HONORS THESIS
GEOMETRIC OBJECTS WITH A MORE COMBINATORIAL FLAVOR
Submitted by
Andreea Erciulescu
Department of Mathematics
In partial fulfillment of the requirements
for the degree of Bachelor of Science, Honors
Colorado State University
Fort Collins, Colorado
Spring 2011
COLORADO STATE UNIVERSITY
April 18, 2011
Committee on Undergraduate Honors Work
Advisor: Professor Anton Betten
Professor Tim Penttila
ii
ABSTRACT OF HONORS THESIS
GEOMETRIC OBJECTS WITH A MORE COMBINATORIAL FLAVOR
We are studying geometric objects, defined over finite fields, with a more combinatorial flavor and present the results of the investigation of classification problems
in geometry and combinatorics. Objects called BLT-sets, living in a vector space
over a finite field, are of great interest to finite geometry, as they provide access
to most of the objects that have been studied for a long time (translation planes,
generalized quadrangles, flocks). On the other hand, there are objects that are
invariant under a Finite group. An example are the Coxeter groups, which act
as symmetry groups of root systems, and have been classified. They turn out as
symmetry groups of BLT-sets.
Andreea Erciulescu
Department of Mathematics
Colorado State University
Fort Collins, Colorado 80523
Spring 2011
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TABLE OF CONTENTS
1 Introduction
1.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
2 Coxeter Groups
2.1 Root Systems/Classification results . . . . . . . . . . . . . . . . . .
2.2 Coxeter graphs of root systems . . . . . . . . . . . . . . . . . . . .
2.3 B − N pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
4
8
10
3 BLT sets
3.1 Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Ovals and conics in a finite projective plane . . . . . . .
3.1.2 Analogy between Euclidean planes and translation planes
3.1.3 Generalized Quadrangles . . . . . . . . . . . . . . . . . .
3.1.4 Flocks of quadratic cones . . . . . . . . . . . . . . . . . .
3.2 BLT sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 1
INTRODUCTION
In this thesis we study geometric objects that live in a vector space over a
finite field, and geometric objects that are invariant under a finite group. Examples
of the first category are objects called BLT-sets that are of great interest to finite
geometry, and that are studied intensely using computer search. These objects
are very important, as they provide access to most of the objects that have been
studied for a long time (translation planes, generalized quadrangles, flocks). They
were defined by Bader, Lunardon and Thas (hence the name) in 1990 in connection
to the study of flocks of a quadratic cone [3]. [5] defines quadrics and generalized
quadrangles with specific examples and important properties in 1997, although
the concept of generalized quadrangles and their properties are due to Payne and
Thas. Independently they showed the existence of a generalized quadrangle of
order (q 2 , q). In 1976 Walker and Thas showed that to each flock of an irreducible
quadric of P G(3, q) corresponds a translation plane of order q 2 . In 1980, Payne
showed the existence from a set of q upper triangular 2x2 matrices over GF (q) of
a certain tpe, known as a q − clan. Than, in 1988, Gevaert, Johnson and Thas
claimed that flocks of quadratic cones can be used to define translation planes.
Examples of the second category are the Coxeter groups, which act as symmetry
groups of root systems, and have been classified. The reflection groups started
1
to be the area of interest for many scientists with the work of Coxeter, in 1930s
and has been developed by J. Tits around 1960, and has continued to the present.
Coxeter groups are groups with particular presentations coming from the geometry
of polyhedral tessellations and have been classified [15]. This gives the subject
both an algebraic and a (combinatorial and differential) geometric flavor, hence
important in design of experiments and coding theory. These groups are very
important in part because of their connection to Lie Algebras [7]. We can think
of them as groups of reflections in a finite dimensional real vector space. Coxeter
groups are very special among groups, since they arise as crucial auxiliary objects
in so many different circumstances and branches of mathematics. As it turns out,
Coxeter groups turn up as symmetry groups of BLT-sets, and this is where the two
areas come together. The study of BLT-sets via their symmetries begins in [2].
