A Tactical Decomposition for Incidence Structures

Combinatorics '90 A. Barlotti et at. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved.

31

A TACTICAL DECOMPOSITION POR INCIDHNCE STRUCTURES MeterBettcn and Mathlaa Bram Math. Sem. Univ. Ludewig-Meyn-Str. 4, D 2300 Kiel Stettiner Str. 6, D 2080 Pinneberg For every finite incidence s t r u c t u r e we define parameters by s o m e ordering process. Refining this process will give a tactical decomposition of t h e incidence matrix which w e call TDO (tactical decomposition by ordering ). This decomposition generally is coarser than t h e tactical decomposition defined by t h e a u t o m o r p h i s m group ( T D A ) ; b u t it has t h e advantage t h a t it can b e computed rather quickly. We point o u t t h a t t h e TDO in combination with t h e TDO's of t h e point derivations is a s t r o n g invariant. For instance it suffices to characterize a l l 5250 linear spaces on 10 points.

1 ) Studying incidence s t r u c t u r e s one often assumes t h a t there is a large g r o u p

of automorphisms. For instance one may suppose t h a t t h e automorphism g r o u p a c t s transitively on points ( homogeneity). One also likes to have regularity conditions, f o r instance t h a t every block has t h e same number of points and every point is incident with t h e same number of blocks. But since m o s t incidence s t r u c t u r e s have small automorphism groups or a r e rather irregular we are interested a l s o in these cases. We observe t h a t irregularity has t h e advantage of providing s o m e information concerning t h e structure. In t h e following we describe a sequence of parameters which arises from irregularity in a natural way.

2 ) We explain t h e procedure by an explicit example: Let there be given t h e following incidence matrix 1

2

X

3

4

x x x x

X

X X

5

6

x x x

X

7

8

x x x

x x x x x x x

9

X

X X

D.Betten. M.Braun

38

having 9 points and 8 blocks. This is a linear space o n 9 points, where t h e 9 blocks of length 2 have been omitted. We collect all blocks with maximal length ( in t h e example, one block of length 4 ) to one block domain. By permutation of blocks w e put this domain a t t h e top. We separate by a horizontal line t h e next block domain consisting of 7 blocks of length 3. 1

2

3

4

x x x x

X X

X X

.

5

6

x x x

X

7

8

x x x

x x x x x x x

X

9

X X

In general there are more than t w o block domains which w e o r d e r by t h e length of their blocks. We call t h e length of a block t h e block t y p e of t h e f i r s t kind , and we a l s o call t h e block domains j u s t defined t h e block domains of t h e f i r s t k i n d . Now we look a t points and the way in which their incidences a r e distributed to t h e various block domains j u s t defined. For instance point no. 3 has one incidence in t h e upper block domain and one incidence in t h e lower block domain, whereas point no. 4 has one incidence in t h e upper domain and t w o incidences in t h e lower domain. So, point no. 3 and point no. 4 have different _type. We now collect all points having t h e same type to one point domain and we order t h e different point domains lexicographically, separating t h e various domains by vertical lines. 1

2

4

3

8

x x xE

-

X X

X X

We speak of Doint

types

9

K

x x

X

x x

K

X

X -

and p o i n t domains of t h e f i r s t kind.

Now we look a t t h e blocks again and t h e way in which their incidences are distributed to t h e point domains of first kind. W e call t h e s e distributions

A tactical decomposition for incidence structures

39

the block types of second kind. In every block domain of first kind we collect all blocks having the same type of second kind and order these domains of second kind lexicographically, separated horizontally by a line. 1

2

4

S

3

6

7

8

9

Now it is clear how to go o n : We define point types and point domains of second kind, then block types and block domains of third kind and so on. This procedure will stabilize because t h e number of points and blocks is finite. In t h e example t h e next stages are: 1

2

4

3

.

4

6

7

8

0

1

-+

2

1 4

4 6

7

7

3

3

5

s

8

8

2

4

3

5

7

6

8

4

3

5

6

7

8

9

2 1

6

1

2

7'

9

The tactical decomposition by ordering ( TDO

1

2

4

3

5

7

6

8

9

):

The stable situation we reach by the procedure described above defines a de-

D. Betten, M.Braun

40

composition of t h e incidence matrix into rectangles, where every rectangle is a configuration. By a configuration we mean a n incidence s t r u c t u r e consisting of vi points, bi blocks with every block incident with kij points and every point incident with rij blocks. If we define by f . . = v i - r.. = b: k. t h e num11 'I I 11 ber of incidences in t h e rectangle ( i , j ) , we g e t t h e following scheme, giving t h e parameters of t h e TDO:

* . . .

.

.

. . .

.

