Factoring Cartan matrices of group algebras
Thesis, University of Chicago, 2003
Brian W. Johnson

Abstract.
The Cartan matrix of a group algebra generally contains 
incomplete information about the decomposition numbers.
For the groups SL(2,q), we show that the information in
the Cartan matrix, along with the knowledge that every
decomposition number is either zero or one, is sufficient
to recover the complete decomposition matrix. In the
process of proving this, we obtain a new combinatorial
description of the decomposition numbers.