1.1
Introductory remarks
Let q denote a prime power, let GF (q) denote a finite field with q elements,
and let d denote a positive integer. Let V = GF (q)2d denote the vector space over
GF (q) of dimension 2d, consisting of column vectors with entries in GF (q).
Define a map Q : V → GF (q) as follows:
For u = (u1 , u2 , .., u2d )T ∈ V let Q(u) =
P
u2i−1 u2i , i = 1..d.
The form Q is a quadratic form on V , that is, Q(λu) = λ2 Q(u), for λ ∈
GF (q), u ∈ V , and
f (u, v) = Q(u + v) − Q(u) − Q(v), for u, v ∈ V
is a bilinear form on V . The form Q is usually called hyperbolic quadric. Note that
for vectors u, v ∈ V we have
2
f (u, v) =
P
(u2i−1 v2i + u2i v2i−1 ).
A vector v ∈ V is called isotropic, if Q(v) = 0. A subspace U of V is called
isotropic, if Q(u) = 0 for every u ∈ U , and it is called maximal isotropic, if it is
maximal (with respect to inclusion) in the set of all isotropic subspaces of V . It
turns out that the dimension of every maximal isotropic subspace if d. Observe
that if u, v ∈ V belong to the same isotropic subspace of V , then Q(λu + λv) = 0
for every λ, µ ∈ GF (q). Furthermore,
f (u, v) = Q(u + v) − Q(u) − Q(v) = 0.
Conversely, if u and v are isotropic with f (u, v) = 0, then hu, vi is an isotropic
subspace of V . Indeed, for λ, µ ∈ GF (q) we have
Q(λu + µv) = λ2 Q(u) + µ2 Q(v) + λµf (u, v) = 0.
3
Chapter 2
COXETER GROUPS
The first two sections of this chapter refer to [15].
2.1
Root Systems/Classification results
Some of the objects of our current interest are collections of vectors in some
inner vector space and reflections associated to each of them, creating very nice
structures; each point acts on both an operation to be performed and an object to
which operations can be applied.
A root system is a reformulation, in terms of linear algebra, of the concept
of a finite Euclidean reflection group. More exactly, it is a translation into linear
algebra of the geometric configuration formed by the reflecting hyperplanes associated with a reflection group. Finite reflections groups are analyzed with great
efficiency using linear algebra.
A root system is a set of vectors satisfying certain axioms motivated by reflecting hyperplanes of a finite Euclidean reflection group. The geometric configuration
formed by these hyperplanes gives a significant amount of information about the
structure of the reflection group. This group permutes the hyperplanes, and this
4
action shapes the configuration of the hyperplanes and provides the link between
the resulting geometric pattern and the structure of the reflection group.
If in two dimensions, drawing a picture is enough to analyze the geometric
patterns, in higher dimensions one turns to linear algebra to make it possible; the
reflecting hyperplanes are replaced by set of vectors.
Let W be a finite reflection group in the Euclidean space, say E, and replace
each reflecting hyperplane of W by its two orthogonal vectors of unit length. Let
∆ be the resulting vectors in the Euclidean space, described as
∆ = {α|sα is the reflection inW associated to α and kαk = 1}.
Using this set, W can be described as the group generated by {sα |α ∈ ∆} and
the reflecting hyperplanes of W consist of {x ∈ E|(x, α) = 0, α ∈ ∆}.
The vectors of ∆ satisfy the following properties:
1. If α ∈ ∆, then λα ∈ ∆ if and only if λ = ±1;
2. If α, β ∈ ∆, then sα · β ∈ ∆. This describes the premutation of ∆ under the
action of W .
Since the dihedral groups are the only reflection groups in the plane, any root
system ∆ in the plane must have some Dn as its associated Weyl group W (∆).
5
Examples in R2
Root system Dl
Definition 2.1.1 (Root System). Let {1 , 2 , ..., l } be an orthonormal basis of Rl ,
and let ∆ = {±i ± j |i 6= j}. This is a root system, and its associated reflection
group is W (∆) = (Z/2Z)l−1 × Σl , where
Σl = the permutation group on {1 , 2 , ..., l }
(Z/2Z)l−1 = sign changes on an even number of {1 , 2 , ..., l }
The symmetric group Σl acts on the l − 1 copies of (Z/2Z) by permuting factors.