Here I is t h e number of final point types and J is t h e number of final block types. There a r e t w o extremes: a.) In t h e TDO all final point domains and all final block domains have only one element ( point or block ) as in t h e example above. In t h i s case t h e TDO and t h e incidence s t r u c t u r e coincide, when the f..= t de'1 n o t e t h e incidences. b.) The incidence matrix is a configuration from t h e beginning. Then of course t h e TDO is of no help. Usually t h e procedure of refinement will s t o p somewhere in between. If t w o incidence matrices are isomorphic, then t h e related TDO's a r e congruent. Therefore, if f o r t w o matrices t h e related TDO's a r e not congruent, then t h e incidence matrices cannot be isomorphic. 3 ) We will use t h e notion of TDO in order to study linear spaces o n small point number. A linear space ( P , B ) consists of a set P of points and a family B q of different of subsets of P called blocks, such t h a t f o r every pair p points there is exactly one block b c B with p , q c b . In order to exclude degenerate c a s e s we assume I P I 2 2 and I b I 2 2 for every b c B. The number N ( n ) of pairwise non-isomorphic linear spaces o n n points, 2 5 n 5 1 0 , is

*

n N(n)

2 1

3 2

4 3

5 5

6 10

7 24

8 69

9 384

10

5250

The numbers for n 9 can b e found in C 11; t h e number N( 10) h a s been calculated by David Glynn ( Christchurch, New Zealand ), who a l s o determined

41

A tactical decomposition for incidence structures

t h e size of t h e automorphism group in every case. Calculating t h e TDO's with t h e computer s h o w s t h a t f o r n i 8 all isomorphism types can be distinguished by t h e TDO's. The first situation where t h e TDO d o e s not suffice occurs f o r n = 9 . Example: (i)

1

2

3

x x x X

X

4

5

6

x x x X

X

X

7

9

2

3

a

9

x x x

X

x

8

X

X 7

(ii)

6

The 2-blocks a r e omitted. These t w o examples are n o t isomorphic b u t they have t h e s a m e TDO: 0

Another example consists of t h e following linear spaces on 9 points: (

(iv)

iii )

x x X

x

X

x

X

x

x x x x x X x x X x x x x x x x x x x x X

xx

x x x x x x x x x x x X X x x X xx x x x x x x < x

< < <


xx

XX

xx xx xx xx

XX

xx

( V )

X

X

xx xx xx xx xx

x X

x

x

x x

x

x x x X x x x X x x X xx x x X x xx x x xx xx x x xx xx x x xx xx x

42

D.Betten. M.Braun These three linear spaces are pairwise non-isomorphic as can be seen from the cycles i n the 2-block domains. B u t in all three cases we get the same TDO:

This example shows that the tactical decomposition defined by the orbits under the automorphism group ( TDA) may be finer than t h e TDO. 4)Since the TDO is not sufficient for the isomorphism problem, we look further invariants. One easily sees that the two linear spaces ( i ) and ( i i ) be distinguished by taking point derivations. By t h i s we mean omitting point and all blocks incident with this point. In example ( i ) all points equivalent and derivation gives :

for can one are

B u t in example ( i i ) w e can find a point having the derivation:

This example proposes to use besides the TDO also the derived structures for all points.-Here we do not look a t the isomorphism type of the derived structures but we calculate their TDO's again. In addition we take into account t h e TDO of t h e total structure already found. The TDO together with t h e TDO's of t h e point derivations is now a stronger invariant. 5 ) Observation: TDO together with TDO's of point derivations distinguish all linear spaces o n n points, ns10. Namely with help of t h e computer, using TDO and TDO's for point derivations we get 384 linear spaces on 9 points and 5250 linear spaces on 10 points. These numbers coincide with the numbers listed above.

We note that the computer constructs and distinguishes 232923 linear spaces o n 11 points. Question: Is t h e number of isomorphism types bigger?

A tactical decomposition for incidence structures

43

Remark: Of course this stronger invariant ( TDO and TDO's f o r point derivations ) will n o t suffice in general. The smallest example w e know, where t h i s invariant d o e s not suffice, a r e t h e t w o Steiner Systems S ( 2,3,13) o n 13 points ( C 2 1 page 27, table 1 ). References C 11

j. Doyen, Sur l e nombre d'espaces lineaires non isomorphes d e n points, Bull. SOC. Math. Belg. 19 ( 1967) , pp. 421-437

J. W. di Paola and H. Gropp, Hyperbolic Graphs from Hyperbolic Planes, Congressus Numerantium 68 ( 1989 ) , pp. 23-44 C 3 1 M. Braun, Erzeugung Linearer Raume auf kleinen Punktanzahlen am Computer, Diplomarbeit Kiel 1990 C41 W. Page and H. L. Dorwart, Numerical patterns and geometrical configurations, Math. Mag. 57, (1984), pp. 82-92 C21