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A diagram is like a blueprint describing a possible combination of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d1101701.png" />-dimensional incidence structures in a higher-dimensional geometry (cf. [[Incidence system|Incidence system]]). In this context, a geometry of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d1101703.png" /> is a connected [[Graph|graph]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d1101704.png" /> equipped with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d1101705.png" />-partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d1101706.png" /> such that every maximal clique of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d1101707.png" /> meets every class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d1101708.png" />. The vertices and the adjacency relation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d1101709.png" /> are the elements and the incidence relation of the geometry, the cliques of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017010.png" /> are called flags, the neighbourhood of a non-maximal flag is its residue, the classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017011.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017012.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017014.png" /> or by some other convention, one indicates which class the residues of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017015.png" /> should belong to. For instance, a simple stroke
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A diagram is like a blueprint describing a possible combination of  $  2 $-
 +
dimensional incidence structures in a higher-dimensional geometry (cf. [[Incidence system|Incidence system]]). In this context, a geometry of rank $  n $
 +
is a connected [[Graph|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
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
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<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
  
are normally used to denote, respectively, the class of projective planes (cf. [[Projective plane|Projective plane]]), the class of generalized quadrangles [[#References|[a5]]] (cf. also [[Quadrangle|Quadrangle]]) and the class of generalized digons (generalized digons are the trivial geometry of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017016.png" />, where any two elements of different type are incident). The following two diagrams are drawn according to these conventions.
+
are normally used to denote, respectively, the class of projective planes (cf. [[Projective plane|Projective plane]]), the class of generalized quadrangles [[#References|[a5]]] (cf. also [[Quadrangle|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)
 
1)
Line 21: Line 48:
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
  
They both represent geometries of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017017.png" />, where the residues of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017018.png" /> are projective planes and the residues of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017019.png" /> are generalized digons. However, the residues of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017020.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017021.png" /> is said to be flag-transitive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017022.png" /> acts transitively on the set of maximal flags of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017023.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017024.png" />-dimensional projective geometries are the only examples for the above diagram 1). On the other hand, the following has been proved quite recently [[#References|[a7]]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017025.png" /> is a flag-transitive example for the diagram 2) and if all rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017026.png" /> residues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017027.png" /> are finite of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017029.png" /> is either a classical [[Polar space|polar space]] [[#References|[a6]]] (arising from a non-degenerate alternating, quadratic or Hermitian form of Witt index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017030.png" />) or a uniquely determined very small geometry with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110170/d11017031.png" />.
+
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 [[#References|[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|polar space]] [[#References|[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 [[#References|[a3]]] for a survey of related theorems.
 
See [[#References|[a3]]] for a survey of related theorems.

Revision as of 17:33, 5 June 2020


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=12876
This article was adapted from an original article by A. Pasini (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article