Incidence structures and cellular algebras (revised version)

101 Pages·2000·0.579 MB·English

by  Christian Pech

Technische Universit¨at Dresden
Fachrichtung Mathematik
Institut fu¨r Algebra
Incidence structures and cellular algebras
revised Version
Diplomarbeit
zur Erlangung des ersten akademischen Grades
Diplommathematiker
vorgelegt von: Christian Pech,
geboren am 22.11.73
in Dresden
Tag der Einreichung: 15.01.98
Betreuer: Prof. Dr. M. Klin,
Prof. Dr. R. P¨oschel
CONTENTS 2
Contents
1 Introduction 3
2 Preliminaries 7
2.1 Permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Cellular algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
A. Definitions and notations . . . . . . . . . . . . . . . . . . . 8
B. Symmetries of cellular algebras . . . . . . . . . . . . . . . 14
C. Some simple propositions about cellular algebras . . . . . . 15
D. Some distinguished classes of cellular algebras . . . . . . . 17
2.3 Incidence structures . . . . . . . . . . . . . . . . . . . . . . . . . . 20
A. Definitions and notations . . . . . . . . . . . . . . . . . . . 20
B. Symmetries of incidence structures . . . . . . . . . . . . . 22
C. t-structures and t-designs . . . . . . . . . . . . . . . . . . 22
D. Some distinguished classes of designs . . . . . . . . . . . . 24
3 Computations in cellular algebras 25
3.1 Permutation groups and centralizer algebras . . . . . . . . . . . . 25
3.2 Generating matrices . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Flag algebras 30
4.1 Flag-algebras of symmetric 2-designs . . . . . . . . . . . . . . . . 44
4.2 Flag-algebras of quasi-symmetric designs . . . . . . . . . . . . . . 45
4.3 Flag-algebras of quasi-balanced designs . . . . . . . . . . . . . . . 57
5 Applications of flag algebras 67
5.1 Hadamard-3-designs . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 The other parameter class of quasi-symmetric 3-designs . . . . . . 77
5.3 2-designs with λ = 1 and v < b . . . . . . . . . . . . . . . . . . . 79
5.4 Generalized quadrangles . . . . . . . . . . . . . . . . . . . . . . . 84
5.5 Generalized triangles . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.6 Graphs associated to [L,N] . . . . . . . . . . . . . . . . . . . . . 92
1 INTRODUCTION 3
1 Introduction
A combinatorial object can be interpreted as a basic set provided with a number
of relations. For instance a graph is defined as a set of vertices equipped with a
binary relation – the adjacency relation.
Between the permutations of the basic set and the relations a Galois correspon-
dence can be introduced. In particular, to each set of permutations the set of
all invariant relations and to each set of relations the set of all automorphisms is
associated. The closed objects of this Galois connection were first described by
M. Krasner in [Kra-38]. From one hand these are just the permutation groups
and from the other hand these are the so called Krasner-clones. This fact makes
it possible to characterize symmetries of combinatorial objects either through
permutation groups or (dually) through Krasner-clones.
Intheinvestigationoftransitivepermutationgroupsthebinaryinvariantrelations
playaprominentrole. InspiredbytheworkofI.SchuraboutS-rings, H.Wielandt
in [Wie-36] introduced the concept of the V-ring (German Vertauschungsring).
In the literature these rings are frequently called centralizer algebra. The V-ring
consists of exactly those complex matrices that commute with all permutational
matrices of the given group (see Section 3.1). It can be easily showed that the
adjacency matrices of the 2-orbits of the group form a basis of the V-ring. On
the other hand each binary invariant relation can be expressed as the union of
2-orbits.
The high importance of the V-rings in permutation group theory lead to the
development of a separate structural theory. In [WeiL-68] B. Yu. Weisfeiler and
A. A. Leman introduced cellular algebras. These are complex, self-adjoint matrix
algebras that are closed (in addition to the usual operations) with respect to the
so called Schur-Hadamard product (compare Section 2.2). It is easy to see that
each V-ring represents a cellular algebra. However, the opposite is not true in
general.