The fundamental system is given by Σ = {1 − 2 , ..., l−1 − l , l−1 + l }. We are
allowed to permute {1, 2, ..., l}, as well as to change the signs on any even number
of {1 , 2 , ..., l }.
Root system An
The central example of a finite reflection group is the symmetric group Sn+1
action on Rn+1 by permutation of coordinates. Transpositions act as orthogonal
6
reflections across hyperplanes of Rn+1 . The diagonal line L in Rn+1 is fixed by
Sn+1 so Sn+1 acts on the orthogonal complement L⊥ , and L⊥ can be identified
with Rn . The associated roots system is type An .
Definition 2.1.2 (Convex hull). The convex hull of a set of points in n dimensions is the intersection of all convex sets containing this set. For n points ri , for
i = 1..n, the convex hull C is described as the set
P
P
{ λi ∗ ri , for all i = 1 . . . n : λi ≥ 0 for all i and
λi = 1} .
Computing the convex hull is a problem in computation geometry but this definition is helpful in our further examples.
Definition 2.1.3 (Simplex). A simplex is the generalization of a tetrahedral region
of space to n dimensions. The boundry of a n − simplex has n + 1 0 − f aces
(polytope vertices), n(n + 1)/2 1 − f aces (polytope edges), and n+1
i − f aces.
i+1
In one dimension, the simplex is the line segment. In two dimensions, the simplex
is the convex hull of the equilateral triangle. In three dimensions, the simplex is
the convex hull of the tetrahedron.
If we project the standard basis of Rn+1 to Rn and take its convex hull, we
get a regular n − simplex. This exhibits Sn+1 as the group of symmetries of
the regular n − simplex. A fundamental domain on Rn+1 is the convex region
{x ∈ Rn+1 |x1 ≥ x2 ≥ . . . ≥ xn+1 }. Projecting this region to Rn , we get a simplicial
cone, and then intersecting with Sn−1 , we get a spherical (n − 1) − simplex.
Definition 2.1.4 (n-simplex). The standard n − simplex (or unit n − simplex)
is the subset of Rn+1 given by
∆n = {(t0 , t1 , . . . , tn ) ∈ Rn+1 |
P
ti = 1 and ti ≥ 0, for all i = 0 . . . n}.
7
The n + 1 vertices of the standard n − simplex are the points {ei } ∈ Rn+1 ,
where ei = (0, 0, . . . 0, 1 (in the ith position), 0, . . . , 0 ). There is a canonical
map from the standard n − simplex to an arbitrary n − simplex with vertices
(v0 , v1 , . . . , vn ) given by
(t0 , t1 , .., tn ) →
P
ti ∗ vi , for i = 0..n.
The n − tuple(t0 , t1 , .., tn ) are the coordinates of a point in the n − simplex. If
the map is affine transformation, than the n − simplex is called affine n − simplex.
Note that the graph of an n − simplex is the complete graph of n + 1 vertices.
All the possible roots systems have been classified and there are four infinite
families and five ’exceptional’ root systems. Also, there are different ways of encoding the information from the root systems, such as Cartan matrices, Coxeter
graphs and Dynkin diagrams.
2.2
Coxeter graphs of root systems
F4
Definition 2.2.1 (Cartan Matrix). The Cartan Matrix is defined as the n × n
matrix whose entry in the ith row and the j th column is ri × rj . The entries
are called Cartan integers. The Cartan Matrix determines the root system up to
isomorphism.
8
Just as the root system can be turned into a Cartan matrix, the matrix as
well can be turned into a pictorial form that we can really represent. Given a n × n
Cartan matrix we can construct a combinatorial graph:
1. we start with n points;
2. the n rows and columns of the matrix correspond to n simple roots;
3. so each of these points correspond to a simple root as well.
