On combinatorial and projective geometry

Abstract

Cross ratios constitute an important tool in classical projective geometry. Using the theory of Tutte groups as discussed in [6] it will be shown in this note that the concept of cross ratios extends naturally to combinatorial geometries or matroids. From a thorough study of these cross ratios which, among other observations, includes a new matroid theoretic version and proof of the Pappos theorem, it will be deduced that for any projective space M=ℙn(K) of dimension n≥2 of M over some skewfield K the inner Tutte group is isomorphic to the commutator factor group K*/[K*, K*] of K*≔K∖{0}. This shows not only that in case M=ℙn(K) our matroidal cross ratios are nothing but the classical ones. It can also be used to correlate orientations of the matroid M=ℙn(K) with the orderings of K. And it implies that Dieudonne's (non-commutative) determinants which, by Dieudonne's definition, take their values in K*/[K*, K*] as well, can be viewed as a special case of a determinant construction which works for just every combinatorial geometry.