Another important element of this development is the theory of association
schemes. The basis of this theory was laid already in the 1930th by R. C. Bose
andK.R.Nair. OutofthisevolvedtheinvestigationoftheBose-Mesneralgebras,
which – from todays point of view – are a special class of cellular algebras.
For a long time the theories of the V-rings and of the association schemes de-
veloped independently. In [Hig-70] both concepts were brought together by
D. G. Higman. He introduced coherent configurations (as a basic set equipped
with a system of binary relation that fulfill certain axioms) and their associated
coherent algebras.
Coherent and cellular algebras comprise more or less the same class of algebraical
structures. Intheliteraturebothtermsareusedsynonymously. Thistreatiseuses
1 INTRODUCTION 4
the terminology of cellular algebras.
The coherent algebras represent at the same time algebraical and combinatorial
structures. This makes them to one of the most important tools in algebraic
combinatorics. They are especially useful for the description of symmetries of
combinatorial objects.
It was D.G. Higman who, in the 1980th, investigated some classes of low-rank co-
herent algebras and who observed the close relation to certain classes of incidence
structures.
Another interesting aspect of coherent configurations (in connection with inci-
dence geometries) was presented by P. H. Zieschang in [Zie-95]. Homogeneous
coherent configurations are called by him generalized groups. This makes it pos-
sible to interprete regular finite buildings as generalized groups in the same way
as thin buildings can be interpreted as groups (the Coxeter groups). An impor-
tant special case is formed by the dihedral configurations. These are coherent
configurations that are generated by two generalized involutions. They are in
a one to one correspondence with a class of incidence structures – the Coxeter
geometries ([Zie-97]). In particular, for a given Coxeter geometry the associated
dihedral configuration is the smallest coherent configuration that contains both:
the collinearity relation R and the concurrency relation R on the flags of the
L N
geometry.
Algebras that are generated by the adjacency matrices L = A(R ) and N =
L
A(R ) have been treated much earlier. An especially successful application of
N
such algebras was presented by R. Kilmoyer and L. Solomon in [KilS-73]. There
a very elegant proof of the classical theorem of W. Feit and G. Higman was given.
Thistheoremsaysthatgeneralizedn-gonsmayonlyexistforn ∈ {2,3,4,6,8,12}.
Indeed, the generalized n-gons are special Coxeter geometries and the algebras
that were used in [KilS-73] are the associated dihedral algebras.
Finally, it is important to understand that there are numerous ways to represent
one and the same incidence structure as a combinatorial object in the above
sense. Apart from the classical definition it is possible to use the set of flags
as basic set and to equip it with the collinearity- and the concurrency relation.
Then the smallest coherent algebra containing L and N is called flag algebra of
the incidence structure.
Because of the close relation of coherent algebras with the V-rings of permutation
groups, it is possible to use the flag algebras for the examination of symmetries
of incidence structures. Here the term “symmetry” is to be understood in a
combinatorial sense as approximation of the usual symmetry concept since an
incidence structure with a small flag algebra needs not to have many automor-
phisms. However, if its automorphism group is large, one can expect that the
flag algebra has a low rank.
1 INTRODUCTION 5
The main topic of this treatise is, to classify incidence structures by properties
of their flag algebras, to find classes of especially symmetric incidence structures
and to examine their flag algebras.
In Section 2 the usual terminology for permutation groups, cellular algebras and
incidence structures is provided.
Section 3 contains descriptions of the most important combinatorial problems
for cellular algebras such as the construction of the first standard basis, the
computation of the structure constants and the cellular subalgebras etc. For
V-rings and for cellular algebras generated by matrices known methods for the
solutions of these tasks are given ([FarKM-94]).
