Note on a p-block of a finite group with Abelian defect group
Let G be a finite group and B a p-block of G with abelian defect group D. Let T(b) be the inertial group of a root b of B in \(C_ G(D)\). Let \(B_ 1\) be the block corresponding to b in \(C_ G(D_ 1)\), where \(D_ 1=C_ D(T(b))\). The author has earlier proved that there are strong numerical (and other) relations between the blocks B and \(B_ 1\). Here this study is continued to include information about lower defect group multiplicities and in particular the elementary divisors of the Cartan matrices. There is also a result about a vertex preserving bijection between a set of indecomposable modules in \(B_ 1\) and a set of indecomposable modules in B.
elementary divisors of the Cartan matrices
scientific article; zbMATH DE number 4165158