is a nonnegative <integer> for each <simple root>.
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