Cartan matrix of an algebra

Summary

The Cartan matrix of a finite-dimensional algebra records how the simple modules occur as composition factors of the indecomposable projective modules. Its rows are indexed by the indecomposable projectives and its columns by the simple modules. In the modular representation theory of finite groups it is a basic invariant of the group algebra. It is a different notion from the Cartan matrix of a semisimple Lie algebra, which shares the name.