Moreover, given the Cartan matrix, its associated Dynkin diagram is its Coxeter graph, decorated with an arrow on double and triple edges, drawn from the
larger simple root to the smaller one, on the edge.
Let G be a Coxeter group with root system ∆ with base Π = {r1 , r2 , . . . , rn }
and denote si the fundamental reflection along ri .
Proposition 2.2.2 (Order of a group element). If ri , rj ∈ Π, then there is an
integer pij ≥ 1 such that
(ri ,rj )
kri kkrj k
= −cos pπij . pij is the order of si sj as a group
element.
Example 2.2.3 (Dynkin diagram - Cartan matrix). As an example, consider the
following Dynkin diagram and Cartan matrix for F4:
Dynkin diagram
Cartan matrix
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2.3
B − N pairs
Preliminaries
Let q denote a prime power, let GF (q) denote a finite field with q elements,
and let d denote a positive integer. Let V = GF (q)2d denote the vector space over
GF (q) of dimension 2d, consisting of column vectors with entries in GF (q).
Definition 2.3.1 (Distance-regular graph [14]). A distance-regular graph is a
regular connected graph with degree k and diameter d for which following holds.
Let Si (X) = {Z ∈ vertex − set|δ(X, Z) = i}. There are natural numbers
b0 = k, b1 , ..., bd−1 , c1 = 1, c2 , ..., cd ,
such that for each pair (X, Y ) of vertices satisfying δ(X, Y ) = i we have
1. the number of vertices in Sj−1 (X) adjacent with Y is cj (1 ≤ j ≤ d);
2. the number of vertices in Sj+1 (X) adjacent with Y is bj (1 ≤ j ≤ d − 1).
Definition 2.3.2 (Dual polar graph Dd (q) on V [8]). The vertex-set V (Dd (q)) of
Dd (q) is the set of all maximal isotropic subspaces of V . Vertices X, Y ∈ V (Dd (q))
are adjacent in Dd (q) if and only if the dimension of X ∩ Y is d − 1. Let δ denote
the path-length distance function on Dd (q). It is easy to see that δ(X, Y ) = i if
and only if dim(X ∩ Y ) = d − i. The graph is bipartite with diameter d and with
Q d−i−1
(q
+ 1) vertices, for i = 0..(d − 1).
10
Let
d−i
bi = q i q q−1−1 , ci =
q i −1
q−1
and ki =
b0 b1 ..bi−1
c1 c2 ..ci
for 0 ≤ i ≤ d. The graph Dd (q) is regular with valency b0 = k1 .
Definition 2.3.3 (Isometry/Orthogonal group). Let GL(V ) denote the general
linear group of V . Then σ ∈ GL(V ) is called isometry of V , if Q(σ(v) = Q(v)
for every v ∈ V or when f (u, v) = f (σ(u), σ(v)) for u, v ∈ V , where f (u, v) is the
symmetric billinear form on V .
The group of all isometries of V is called the orthogonal group for Q, and is de+
+
noted by O2d
(q). Note that every σ ∈ O2d
(q) acts on V (Dd (q)) as an autmorphism
of Dd (q). The full automorphism group G of Dd (q) acts distance-transitively on
V (Dd (q)), that is, for X, Y, Z, W ∈ V (Dd (q)) with δ(X, Y ) = δ(Z, W ) there exists
σ ∈ G such that σ(X) = Z and σ(Y ) = W .
Pick X, Y ∈ V (Dd (q)) and let GX and GY denote the stabilizers of X and Y
in G, respectively. Since G acts distance-transitively on V (Dd (q)), the orbits of
GX are precisely the sets Si (X), 0 ≤ i ≤ d.
These distance regular graphs play an important role in the combinatorics
from construction of the (B, N ) pairs [18].
A (B, N ) pair is a pair of subgroups B and N of a group G such that the
following axioms hold:
1. G is generated by B and N .
11
2. The intersection H, of B and N , is a normal subgroup of N .
3. The group W = N/H is generated by a set of elements wi of order 2, for i
in some non-empty set I.
