On combinatorial and projective geometry |
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Authors: | Andreas W M Dress Walter Wenzel |
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Institution: | (1) Fakultät für Mathematik, Universität Bielefeld, Postfach 8640, 4800 Bielefeld 1, F.R.G. |
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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=
n
(K) of dimension n2 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=
n
(K) our matroidal cross ratios are nothing but the classical ones. It can also be used to correlate orientations of the matroid M=
n
(K) with the orderings of K. And it implies that Dieudonné's (non-commutative) determinants which, by Dieudonné'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.Research supported by the DFG (Deutsche Forschungsgemeinschaft). |
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Keywords: | |
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