Weyl Character Formula: Closed-Form Irreducible Representation Characters — E8 Intelligence Research

FINDING: The Weyl character formula provides a closed-form expression for characters of irreducible representations of compact Lie groups, encoding root system structure via alternating sums over the Weyl group. | MATH: For a compact Lie group \(G\) with maximal torus \(T\), Lie algebra \(\mathfrak{g}\), and weight lattice \(P\), the character of irreducible representation \(V_\lambda\) (highest weight \(\lambda\)) is: \[ \chi_\lambda = \frac{\sum_{w \in W} (-1)^{\ell(w)} e^{w(\lambda+\rho)}}{\sum_{w \in W} (-1)^{\ell(w)} e^{w(\rho)}} = \frac{A_{\lambda+\rho}}{A_\rho} \] where \(W\) is the Weyl group, \(\ell(w)\) is the length (number of simple reflections), \(\rho = \frac{1}{2}\sum_{\alpha \in \Delta^+} \alpha\) (half-sum of positive roots), and \(A_\mu = \sum_{w \in W} (-1)^{\ell(w)} e^{w(\mu)}\). The denominator \(A_\rho = \prod_{\alpha \in \Delta^+} (e^{\alpha/2} - e^{-\alpha/2})\) (Weyl denominator identity). Freudenthal's formula gives multiplicities recursively: \[ m_\lamb Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com