4. If wi is one of the generators of W and w is any element of W , then wi Bw
is contained in the union of Bwi wB and BwB.
5. No generator wi normalizes B [13].
B is an analogue of the upper triangular matrices of the general linear group
GLn (K), H is an analogue of the diagonal matrices, and N is an analogue of the
normalizer of H. For our convenience, B is the stabilizer of a maximal flag and
N represents the monomial matrices. Hence H represents the monomial matrices
that stabilize the maximal flag.
Results for D6
D6 turns out to be isomorphic to the quotient group
12
N
.
H
Chapter 3
BLT SETS
A BLT -set is a set X of q + 1 points of the generalized quadrangle Q(4, q),
q odd, such that no point of Q(4, q) is collinear with more than 2 points of X.
BLT-sets are named after Laura Bader, Guglielmo Lunardon and Jef Thas who
first studied them in 1990 and are closely related to flocks of the quadratic cone,
elation generalised quadrangles and certain translation planes.
We will continue describing the relationships listed in the figure above.
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3.1
Projective Geometry
Basic Definitions:
Definition 3.1.1 (Projective Plane [9]). A projective plane is a geometric construction that extends the concept of a plane by adding a line at infinity. That
is, if in the plane there exist parallel lines that do not intersect, in the projective
plane the parallel lines intersect in a point at infinity. This system of points and
lines, together with a relation of incidence, satisfy the following axioms:
1. Any two distinct points are incident with just one line;
2. Any two distince lines are incident with just one point;
3. There exist four points, no three of which are incident with the same line.
Definition 3.1.2 (Finite Projective Plane [5]). A finite projective plane of order
n has the following properties:
1. Every line contains n + 1 points;
2. Through every point pass n + 1 lines;
3. There are exactly n2 + n + 1 points and n2 + n + 1 lines.
Fano Plane [12] is the finite projective plane with the smallest possible number
of points and lines: 7 each. It has order 2.
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Theorem 3.1.3 (Bruck’s Theorem [5]). Let α be a projective plane of order n
and β be a proper subplane of α of order m. Then either m2 = n or m2 + m ≤ n.
Proof. If L is a line of β, than L has m + 1 points of β and n − m points outside
the subplane. Since any two lines of β intersect in β, and there are m2 + m + 1
lines in β, there are (m2 + m + 1) ∗ (n − m) points outside of β that are incident
with a line of β. But there are n2 + n + 1 points in α, so
n2 + n + 1 ≥ m2 + m + 1 + (m2 + m + 1)(n − m) ⇔ 0 ≥ (n − m2 )(m − n).
Since m < n ⇔ m − n < 0, this implies that n ≥ m2 . Equality holds when
evey point of α is incident with a line of β, or dually, when every line of α meets
β in at least one point. In this case, β is called Baer subplane.
Suppose n > m2 and let P be a point of α that does not lie on any line
of β. So every line through P containts at most one point of β and hence, the
number of lines through P is at least as great as the number of points of β ⇔n+1
≥ m2 + m + 1 ⇔n ≥ m2 + m.
15
This theorem gives an important restriction on the existence of projcetive
planes inside other projective planes. For instance, the projective plane of order 3
cannot contain the Fano plane, as 3 ≤ 22 and 3 ≥ 22 + 2.
3.1.1
Ovals and conics in a finite projective plane
In finite projective planes lie geometric objects like arcs, ovals and hyperovals.
Definition 3.1.4 (Arc). An arc in a projective plane α of order n is a set of points,
no three collinear.
Claim 3.1 (Claims about arcs).
1. If A is an arc, then |A| ≤ n + 2.
Proof. Choose P , a point on the arc. Each of the n + 1 lines on P contains at
most one other point on A. Equality holds when each line meets A in either
0 or 2 points.
2. If A is an arc and n is odd and n > 1 then |A| ≤ n + 1.
Proof. Choose Q outside the arc. If |A| = n + 2 then exactly
n+2
2
lines on Q
intersect A, but for n odd, this is not an integer.
