Let $L(\lambda)$ be the irreducible representation with highest weight $\lambda \in P^+$. The character is: $\chi_{L(\lambda)}(e^H) = \frac{\sum_{w \in W} \epsilon(w) e^{w(\lambda + \rho) - \rho}}{\sum_{w \in W} \epsilon(w) e^{w(\rho) - \rho}} = \frac{\sum_{w \in W} \epsilon(w) e^{w(\lambda + \rho)}}{\prod_{\alpha \in \Phi^+} (e^{\alpha/2} - e^{-\alpha/2})}$ where $\rho = \frac{1}{2}\sum_{\alpha \in \Phi^+} \alpha$ is half the sum of positive roots, $W$ is the Weyl group, and $\epsilon(w) = \det(w) = \pm 1$.

The dimension of $L(\lambda)$ is obtained by evaluating the character at the identity: $\dim L(\lambda) = \prod_{\alpha \in \Phi^+} \frac{\langle \lambda + \rho, \alpha \rangle}{\langle \rho, \alpha \rangle}$

This gives an explicit combinatorial formula for dimensions.

Multiplicity formula: In the weight space decomposition $L(\lambda) = \bigoplus_\mu V_\mu$, the multiplicity of weight $\mu$ is: $\text{mult}_\lambda(\mu) = \sum_{w \in W} \epsilon(w) P(w(\lambda + \rho) - (\mu + \rho))$ where $P$ is the Kostant partition function (number of ways to write a vector as a sum of positive roots).

Freudenthal's formula: Recursive formula for weight multiplicities: $(\langle \lambda + \rho, \lambda + \rho \rangle - \langle \mu + \rho, \mu + \rho \rangle) \text{mult}(\mu) = 2\sum_{\alpha \in \Phi^+} \sum_{k \geq 1} \langle \mu + k\alpha, \alpha \rangle \text{mult}(\mu + k\alpha)$