Projective indecomposable characters and Cartan invariants

For each irreducible Brauer character $\phi$, let $S_{\phi }$ be the corresponding simple $kG$-module and let $P_{\phi }\to S_{\phi }$ be its projective cover. The module $P_{\phi }$ is an indecomposable projective $kG$-module, and these exhaust the indecomposable projectives up to isomorphism.

Choose a projective $\mathcal{O}G$-lattice ${\widehat{P}}_{\phi }$ whose reduction is $P_{\phi }$. Its ordinary character ${\mathrm{\Phi }}_{\phi }$ is called the projective indecomposable character attached to $\phi$. The existence of such a projective lift and the independence of its ordinary character are proved in Brauer reciprocity ↗.

For irreducible Brauer characters $\phi ,\psi$, the multiplicity of $S_{\psi }$ as a composition factor of $P_{\phi }$ is denoted $c_{\phi \psi }$. The matrix $C=(c_{\phi \psi })$ is the Cartan matrix of $G$ at $p$.