Decomposition matrix (modular representation theory, $D$)

The decomposition matrix records the multiplicities of simple modular representations in reductions of ordinary representations. If its rows are indexed by ordinary irreducible characters and its columns by irreducible Brauer characters, its entry $d_{\chi \varphi }$ is the decomposition number of $\varphi$ in the reduction of $\chi$.

The Cartan matrix records composition-factor multiplicities in the projective indecomposable modules. For a split modular group algebra, it iswhere $D$ is the decomposition matrix.