The last two sections form the kernel of the treatise. Section 4 contains the
theoretical considerations. As was mentioned above, incidence structures can be
interpreted as relational structure on their flags. This allows to introduce such
terms as “L-path”, “N-path”, “L-distance”, “N-distance”, “L-connected” and
“N-connected”. This approach follows the methods of algebraical graph theory
for the investigation of adjacency algebras of graphs (see e.g. [Big-93]). Using
these concepts matrices are constructed that have to be contained in each flag
algebra. With these matrices it is possible to give a lower bound for the rank of
flag algebras. Incidence structures that attain this lower bound are called “flag
minimal”. Independently, incidence structures whose flag algebra is dihedral are
called “dihedral designs”. It is observed that each dihedral design must belong
to one of the following classes:
1) symmetric 2-designs,
2) quasi-symmetric designs with intersection numbers (0,µ) or
3) quasi-balanced designs with intersection numbers (0,λ;0,µ) (see Section
2.3.D for a definition).
Especially remarkable are therefore the flag minimal dihedral designs.
Forthecaseofdesignsabasisoftheself-adjointcomplexmatrixalgebragenerated
by L and N is given. Following this the first standard basis of the flag algebra
for flag minimal dihedral designs is constructed. It turns out that these designs
play the same role in the theory of flag algebras as the distance regular graphs
do in the algebraic graph theory. In particular, the elements of the first standard
basis can be expressed only using the two distance concepts “L-distance” and
“N-distance”.
The three above mentioned classes of designs are considered in Sections 4.1, 4.2
and 4.3
In Section 4.1 results from [Smi-88] and [FarKM-94] about the flag algebras of
symmetric 2-designs are collected.
1 INTRODUCTION 6
Forquasi-symmetricdesignsinSection4.2,again,theknownresultsfrom[KliMMZ-97]
about the flag algebras of 2-designs with λ = 1 are given. In addition to this two
other classes of flag minimal dihedral designs are described: the quasi-symmetric
3-designs and the affine point-hyperplane designs. For these classes of algebras
generating relations are elaborated.
Section4.3startswithaconsiderationofdisconnecteddesigns. Thiswasnotdone
before since the 2-designs are self-evidently both: L-connected and N-connected.
A proof is presented that in general for designs the to concepts L-connected and
N-connected are equivalent. This is followed by a complete description of the
disconnected flag-minimal dihedral designs using connected flag minimal dihedral
designs. Finally, a proof for the fact that generalized n-gons are flag-minimal and
dihedral is given. In principle this was already known from [KilS-73]. However,
no evident proof is given there that the considered algebras are coherent.
In Section 5 the combinatorial standard problems for some of the flag algebras
from Section 4 are treated. Using a special kind of term rewriting systems the
structure constants for whole classes of algebras are computed. Each constant
is then a function. The cellular subalgebras are constructed using the method
from Section 3. Using the structure constants it is possible to search for other
types of combinatorial objects such as distance regular graphs (drg), strongly
regular graphs (srg) or directed strongly regular graphs (dsrg). In this way a
few infinite families of directed strongly regular graphs are discovered. At the
end of the section an attempt is made to generalize these series of dsrgs to other
(non flag-minimal) classes of designs. The result is, that two of the families
can be extended to all 2-designs. Another series can be generalized to all quasi-
symmetric 2-designs with intersection numbers (0,µ) and with k = 2µ or µ = 1.
The supplement contains MAPLE-implementations of the following combinato-
rial algorithms:
• Calculation of the structure constants and construction of the cellular sub-
algebras,
• Search for directed strongly regular graphs.
For those algebras where the automatic construction of the cellular subalgebras
was not successful, protocols of the hand-computations are added.
Hereby I declare that I wrote this master thesis independently. All results were
obtained basing on the sources that are mentioned in the section “References”.
Moreover, each citation is marked appropriately in the text.
2 PRELIMINARIES 7
2 Preliminaries
2.1 Permutation groups
Definition 2.1 (Permutation) Let Ω be a set of n elements. A bijective map
π : Ω −→ Ω is called permutation of Ω. The image of an element x ∈ Ω under a
permutation π of Ω will be denoted as xπ.
