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.

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.

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).

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.

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.

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.

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).

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.

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).

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,

Here are a few references related to the project.

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