Definition 3.1.5 (Generalized definition of arc). A (k, n)-arc in a projective plane
α of order q is a set of k points with some n but no n + 1 points on a line. A
(k, 2)-arc is simply called k-arc.
Theorem 3.1.6 (Bose: (k,n)-arcs).
1. k ≤ q + 2 and equality holds when q is
even.
2. (q + 1) − arcs are called ovals
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3. (q+2)−arcs are called hyperovals (adding the nucleus, that is the intersection
of the tangents), for q even.
In finite Desarguesian planes, the canonical example of oval is a conic in
P G(2, q).
Theorem 3.1.7 (Desargues’ Theorem [10]). If the three straight lines joining the
corresponding vertices of two triangles ABC and A0 B 0 C 0 all meet in a point (the
perspector), then the three intersections of pairs of corresponding sides lie on a
straight line (the perspectrix). Equivalently, if two triangles are perspective from
a point, they are perspective from a line.
Definition 3.1.8 (Projective geometry). Let V be a finite vector space of dimension n + 1 over GF (q). Then the geometric structure P G(n, q) is the projective
geometry of dimension n over GF (q). Subspaces of dimension 0, 1 and 2 are
called points, lines and planes respectively. If n = 2 then any two distinct lines
in P G(2, q) intersect because any two distinct 2-dimensional subspaces in a 3dimensional space share a one-dimension subspace.
17
Why classify projective planes - incomplete problem?
Considering Desargues’ Theorem, the projective planes reduce to the field plane
P G(2, F ), a classical example based on the projective geometry of dimension 2. In
the finite case, this means that F = GF (q).
Theorem 3.1.9 (Segre’s Theorem: [9]). A set of q + 1 points in P G(2, q), q odd,
not containing three collinear points must be a conic.
Definition 3.1.10 (s-fold / blocking set). An s − f old blocking set K in P G(2, q)
is a set of points such that every line of P G(2, q) intersects K in at least s points.
A 1 − f old blocking set is simply called a blocking set. A blocking set is said to be
trivial if it contains a line of P G(2, q). An s − f old blocking set is called minimal
or irreducible when no proper subset of it is an s − f old blocking. A minimum
blocking set is a blocking set of smallest cardinality.
Theorem 3.1.11 (The size of a minimal blocking set). Let k be the size of a
minimal blocking set in a projective plane of order n. Then
n+
√
n+1≤k ≤n·
√
n+1
Equality holds on the left hand side if n is a square and the blocking set is a Baersubplane. Equality holds on the right hand side if n is a square and the blocking
set is a unital [4].
Observe that (k, n) − arcs (used when n is small compared to q) and s − f old
blocking sets (used when s is small) with n + s = q + 1 are in fact each others
complement.
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3.1.2
Analogy between Euclidean planes and translation planes
Euclidean planes:
1. points (x, y) are contained in 2-dimensional vector spaces over R ;
2. lines through the origin are 1-dimensional vector spaces;
3. the other lines are translates of the lines through the origin.
Translation planes:
Let V be a vector space which is a direct sum of two copies of a subspace V 1 so
that V = V 1 ⊕ V 1.
1. each element of V can be written as (X, Y ); X, Y ∈ V 1;
2. if dim(V 1) = r ⇒ dimV = 2r and the sets of vectors for which X = 0 or
Y = 0 are r-dimensional subspaces isomorphic to V 1.
3.1.3
Generalized Quadrangles
Definition 3.1.12 (Quadrangle). A quadrangle is a set of four points no three of
which are collinear. The third axiom listed above in the definition of the projective
plane shows their existence in projective planes.
One could construct subplanes, by completeting quadrangles in a projective
plane as follows: Consider P1 ,P2 ,P3 ,P4 be the points of a quadrangles in the projective plane α, the lines P1 P2 ,P1 P3 ,P1 P4 , P2 P3 ,P2 P4 ,P3 P4 , the points of intersection
in α of these lines, the lines determined in α by this larger set of points and the
points of intersection in α of this larger set of lines and so on.
