Dowling group geometries and the critical problem

Abstract This paper relates the critical problem of Crapo and Rota [“On the Foundations of Combinatorial Theory: Combinatorial Geometries”, M.I.T. Press, Cambridge, MA 1970] 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.

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