Dowling group geometries and the critical problem (Q1092912)

This paper relates the critical problem of Crapo and Rota to Dowling group geometries. If A is a finite group, \(Q_ r(A)\) is the rank r Dowling group geometry over A and M is a rank r matroid embeddable as a minor of \(Q_ r(A)\), then it is shown that the critical exponent of M over A is well defined and is determined by an evaluation of the characteristic polynomial of M. Classes of tangential k-blocks obtained from Dowling group geometries are also displayed. A consequence of the theory is that for the first time all cases of Hadwiger's conjecture can be stated as critical problems.