Lemma 2.2 Let Ω be a set of n elements. Then the set of all permutations of
Ω forms a group S(Ω) — the full symmetric group of Ω. The operations in S(Ω)
are the usual composition of maps and the inversion of maps according to:
• x(πφ) = (xπ)φ,
• xπ−1 = y ⇐⇒ yπ = x.
(cid:50)
Definition 2.3 (Permutation Group) Let Ω be a finite set (|Ω| = n). A
subgroup U ≤ S(Ω) is called permutation group of Ω. Additionally define
1) the degree of U as deg(U) := n and
2) the order of U as ord(U) := |U|.
Definition 2.4 (Orbit) Let G ≤ S(Ω) be a permutation group. Let x ∈ Ω
then
xG := {y ∈ Ω | ∃π ∈ G : xπ = y}
is called the orbit that is generated by x (or just orbit of x).
Lemma 2.5 Let G ≤ S(Ω) be a permutation group. For x,y ∈ Ω define x ∼
y ⇐⇒ y ∈ xG. Then ∼ is an equivalence relation of Ω. Thus Ω splits into
orbits of G. The set of all orbits of G on Ω will be denoted as orb(G,Ω).
(cid:50)
Definition 2.6 (Action) Let G be a finite group (not necessarily a permuta-
tion group). Let Ω be a finite set. Let
Φ : G×Ω −→ Ω.
The triple (G,Ω,Φ) is called action of G on Ω if:
2 PRELIMINARIES 8
1) ∀x ∈ Ω ϕ(e,x) = x (where e is the identity element of G),
2) ∀x ∈ Ω,∀g,h ∈ G ϕ(gh,x) = ϕ(h,ϕ(g,x)).
Remarks:
1) Let G be finite group and let (G,Ω,Φ) be an action of G on the finite set
Ω. Then Φ induces a homomorphism Φ of G into S(Ω) according to:
(cid:101)
xΦ(g) := Φ(g,x).
(cid:101)
2) The image of G under Φ is called permutation representation of G on Ω
(with respect to Φ). (cid:101)
3) Let K = ker(Φ) := {g ∈ G | Φ(g) = id} be the kernel of Φ. Then (G,Ω,Φ)
will be called(cid:101)faithful if |K| =(cid:101) 1. (cid:101)
4) If (G,Ω,Φ) is faithful then it will be identified with Φ(G). If in addition Φ
is clear from the context then the action (G,Ω,Φ) is(cid:101)denoted as (G,Ω) —
the usual notation for a permutation group on Ω.
Notation: Let U and G be groups such that U ≤ G. Then N (U) denotes the
G
normalizer of U in G. If G = S(Ω) then instead of N (U) the form N(U,Ω)
S(Ω)
or even N(U) will be used.
Amoredetailedintroductiontopermutationgroupscanbefounde.g. in[Wie-64],
[KliPR-88] or [Ker-91].
2.2 Cellular algebras
A. Definitions and notations
Following some definitions for different types of linear operator algebras will be
given. As the terminology in this field is to be considered as classical, termini
such as e.g. vector space, linear operator or adjoint operator will be used without
further introduction. A rather extensive treatment of the matter can be found in
[Gre-63] and [Koc-62].
Definition 2.7 (B-Algebra) Let V be a finite dimensional vector space over
the complex number field C. A set W of linear operators from V into itself is
called B-algebra if it complies to the following axioms:
1) The identity operator I is element of W,
2 PRELIMINARIES 9
2) ∀A,B ∈ W : the sum A+B ∈ W,
3) ∀A ∈ W, ∀λ ∈ C : the scalar multiple λA ∈ W,
4) ∀A,B ∈ W : the operator product A·B ∈ W,
5) ∀A ∈ W : the adjoint operator A∗ ∈ W.
