= Solution
Let $\mathfrak g$ be a complex <semisimple Lie algebra>, choose a <Cartan subalgebra> and a <positive root> system $R^+$, and let $\lambda$ be a dominant integral <highest weight>. Write $L(\lambda)$ for the corresponding finite-dimensional irreducible representation, $\alpha^\vee=2\alpha/(\alpha,\alpha)$ for a <coroot>, and $\rho=\tfrac12\sum_{\alpha\in R^+}\alpha$ for the <Weyl vector>. The <Weyl dimension formula> is
$$
\boxed{\dim L(\lambda)=\prod_{\alpha\in R^+}
\frac{\langle\lambda+\rho,\alpha^\vee\rangle}{\langle\rho,\alpha^\vee\rangle}.}
$$
The bracket denotes the natural pairing of <weights> with <coroots>. The normalization of an invariant inner product cancels from each ratio, so the formula can equivalently use roots instead of <coroots> in both numerator and denominator.
Back to article page
