Modular Representation Theory - Decomposition Matrix and Cartan Matrix

Decomposition Matrix and Cartan Matrix

The composition factors of the projective indecomposable modules may be calculated as follows: Given the ordinary irreducible and irreducible Brauer characters of a particular finite group, the irreducible ordinary characters may be decomposed as non-negative integer combinations of the irreducible Brauer characters. The integers involved can be placed in a matrix, with the ordinary irreducible characters assigned rows and the irreducible Brauer characters assigned columns. This is referred to as the decomposition matrix, and is frequently labelled D. It is customary to place the trivial ordinary and Brauer characters in the first row and column respectively. The product of the transpose of D with D itself results in the Cartan matrix, usually denoted C; this is a symmetric matrix such that the entries in its j-th row are the multiplicities of the respective simple modules as composition factors of the j-th projective indecomposable module. The Cartan matrix is non-singular; in fact, its determinant is a power of the characteristic of K.

Since a projective indecomposable module in a given block has all its composition factors in that same block, each block has its own Cartan matrix.