# Diagram geometry

A diagram is like a blueprint describing a possible combination of $2$- dimensional incidence structures in a higher-dimensional geometry (cf. Incidence system). In this context, a geometry of rank $n$ is a connected graph $\Gamma$ equipped with an $n$- partition $\Theta$ such that every maximal clique of $\Gamma$ meets every class of $\Theta$. The vertices and the adjacency relation of $\Gamma$ are the elements and the incidence relation of the geometry, the cliques of $\Gamma$ are called flags, the neighbourhood of a non-maximal flag is its residue, the classes of $\Theta$ are called types and the type (respectively, rank) of a residue is the set (respectively, number) of types met by it. Given a catalogue of "nice" geometries of rank $2$ and a symbol for each of those classes, one can compose diagrams of any rank by those symbols. The nodes of a diagram represent types and, by attaching a label to the stroke connecting two types $i$ and $j$ or by some other convention, one indicates which class the residues of type $\{ i, j \}$ should belong to. For instance, a simple stroke

a double stroke

and the "null stroke"

are normally used to denote, respectively, the class of projective planes (cf. Projective plane), the class of generalized quadrangles [a5] (cf. also Quadrangle) and the class of generalized digons (generalized digons are the trivial geometry of rank $2$, where any two elements of different type are incident). The following two diagrams are drawn according to these conventions.

1)

2)

They both represent geometries of rank $3$, where the residues of type $\{ 1,2 \}$ are projective planes and the residues of type $\{ 1,3 \}$ are generalized digons. However, the residues of type $\{ 2,3 \}$ are projective planes in 1) and generalized quadrangles in 2). The following is a typical problem in diagram geometry: given a diagram of such-and-such type, classify the geometries that fit with it, or at least the finite or flag-transitive such geometries (a geometry $\Gamma$ is said to be flag-transitive if ${ \mathop{\rm Aut} } ( \Gamma )$ acts transitively on the set of maximal flags of $\Gamma$). In the "best" cases the answer is as follows: the geometries under consideration belong to a certain well-known family or, possibly, arise in connection with certain small or sporadic groups. For instance, it is not difficult to prove that $3$- dimensional projective geometries are the only examples for the above diagram 1). On the other hand, the following has been proved quite recently [a7]: If $\Gamma$ is a flag-transitive example for the diagram 2) and if all rank- $1$ residues of $\Gamma$ are finite of size $\geq 2$, then $\Gamma$ is either a classical polar space [a6] (arising from a non-degenerate alternating, quadratic or Hermitian form of Witt index $3$) or a uniquely determined very small geometry with ${ \mathop{\rm Aut} } ( \Gamma ) \cong { \mathop{\rm Alt} } ( 7 )$.

See [a3] for a survey of related theorems.

#### References

 [a1] F. Buekenhout, "Diagrams for geometries and groups" J. Combin. Th. A , 27 (1979) pp. 121–151 [a2] F. Buekenhout, "Foundations of incidence geometry" F. Buekenhout (ed.) , Handbook of Incidence Geometry , North-Holland (1995) pp. 63–105 [a3] F. Buekenhout, A. Pasini, "Finite diagram geometries extending buildings" F. Buekenhout (ed.) , Handbook of Incidence Geometry , North-Holland (1995) pp. 1143–1254 [a4] A. Pasini, "Diagram geometries" , Oxford Univ. Press (1994) [a5] S.E. Payne, J.A. Thas, "Finite generalized quadrangles" , Pitman (1984) [a6] J. Tits, "Buildings of spherical type and finite BN-pairs" , Lecture Notes in Mathematics , 386 , Springer (1974) [a7] S. Yoshiara, "The flag-transitive geometries of finite order" J. Algebraic Combinatorics (??)
How to Cite This Entry:
Diagram geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagram_geometry&oldid=46645
This article was adapted from an original article by A. Pasini (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article