Graphs, Root Systems, and Cohomology
Summer 2009 REU
Led by Prof. Leonard Chastkofsky
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A (0,2)-graph Is a graph such that any 2 vertices have either 0 or 2 common neighbors. These are regular and graphs arise in a wide variety of contexts in Mathematics. One important class of examples are the hypercubes. Another construction comesas certain quotients of graphs associated to finite projective planes. Recently, another construction was discovered related to root systems of Lie Algebras. The goal of the REU was to explore these graphs.
The REU started out with Dr. Chastkofsky giving a series of lectures on the background needed to explore this topic. The topics were graph theory, group theory, root systems and cohomology. The elements from these areas relevant to the research were explained. Although a comprehensive study of these areas could of course nut be undertaken in such a short time frame, enough of the theory was developed for the students to explore the topic and perform computations. The lectures generally consisted of a lecture in theory in the morning, with a lecture introducing the students to computer algebra systems in the afternoon. The students were introduced to Mathematica and GAP. In practice, they ended up doing most of their computations with GAP, and they proved adept at learning the rudiments of the program. The first week was spent in lectures and the students absorbing them. For the second and third weeks, the lectures continued, with the students being encouraged to start thinking about problems. During this period,there were several guest lectures given by specialists in the relevant topics. Dr. Clint McCrory gave a lecture introducing simplicial homology. Dr Dan Nakano gave a lecture introducing representation theory. Dr. Brian Boe gave a lecture on reflection groups and root systems. Dr. Robert Varley gave a lecture on cohomology and Laplacians. In addition, the graduate assistant, Irfan Bagci, gave a couple of lectures on Lie Algebra cohomology. These lectures were introductory in nature, designed to give the students an idea of what the subject was about. During the third week, a series of specific research problems were laid out.
For the last 4 weeks, the students were asked to choose specific problems to work on. They were encouraged to collaborate, but most wound up working on their own specific problems, although some of the problems were closely related. The discovery of the graphs related to root systems was made in connection with an investigation into cohomology of algebras. This suggested that a complex can be associated with these graphs. One of the themes of the students' research was to explore this. For example, the complex associated with a hypercube is the standard n-complex.Andries Brouwer has compiled a complete list of (0,2)-graphs of small valency. Elisabeth Palchak, Jennifer Wise and Jane Rieck worked together in examining complexes for some of these graphs in detail. They looked at series of related graphs, and investigated how their complexes are related. They verified that the complex associated with the Cartesian product of 2 graph is the join of their complexes, and that the complex associated with a graph quotient is a quotient of the complex. Their write-up includes detailed descriptions of the complexes associated with many of these graphs. Although a geometrically realizable complex can be associated with these graphs, these are not always CW-complexes. Indeed, a boundary map cannot always be defined. Lee Troupe worked on the problem of investigating when such a boundary map can be defined. He found the smallest case where it cannot be defined, and proved that on the other hand, such a boundary map can always be defined for graphs associated to finite projective planes. The hypercube graphs also arise as graphs associated with root systems. In this context, they are associated with sets of root summing to alpha_0, the highest root of the root system. It is natural to examine graphs associated with roots summing to a multiple of alpha_0. For example, how many vertices does such a graph have, and what is its automorphism group? For the case of type A_n and 2 times alpha_0, a precise formula can be given for the number of vertices, involving a second order recursion. Alexander Garver wrote up the proof for the argument giving this formula, and started investigating the case of 3 times alpha_0. Here the situation is much more complicated, and probably involves an eigth-order recursion. Alexander made some empirical investigations of this case, and made some progress in understanding it. The automorphism groups of hypercubes are known to be semi-direct products of a symmetric group with an elementary abelian 2-group. The graphs associated to roots summing to a 2 times alpha_0 have much less symmetry. Brian Bonsignor investigated their automorphism groups. In the end, a complete and explicit description for these was found. The automorphism groups of hypercubes are also the Weyl groups of root systems of type B_n. David Chen's problem was to find a canonical isomorphism between these groups and to exploit it. For example, the conjugacy classes of the group can be described in a very nice way, using its action on the hypercube. David was able to establish the isomorphism, and used it to give a description of the conjugacy classes that is ain agreement with one given by Geck and Pfeffer. He has written this up, and it has been accepted for publication in the Rose-Hulman Undergraduate Mathematics Journal. The incidence graphs of projective planes are related to the ones the group was studying. These graphs are distance regular, and have only 4 distinct eigenvalues. Lorenzini has developed techniques for finding critical groups of graphs, and these graphs seemed especially suitable for such techniques. Indeed, in many cases, Lorenzini's results give these groups almost immediately. Stuart found that Lorenzini's methods could be extended to cases where the results were not so immediate. Danid Chen and Alex Garver presented their research at the Young Mathematician's Conference at Ohio State in August of 2009.
The reports of the students' research may be found below:
Elisabet Palchak, Jennifer Wise and Jane Rieck, "On Semibiplanes and Cell Complexes" http://www.math.uga.edu/~lenny/REU Summer 2009/StudentWriteUps/Palchak.pdf
Stuart Shirrell, "On Graphs Arising from Projective Planes"
http://www.math.uga.edu/~lenny/REU Summer 2009/StudentWriteUps/ShirrellWriteup.pdf
Lee Troupe, "On Complexes for Graphs Arising as quotients of Projective Planes" http://www.math.uga.edu/~lenny/REU Summer 2009/StudentWriteUps/TroupeReport.dvi
David Chen, "On the Automorphism Group of the Cube and the Weyl group of type B_n"http://www.math.uga.edu/~lenny/REU Summer 2009/StudentWriteUps/ChenRev.pdf"> pdf
Al Garver, "On Construction of certain (0,2)-graphs coming from root systems" http://www.math.uga.edu/~lenny/REU Summer 2009/StudentWriteUps/Garver.pdf
Brian Bonsignore, "On Automorphism Groups of certain (0,2)-graphs coming from root systems" http://www.math.uga.edu/~lenny/REU Summer 2009/StudentWriteUps/BonsignoreWriteUp.pdf
