= Freudenthal multiplicity formula
{c}
{title2=$\bigl(\|\lambda+\rho\|^2-\|\mu+\rho\|^2\bigr)m_\lambda(\mu)=2\sum_{\alpha>0}\sum_{j\geq1}(\mu+j\alpha,\alpha)m_\lambda(\mu+j\alpha)$}
For a finite-dimensional <Irreducible Lie algebra representation> of a complex <semisimple Lie algebra> with <highest weight> $\lambda$, this recursion computes its <weight multiplicities> from $m_\lambda(\lambda)=1$. Here $\rho$ is the <half-sum of positive roots> and the <inner product> is induced by the <Killing form>. Write the <Casimir operator> on the weight-$\mu$ space as $(\mu,\mu+2\rho)I+2\sum_{\alpha>0}F_\alpha E_\alpha$. The <cyclic trace identity between adjacent weight spaces> gives $\operatorname{tr}(F_\alpha E_\alpha|_{V_\mu})=\sum_{j\geq1}(\mu+j\alpha,\alpha)m_\lambda(\mu+j\alpha)$. Taking the trace and using the <Casimir eigenvalue> proves the formula. Only finitely many terms are nonzero.
Back to article page
