Li [5] classified the finite 2- and 3-dimensional basis-homogeneous geometric ... Moreover, he proved that if a finite geometric lattice S£ is basis- ... on which Aut 5£ acts basis-transitively, and the subspaces of °) with vertex set /,. (3) .... by (x, S)t and will be called the i-degree of x in S U x.
BASIS-HOMOGENEOUS GEOMETRIC LATTICES ANNE DELANDTSHEER
ABSTRACT
We classify the finite point—and basis—homogeneous geometric lattices having a line of size greater than 2. As a consequence, the finite basis-homogeneous geometric lattices are known, up to the knowledge of those having only lines of size 2.
1. Introduction In an n-dimensional geometric lattice $£ (or simple matroid of rank n+ 1), the elements (or subspaces) of dimension / ^ n are called i-spaces, the 1-spaces being preferably called lines, the 2-spaces planes and the (n — l)-spaces hyper planes. The subspace generated in JS? by a set P of points is denoted by {P}^, or simply by > when no confusion is possible. The residue Jz?0 of a point 0 of S£ is the (n—l)-dimensional geometric lattice whose /-spaces are the (/+l)-spaces of Jzf containing 0. The lattice $£ is a linear space if n = 2 and a planar space if n = 3. The linear structure lin i f is the 2-dimensional truncation of if consisting of its points and lines only. Any subspace of i f will be identified with the set of its points, so that a proper linear subspace of 5£ can be defined as a set 5 of points of J^, containing at least two lines but not all points, and such that the line through any two points of 5 is entirely contained in 5. All the /-spaces with 2 ^ / < n — 1 are proper linear subspaces of if. The independent m-sets of S£ (that is, the sets of m independent points) will be called edges if m = 2, triangles if m = 3 and bases if m = n+1. An (n+ l)-tuple (JC1} •••,xn+1) is an ordered basis if and only if {xlf ...,* n + 1 } is a basis. The natural problem of classifying the basis-homogeneous geometric lattices (that is, whose automorphism group is transitive on the bases) has received some partial answers relying on the classification of finite 2-transitive permutation groups. Kantor [4] proved that the finite geometric lattices whose automorphism group is transitive on ordered bases are (i) the truncations of Desarguesian projective or affine geometries, or of Boolean lattices, (ii) the lattices associated with the Steiner systems 5(3,6,22), 5(4,7,23) or 5(5,8,24), or (iii) the lattice whose points and lines are those of the Hermitian unital of order 4. Li [5] classified the finite 2- and 3-dimensional basis-homogeneous geometric Received 29 December 1985. 1980 Mathematics Subject Classification 05B35. Part of this work was done while the author was visiting the Ohio State University. J. London Math. Soc. (2) 34 (1986) 385-393 13
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lattices. Moreover, he proved that if a finite geometric lattice S£ is basis-homogeneous but not point-homogeneous, then the point orbits of Aut i f are sublattices i? l 5 ..., S£t on which Aut 5£ acts basis-transitively, and the subspaces of 2. It is easy to define a geometric lattice 5£ whose bases are all subsets B of Lo U ... U L3 such that \B n Lt\ = \B n Li+1\ = 2 and \B n Li+t\
= \B f) Li+3\ = 1
for some /, the subscripts being computed modulo 4. Now J? is basis-homogeneous and Aut S£ s Sym k wr D 8 , where D 8 denotes the dihedral group of order 8. Note that the preceding construction does not lead to a geometric lattice if we require that \B n Lt\ = \B(] Li+2\ = 2 and \B n Lt+1\ = \B0 Li+3\ = 1, because Lx U L 2 U L 3 would
be a subspace generated both by an independent 4-set and by an independent 5-set. More generally, let / = {1,2, ...,t} with t ^ 2 and let J be a point- and basishomogeneous matroid of rank s ^ 1 with point set /, in other words (1) if s = 1, the bases of ^ are all 1-subsets of/, (2) if s = 2, the bases of J are the edges of a complete regular multipartite graph K r, r, r (r > °) w i t h vertex set /, (3) if s ^ 3, J is any point- and basis-homogeneous (s— l)-dimensional geometric lattice. Given a positive integer M, we say that a /-tuple fi = (/?l5 ...,fit) is good if and only if / = {ieI: fa = M+1} is a basis of J and fi{ = M for every i;$J. Now let G be a finite Af-dimensional edge- and basis-homogeneous geometric lattice with line size greater than 2, and let (£1; ...,G t be / disjoint copies of (£, called components. We define i f to be the geometric lattice whose points are those of [Jiei&t and whose bases are the sets B = [jieI Bt, where Bt is a set of fit independent points in (£f and (fi1,...,fit) is a good /-tuple. A set S of points of U i e / ^ { is independent in
