GEOMETRIC OBJECTS WITH A MORE COMBINATORIAL FLAVOR
Andreea Erciulescu
Mentor Anton Betten
Department Mathematics, Colorado State University, Fort Collins
BLT sets
Abstract
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
Why BLT sets?
A BLT -set is a set X of q + 1 points of the generalized quadrangle Q(4, q), q odd, such
Finite geometry, as they provide access to most of the objects that have been studied for a long time (translation planes, generalized quadrangles,
that no point of Q(4, q) is collinear with more than 2 points of X. BLT-sets are named after
flocks). On the other hand, there are objects that are invariant under a Finite group. An example is the Coxeter groups, which act as symmetry
Laura Bader, Guglielmo Lunardon and Jef Thas who first studied them in 1990 and are
groups of root systems, and have been classified. They are very important because their connection to Lie Algebras. Coxeter groups turn out
as symmetry groups of BLT-sets. For example, the automorphism group of BLT sets in characteristic 23 and 47 is a Coxeter group of type F4 or
order 1152 (or closely related to this group). Similar behavior can be found with other examples. In terms of geometry, we consider an example
of a transitive BLT-set in the Finite Field of order 67 and present the results of two pairs of elements in the 2-dimensional projective linear groups
closely related to flocks of the quadratic cone, elation generalised quadrangles and certain
translation planes.
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.
Example of a transitive BLT -set in the finite field of order 67
corresponding to the generators for the groups of order 17 and 4, respectively.
Consider the irreducible polynomial of degree 2 over Fq , x2 + a1x + a0. Then
Root Systems - Classification results
S=
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
0
1
, < S >∼
= Cq+1 ≤ P GL(2, q),
−a0 −a1
and it has order
configuration formed by the reflecting hyperplanes associated with a reflection group.
(
Examples in R2
q 2 − 1, in the vector space
q + 1, inP G(1, q)
Let A = S 4, of order 17 and let i and j be prime numbers, none equall 17. Then the question
Projective Geometry
Desargues’ Theorem: If the three straight lines joining the corresponding vertices of two
is to solve for i and j such that
O+(4, 67) ∼
= Ai × Aj , where Ai ∼
= P GL(2, 67) and Aj ∼
= P GL(2, 67).
triangles ABC and A’B’C’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.
2
A finite projective plane of order n is defined as a set of n + n + 1 points with the properties
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
that:
Root systems Dl
order 4. The 4 induces a 4 on the 17.
Let {1, 2, ..., l } be a orthonormal basis of Rl , and let ∆ = {±i ± j |i 6= j}. This is a root
1. Any two points determine a line,
system, and W (∆) = Z/2Z l−1 ± Σl , where
2. Any two lines determine a point,
3. Every point has n + 1 lines on it, and
3. Using an eigenvalue technique, we are able to identify a one-dimensional subspace that
4. Every line contains n + 1 points.
Σl = the permutation group on {1, 2, ..., l }
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
Z/2Z l−1 = sign changes on an even number of {1, 2, ..., l }.
of the 4-dimensional orthogonal group O+(4, q).
The fundamental system is given by Σ = {1 − 2, ..., l−1 − l , l−1 + l }. We are allowed to
4. The group is the direct product of two 2-dimensional projective linear groups. The
permute {1, 2, ..., l}, as well as to change the signs on any even number of {1, 2, ..., l }.
isomorphism is explicit.
Coxeter graphs of root systems
Dl
!
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.
F4
Fano Plane is the finite projective plane with the smallest possible number of points and
lines: 7 each.
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
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.
projective geometry of dimension 2. In the finite case, this means that F = GF (q).
B-N pairs
A (B, N ) pair is a pair of subgroups B and N of a group G such that the following axioms
hold:
Future Directions
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. Two flocks of K are equivalent if there is a collineation fixing
1. |O+(8, 2)| = 384 = 192 ∗ 2 and |D4| = 192, so we could construct the corresponding BN
pair using the stabilizer chain of fundamental roots.
K and taking the first flock to the second. A flock is linear if the planes all share a line. A
2. The BLT group mentioned above doesn’t generalize. One should try to find an alge-
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,
2. The intersection H of B and N is a normal subgroup of N .
braic equation for the points. In other words, once the points are in (x, y) coordinates
the number of points of B collinear with P is at most 2. Two partial BLT -sets are equivalent
over GF (672) using the model, one can put z = 1 and then try to find a low degree
3. The group W = N/H is generated by a set of elements wi of order 2, for i in some
if they are in the same orbit of the automorphism group P (5, q). A partial BLT -set of Q(4, q)
homogeneous polynomial (in x, y, z) whose zero set is the points. Another idea comes
has size at most q + 1; if equality occurs it is a BLT-set.
from the fact that the set needs to be partitioned. Maybe low degree polynomials ex-
1. G is generated by B and N .
non-empty set I.
Generalized quadrangles
4. If wi is one of the generators of W and w is any element of W , then wiBw is contained in
the union of BwiwB and BwB.
Let’s consider the four dimensional symplectic vector space, known as doily. This is an
5. No generator wi normalizes B.
incidence geometry with 15 points and 15 lines each containing 3 points. It is the smallest
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
thick generalised quadrangle and denoted by W3(2). It has automorphism group S6.
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).
ist for each of the parts of the partition. This would mean the set is the intersection of
varieties.
Acknowledgements
This research was supported by the College of Natural Sciences Undergraduate Research
Institute Summer Fellowship and MAA.
Bibliography
Richard Kane (2001). Reflection Groups and Invariant Theory, Canadian Mathematical Society.
Law and Penttila (2003). Classification of flocks of the quadratic cone over fields of order at most
29, Advances in Geometry, Vol. 3, Special Issue.