Definition 3.1.13 (Generalized quadrangle [9]). A generalized quadrangle of order
(s; t) is an incidence structure of points and lines such that:
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• there is at most one line through two points;
• two lines intersect in at most one point:
• there are s + 1 points on every line where s ≥ 1;
• there are t + 1 lines through every point where t ≥ 1;
• for any point P and line l not containing P there exists a unique line l
through P meeting l.
Observe that the generalized quadrangles are self-dual because two points lie
on at most one line and no point is collinear with all others.
Theorem 3.1.14 (GQ [9]). Let G be a finite GQ with orders s, t.
• G has (s + 1)(st + 1) points and (t + 1)(st + 1) lines;
• s + t divides st(s + 1)(t + 1);
• if s ≥ 1, then t ≤ s2 ;
• if t > 1, then s ≤ t2 .
Let’s consider the four dimensional symplectic vector space, known as doily
[11]. This is an incidence geometry with 15 points and 15 lines each containing 3
points. It is the smallest thick generalized quadrangle and denoted by W3 (2). It
has automorphism group S6 .
20
A BLT -set of lines of W3 (q) is a set L of q + 1 disjoint lines such that no line
of W3 (q) meets more than two lines of L. Since W3 (q) is dual to Q(4, q), we study
BLT -sets of points of Q(4, q) and we will define them later.
Note: The proof of this duality involves Plucker coordinates and Klein correspondance, that makes the connection between spreads of lines and translation
planes [9]. The results show that the points in W3 (q) correspond to lines in Q(4, q)
and the lines in W3 (q) correspond to points in Q(4, q). Hence, BLT -set of lines of
W3 (q) correspond to BLT -sets of points of Q(4, q).
Furthermore, there is one known generalized quadrangle constructed from a
BLT -set, too. That is the Knarr construction that gives a GQ(q 2 , q) [16]. According to the definition above, it has (q 2 + 1)(q 3 + 1) points and (q + 1)(q 3 + 1) lines.
Example 3.1.15 (Example of a BLT -set of lines). Consider the twisted cubic [19],
the fundamental example of a skew curve. It is a smooth, rational curve of degree
3 in a projective 3-space:
21
AG(1, q) → AG(3, q)
t → (t, t2 , t3 )
P G(1, q) → P G(3, q)
(s, t) → (s3 , s2 t, st2 , t3 ) ↔ (x, y, z, t)
It is described by the homogeneous polynomials:

 xt − yz = 0
y 2 − xz = 0
 2
z − yt = 0
The equation of the tangent line is:
∂f
| (x
∂x P
− Px ) +
∂f
| (y
∂y P
− Py ) +
∂f
| (z
∂z P
− Pz ) +
∂f
| (t
∂t P
− Pt ) = 0
Hence the tangent plans are:
π1 : xt0 − yz0 − zy0 + tx0 = 0
π2 : −xz0 + 2yy0 − zx0 = 0
π3 : −t0 y + 2z0 z − y0 t = 0
The set of q + 1 tangent lines to this curve form a BLT set of lines in W (3, q), for
q odd.
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3.1.4
Flocks of quadratic cones
Definition 3.1.16 (Flocks of quadratic cones). A flock of a quadratic cone K in
P G(3, q) is a set of q planes meeting K in sections which partition K, minus its
vertex. A flock is linear if the planes all share a line. In other words, a flock of K
is a partition of the points of
K
V
into q conics [17].
Simplest example: The linear flock, q planes passing through a fixed line skew
to the cone.
Independently, the following theorems were proven a few decades ago, showing
the existence of GQ(q 2 , q) mentioned above.
Theorem 3.1.17 (1976 Walker and Thas). To each flock of an irreducible quadric
of P G(3, q) corresponds a translation plane of order q 2 .
Theorem 3.1.18 (1980 Payne). Given a set of q upper triangular 2x2 matrices
over GF (q) of a certain type, known as a q − clan, there exists a generalized
quadrangle of order (q 2 , q).