Definition 2.8 (Cellular Algebra) Let V be a finite dimensional vector
space over the complex number field C. Let v = {v ,v ,...,v } be a basis of V.
1 2 n
Let further on W be the matrix representation of a B-algebra on V with respect
to v. W is called cellular algebra if it fulfills the following axioms (in addition to
the axioms of a B-algebra):
1) W contains the all-1 matrix J
2) W is closed with respect to the Schur-Hadamard multiplication which is
defined as follows: A = (a ), B = (b )
ij ij
A◦B = C = (c ) where c = a b
ij ij ij ij
Note that for the definition of cellular algebras it is really necessary to fix a base
in V since the Schur-Hadamard product cannot be defined base independently
(it is a combinatorial property of the algebra).
Proposition 2.9 (Schur-Wielandt Principle) Let W be a cellular algebra.
Let A = (a ) ∈ W, 0 (cid:54)= c = a ∈ C for some i and j.
ij ij
Then the matrix
1 a = c
A = (c ) where c = (cid:26) ij
(c) ij ij 0 else
is element of W.
(cid:50):
If A does only have entries c and 0 then nothing has to be proved.
Otherwise let {x = c,x ,...,x } be the set of different entries of A. Define
1 2 k
A(i) = A−x J and
i
A˜ := A◦A(2)◦···◦A(k) ,A˜ = (a˜ ) .
ij
Then a˜ (cid:54)= 0 ⇐⇒ a = c. Moreover all entries of A˜ are equal to c(c−x )(c−
ij ij 2
x )···(c−x ) or 0. Thus we get
3 k
1
˜
A = A
(c) c(c−x )(c−x )···(c−x )
2 3 k
:(cid:50)
2 PRELIMINARIES 10
Proposition 2.10 (Existence of the first standard basis) Let W be a
cellular algebra. Then W contains a basis (cid:104)A ,A ,...,A (cid:105) of 0/1-matrices with
1 2 r
the following properties:
1) A +A +···+A = J,
1 2 r
2) A ◦A = δ A ,
i j ij i
3) For each i ∈ {1,2,...,r} there exists a j ∈ {1,2,...,r} such that A∗ = A .
i j
This basis is called first standard basis of W.
(cid:50):
In this proof the notion “support” of a matrix A = (a ) will be used. It is defined
ij
as the set of all pairs (i,j) such that a (cid:54)= 0.
ij
Because of the Schur-Wielandt principle each element A of W can be expressed
as
A = a A +a A +···+a A
1 (a1) 2 (a2) k (ak)
where A is a 0/1-matrix. From this follows that W has a basis of 0/1-matrices,
(ai)
suppose this is
W = (cid:104)B ,B ,...,B (cid:105).
1 2 s
Take any two distinct elements B , B for which B ◦B (cid:54)= 0 Define B˜ = B −
i j i j i i
(B ◦B ), B˜ = B −(B ◦B ), B˜ = B ◦B . Then
i j j j i j i j
˜ ˜ ˜
B ,B ,...,B ,B ,B ,...,B ,B ,B ...,B ,B
(cid:110) 1 2 i−1 i i+1 j−1 j j+1 s (cid:111)
is still a (linearly) generating set of W. Repeating this procedure leads to a gen-
erating set of matrices with mutually disjoint support. Because of this property
this set of matrices forms a base (cid:104)A ,A ,...,A (cid:105) of W.
1 2 r
Let A be any of these basis elements. Then
i
r
∗
A = a A .
i (cid:88) k k
k=1
Suppose a = 1 then A = A∗ (the number of non-zero entries of A∗ and of A
i i i i i
must be equal).
Suppose a = 0 and more than one other coefficient — say a and a — are equal
i j k
to 1. Then the support of A∗ is strictly contained in the support of A . But this
j i
is a contradiction to the basis properties of (cid:104)A ,A ,...,A (cid:105).
1 2 r
Consequently A∗ = A for some j.
i j
This completes the proof.
:(cid:50)