GEOMETRIC OBJECTS WITH A MORE COMBINATORIAL FLAVOR
Andreea Erciulescu, Statistics Department, Iowa State University, Ames
Mentor Anton Betten, Mathematics Department, Colorado State University, Fort Collins
Abstract
This research is a synergy between algebra, geometry and combinatorics. 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 is the Coxeter groups, which act as symmetry groups of root systems, and have been classified. The groups act as
permutations on the roots and Coxeter groups turn out as symmetry groups of BLT-sets. For example, the automorphism group of BLT sets in
Why BLT sets?
Encoding the information from the root systems
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.
Cartan matrices, Coxeter graphs and Dynkin diagrams
Example- consider F4
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
Dynkin Diagram
Example of a BLT set of lines
Cartan Matrix
Coxeter Graph of the Root System
of elements in the 2-dimensional projective linear groups corresponding to the generators for the groups of order 17 and 4, respectively.
BLT sets
The set of q + 1 tangent lines to the twisted cubic form a BLT -set in W3(q), for q odd.
AG(1, q) 7−→ AG(3, q)
A BLT -set is a set X of q + 1 points of the generalized quadrangle Q(4, q), q odd, such that no
t 7−→ (t, t2, t3)
B-N pairs
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.
A (B, N ) pair is a pair of subgroups B and N of a group G such that the following axioms
P G(1, q) 7−→ P G(3, q)
(s, t) 7−→ (s3, s2t, st2, t3)
←→
hold:
(x, y, z, t)
1. G is generated by B and N .
The equation of the tangent line is:
∂f
∂x |P (x
− Px ) +
∂f
∂y |P (y
− Py ) +
∂f
∂z |P (z
y 2 − xz = 0
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
π2 : −xz0 + 2yy0 − zx0 = 0





 z 2 − yt = 0
Desargues’ Theorem: If the three straight lines joining the corresponding vertices of two
− Pt ) = 0
Hence the tangent plans are:



 π1 : xt0 − yz0 − zy0 + tx0 = 0
Homogeneous polynomials:



 xt − yz = 0
Projective Geometry
− Pz ) +
∂f
∂t |P (t
π3 : −t0y + 2z0z − y0t = 0
non-empty set I.
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.
5. No generator wi normalizes B.
For our convenience, B is the stabilizer of a maximal flag and N represents the monomial
Groups of Symmetries
triangles ABC and A’B’C’all meet in a point (the perspector), then the three intersections
matrices. Hence H represents the monomial matrices that stabilize the maximal flag.
of pairs of corresponding sides lie on a straight line (the perspectrix). Equivalently, if two
Results for D6
triangles are perspective from a point, they are perspective from a line.
Examples in R2
A finite projective plane of order n is defined as a set of n2 + n + 1 points with the properties
The reflection group - automorphism group of the n-gon, Pn
that:
Dihedral groups - the simplest examples of Coxeter groups
1. Any two points determine a line,
3. Every point has n + 1 lines on it, and
2. Any two lines determine a point,
4. Every line contains n + 1 points.
Dn = Z/nZ × Z/2Z
• if hxi = Z/2Z then xyx = y −1, for any y ∈ Z/nZ
• x = the reflection of Pn and Z/nZ = the rotations of Pn
• if hyi = Z/nZ (i.e. rotation of order n) then x2 = y n = 1
2
n
2
Dn = x, y|x = y = (xy) = 1
Future Directions
Examples in R3
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,
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.
The reflection groups - automorphism groups of 3D figures
the projective planes reduce to the field plane P G(2, F ), a classical example based on the
2. Investigate more BLT groups, some generalize and some don’t.
projective geometry of dimension 2. In the finite case, this means that F = GF (q).
3. Find an algebraic equation for the points, by a finding low degree homogeneous polynomial whose zero set is the points. The set needs to be partitioned and maybe low
Generalized quadrangles
degree polynomials exist for each of the parts of the partition. This would mean the set
Doily, denoted as W3(2):
• four dimensional symplectic vector space
• an incidence geometry with 15 points and 15 lines each containing 3 points
Σ4
Σ4 × Z/2Z ⇔ (Z/2Z)3 × Σ3
A5 × Z/2Z
- ortogonal transformations on R3
Acknowledgements
Root systems
• the smallest thick generalised quadrangle
• automorphism group S6
is the intersection of varieties.
It is a translation into linear algebra of the geometric configuration formed by the reflecting
This research was supported by the College of Natural Sciences Undergraduate Research
Institute Summer Fellowship, CSU and SACNAS, and was conducted while I was a student
in the Mathematics Department, at Colorado State University, Fort Collins.
hyperplanes associated with a reflection group.
Examples in R2
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
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).
29, Advances in Geometry, Vol. 3, Special Issue.
W (∆) is a subgroup of the isometry group of the root system.
Classified: 4 infinite families (An, Bn,Cn,Dn) and 5 exceptions (E6, E7, E8, F4, G2)