|
In combinatorial mathematics, a Dowling geometry, named after Thomas A. Dowling, is a matroid associated with a group. There is a Dowling geometry of each rank for each group. If the rank is at least 3, the Dowling geometry uniquely determines the group. Dowling geometries have a role in matroid theory as universal objects (Kahn and Kung, 1982); in that respect they are analogous to projective geometries, but based on groups instead of fields. A Dowling lattice is the geometric lattice of flats associated with a Dowling geometry. The lattice and the geometry are mathematically equivalent: knowing either one determines the other. Dowling lattices, and by implication Dowling geometries, were introduced by Dowling (1973a,b). A Dowling lattice or geometry of rank ''n'' of a group ''G'' is often denoted ''Qn''(''G''). ==The original definitions== In his first paper (1973a) Dowling defined the rank-''n'' Dowling lattice of the multiplicative group of a finite field ''F''. It is the set of all those subspaces of the vector space ''F'' ''n'' that are generated by subsets of the set ''E'' that consists of vectors with at most two nonzero coordinates. The corresponding Dowling geometry is the set of 1-dimensional vector subspaces generated by the elements of ''E''. In his second paper (1973b) Dowling gave an intrinsic definition of the rank-''n'' Dowling lattice of any finite group ''G''. Let ''S'' be the set . A ''G''-labelled set (''T'', α) is a set ''T'' together with a function α: ''T'' --> ''G''. Two ''G''-labelled sets, (''T'', α) and (''T'', β), are equivalent if there is a group element, ''g'', such that β = ''g''α. An equivalence class is denoted (α ). A partial ''G''-partition of ''S'' is a set γ = of equivalence classes of ''G''-labelled sets such that ''B''1, ..., ''B''''k'' are nonempty subsets of ''S'' that are pairwise disjoint. (''k'' may equal 0.) A partial ''G''-partition γ is said to be ≤ another one, γ *, if * every block of the second is a union of blocks of the first, and * for each ''B''''i'' contained in ''B'' *''j'', α''i'' is equivalent to the restriction of α *''j'' to domain ''B''''i'' . This gives a partial ordering of the set of all partial ''G''-partitions of ''S''. The resulting partially ordered set is the Dowling lattice ''Q''''n''(''G''). The definitions are valid even if ''F'' or ''G'' is infinite, though Dowling mentioned only finite fields and groups. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dowling geometry」の詳細全文を読む スポンサード リンク
|