Freudenthal multiplicity formula ($(\Vert \lambda +\rho {\Vert }^2-\Vert \mu +\rho {\Vert }^2)m_{\lambda }(\mu )=2{\sum }_{\alpha >0}{\sum }_{j\ge 1}(\mu +j\alpha ,\alpha )m_{\lambda }(\mu +j\alpha )$)

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 $\mathrm{tr}(F_{\alpha }{E}_{\alpha }{|}_{V_{\mu }})={\sum }_{j\ge 1}(\mu +j\alpha ,\alpha )m_{\lambda }(\mu +j\alpha )$. Taking the trace and using the Casimir eigenvalue proves the formula. Only finitely many terms are nonzero.