The affine Kac-Moody algebras give rise to rational conformal field theories (RCFTs) called the Wess-Zumino-Witten (WZW) models. An important component of an RCFT is its fusion ring, whose structure constants are given by the associated $S$-matrix. We apply a fixed point property possessed by the WZW models (``fixed point factorization") to calculate non-negative integer matrix representations of the fusion ring, allowing for the calculation of D-brane charges in string theory.