Theorem 3.1.19 (1987 Thas). To a q − clan corresponds a flock of a quadratic
cone of P G(3, q) and conversely. Hence, with each flock of a quadratic cone of
P G(3, q) there corresponds a generalized quadrangle of order (q 2 , q).
In 1988, Gevaert, Johnson and Thas claimed that flocks of quadratic cones
can be used to define translation planes.
The strong connection between flocks and other geometric structures leads us
to introduce a new structure in the projective plane that may be used to determine
the existence of certain flocks of quadratic cones: conic blocking sets.
23
Definition 3.1.20 (A conic blocking set). A conic blocking set is a set of lines in
a Desarguesian projective plane such that all conics meet these lines.
Also with Thas, in 1975, started the interest in hyperbolic quadrics. He showed
that the only flocks of the hyperbolic quadric are the linear flocks and the Thas
flocks. Then in 1989, Bader and Lunardon showed that every flock of the hyperbolic
quadric is linear, a Thas flock or one of the three exceptional flocks.
In [1] it was shown that the exceptional hyperbolic flocks can be connected to
root systems in R4 , as the root systems arise from reflection groups.
3.2
BLT sets
Definition 3.2.1 (A partial BLT -set). A partial BLT -set of Q(4, q) is a set B
of points, such that for any point P of Q(4, q) not in B, the number of points of
B collinear with P is at most 2. Two partial BLT -sets are equivalent if they are
in the same orbit of the automorphism group P ΓO(5, q). A partial BLT -set of
Q(4, q) has size at most q + 1; if equality occurs it is a BLT-set.
Why BLT sets?
BLT -sets of order q would give rise to translation planes of order q 2 (that is, with
q 4 + q 2 + 1 points and the same number of lines). Thus, a BLT set of order 67
would create a really big plane. If the BLT set is linear (it arises from the linear
flock) then the projective plane will be the Desarguesian plane P G(2, q 2 ). Thus,
listing many non-linear BLT sets means listing many non-Desarguesian projective
planes.
The BLT set in characteristic 23 turns out to have as automorphism the
Coxeter groups F4 , of order 1152, or closely related to this group [6].
24
Example 3.2.2 (Example of a transitive BLT -set in the finite field of order 67).
Consider the irreducible polynomial of degree 2 over Fq , x2 + a1 x + a0 . Then
S=
0
1
−a0 −a1
, < S >∼
= Cq+1 ≤ P GL(2, q),
and it has order
q 2 − 1, in the vector space
q + 1, in P G(1, q)
Let A = S 4 , of order 17 and let i and j be prime numbers, none equall 17. Then
the question is to solve for i and j such that
O+ (4, 67) ∼
= P GL(2, 67).
= P GL(2, 67) and Aj ∼
= Ai × Aj , where Ai ∼
1. It was found recently by Betten using computer search on the Open Science
Grid and the Teragrid (of the order 16 years CPU time).
2. This BLT -set has a group to a cyclic group of order 17 extended by a cyclic
group of order 4. The 4 induces a 4 on the 17.
3. Using an eigenvalue technique, we are able to identify a one-dimensional
subspace that is fix under the group, so everything takes place in the perpendicular subspace, which is a 4-dimensional. This subspace is of plus type
and hence the group must be a subgroup of the 4-dimensional orthogonal
group O+ (4, q).
4. The group is the direct product of two 2-dimensional projective linear groups.
The isomorphism is explicit.
25
5. Using all this, we now have two pairs of elements in the 2-dimensional projective linear groups corresponding to the generators for the groups of order
17 and 4, respectively.
6. The associated matrices satisfy two relations. If these relations could be
made to work for general field size q (under a congruence on q), then there
is hope that the BLT -set generalizes.
26
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Penttila. Flocks and partial flocks of hyperbolic quadrics via root systems.
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Penttila. Symmetries of BLT-sets. Designs, Codes and Cryptography, 29(13):41–50, 2003.
[3] Laura Bader, Guglielmo Lunardon, and Joseph A. Thas. Derivation of flocks
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