Decomposition numbers and the decomposition matrix

Let $\chi$ run through the ordinary irreducible characters of $G$, and let $\phi$ run through the irreducible Brauer characters. If $[V_{\chi }]$ is the class of the ordinary irreducible affording $\chi$, the well-defined decomposition map of The decomposition map is independent of the stable lattice has a unique expansion

$d([V_{\chi }])={\sum }_{\phi }{d}_{\chi \phi }[S_{\phi }],$

where $S_{\phi }$ is the simple $kG$-module affording $\phi$. The classes of the simple modules form the integral basis of the Grothendieck group, so the coefficients $d_{\chi \phi }$ are integers. Additivity of Brauer characters on composition series gives the equivalent character identity

${\chi }^0={\sum }_{\phi }{d}_{\chi \phi }\phi \mathrm{.}$

The integers $d_{\chi \phi }$ are the decomposition numbers, and the matrix $D=(d_{\chi \phi })$ is the decomposition matrix of $G$ at $p$.