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VIGRE REU 2003 - LIE THEORY AND WIRELESS ANTENNAE NETWORKS

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Hi, my name is Rupert Venzke and I'm working with the Rutgers University
VIGRE mathematics REU program this summer. I recently completed
an M.A. degree in mathematics at the
University of Pittsburgh and
I will begin study in the mathematics Ph.D program at the
California Institute of
Technology this September. Currently,
I am studying Lie Theory and Algebraic Geometry here with
Dr. Shawn Robinson,
Dr. Chris Woodward, and
Sasha Ovetsky.

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Our research question is a mathematical reformulation of a practical
problem from electrical engineering. Certain communications systems
operate by sending out data in the form of a string of matrices, each
matrix chosen from some initially prescribed finite set called a
constellation. However, it is ultimately inevitable that static will
interfere with such data transmission from time to time. Thus, we would
like to design our matrix "alphabet" in such a way as to minimize the
probability of confusing such matrix signals.

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In practice, the matrices in the constellation are special unitary
matices. A special unitary matrix is a matrix of determinant 1 whose
conjugate transpose is its inverse. Computing the probability of
confusing any two such square matrices A and B requires evaluation of an
integral that is approximated very closely by a fraction whose denominator
has a term of the form |det(A - B)|^(1/n), where n is the number of rows
of A, B. All other things being equal, we attempt to minimize the
probability by maximizing the value of value of |det(A - B)|^(1/n).

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Given a constellation {A1, ..., Ak} of special unitary matrices, we define
the diversity product of the constellation to be the minimum of the values
|det(Ai - Aj)|^(1/n), i and j distinct. In the case of matrices contained
in SU(2), SU(2) can be visualized geometrically as a three manifold unit
sphere S^3 in C^2. In this case, matrices of the constellation are
identified with certain points in S^3 and the diversity product is a
direct measure of the minimum distance between any two such distinct
points of the constellation in S^3. So, given any integer k, the problem
of forming good constellations in SU(2) amounts to spacing k points on S^3
as far apart as possible. More generally, for the case of larger
matrices, the paradigm is that we are attempting to geometrically space
out some number of points on a manifold in a nice way.

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We can explicitly compute some initial values of Z(SU(n), k) =
(1/2)*maximum over constellations of k elements in SU(n) of {minimum
|det(Ai - Aj)|^(1/n)}.
For example, Z(SU(1), k) is just half the length of a side of a regular
k-gon inscribed in a unit circle. Z(SU(2), 2) is 1, since the optimal way
of spacing out 2 points on S^3 is to place them at antipodes. Similarly,
Z(SU(2), 3) = sqrt(3)/2 since sqrt(3) is the edge length of an equilateral
triangle inscribed in a unit circle; Z(SU(2), 4) = sqrt(6)/3 since
2*sqrt(6)/3 is the edge length of a regular tetrahedron inscribed in a
unit sphere; and Z(SU(2), 5) = sqrt(10)/4, half the length of an edge of a
generalized regular 4-d tetrahedron inscribed in unit S^3.

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To simplify the problem, initially only constellations with group
structure were considered. In addition to this structure, because we are
dealing with the reciprocal of terms |det(A - B)| we would like the group
to be designed in such a way that |det(A - B)| is always non-zero.
Thinking of the matrices as operators on an n dimensional complex space,
this condition is really equivalent to requiring that the operators do not
fix any vector of C^n. Such arrangements are called fixed point free
groups and there is a general result classifying fixed point free group
into 6 nice infinite families.

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Of these 6 families, representations from one family in particular,
referred to as the J_m,r family, seemed to do surprisingly well. These
representations are built as certain products of representations of
SL2(F5), a constellation that does well in SU(2). In fact, we found that
the group SL2(F5), known as the binary icosahedral group, can be
identified with the collection of vertices of a regular 4-d polyhedron
called the 600-cell. The 600-cell consists of 600 3-d tetrahedrons glued
together along faces in 4-d in a highly symmetric fashion. When viewed
from this perspective, we see that representations of SL2(F5) actually
give rise to the optimal value of Z(SU(2), 120).

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Next, by considering centers of tetrahedral faces of the 600-cell, we get
a dual polyhedron known as the 120-cell. This figure consists of 120
dodecahedrons pieced together in a nice symmetric way. Because of the
symmetries involved, one would expect the 120-cell to give rise to a whole
new class of good constellations. In fact, what makes a
constellation good, in addition to low diversity, is whether it
has a large number of nodes in contrast with the dimension of
the space. Currently, I am finishing a program to compute such
representations and their corresponding diversities.

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We would now like to produce results for non-group constellations. The
case of constellations in SU(2) has been essentially solved. Our goal
is to now use a technique from algebraic geometry to produce good
constellations for SU(3). We work with Bott-Samelson Varieties. A
variety is a zero-set of some multi-variable polynomial. In this sense, a
variety is like a surface, but much more general than a manifold. To form
Bott-Samelson varieties associated to SU(n), we first identify each
transposition of the symmetric group on n letters with a particular
subgroup of SU(n) isomorphic to SU(2) containing the diagonal special
unitary group T. Then, a Bott-Samelson variety is a product of k such
subgroups, modulo a specific action by T^k. A corresponding Bott-Samelson
map here is the product map from a Bott-Samelson variety to SU(n)/T.
Choosing the subgroups in a way consistent with the shortest decomposition
into transpositions of the longest Weyl group element leads to a nice
Bott-Samelson map. Specifically, the Bruhat decomposition of the image
forms a simplicial complex in which one element of the decomposition is a
dense open subset of the image precisely where the Bott-Samelson map is
1-1. This gives us a method for identifying representations in SU(3) with
collections of representations of SU(2).

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So, we form new constellations in SU(n) from previous ones in SU(2) via a
nice product. However, because of the construction, our new
constellations will have zero diversity. However, in spite of the fact
that our constellations have zero diversity, there is still a nice
inherent symmetry. Our job now is to delete constellations in some
satisfactory manner so as to preserve the symmetry while avoiding a
degenerate case in which zero diversity is got. Using the construction
outlined above,

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Here are a few references
related to the project.

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Last year, I worked on an intriguing REU project in Representation Theory
with Shawn Robinson. If you're interested, follow this mysterious link.