Combinatorial complexes, Bruhat intervals and reflection distances

Abstract: The various results presented in this thesis are naturallysubdivided into three different topics, namely combinatorialcomplexes, Bruhat intervals and expected reflection distances.Each topic is made up of one or several of the altogether sixpapers that constitute the thesis. The following are some of ourresults, listed by topic:Combinatorial complexes:Using a shellability argument, we compute the cohomologygroups of the complements of polygraph arrangements. These arethe subspace arrangements that were exploited by Mark Haiman inhis proof of the n! theorem. We also extend these results toDowling generalizations of polygraph arrangements.We consider certainB- andD-analogues of the quotient complex ?(?n)=Sn, i.e. the order complex of the partition latticemodulo the symmetric group, and some related complexes.Applying discrete Morse theory and an improved version of knownlexicographic shellability techniques, we determine theirhomotopy types.Given a directed graphG, we study the complex of acyclic subgraphs ofGas well as the complex of not strongly connectedsubgraphs ofG. Known results in the case ofGbeing the complete graph are generalized.We list the (isomorphism classes of) posets that appear asintervals of length 4 in the Bruhat order on some Weyl group. Inthe special case of symmetric groups, we list all occuringintervals of lengths 4 and 5.Expected reflection distances:Consider a random walk in the Cayley graph of the complexreflection groupG(r, 1,n) with respect to the generating set of reflections. Wedetermine the expected distance from the starting point aftertsteps. The symmetric group case (r= 1) has bearing on the biologist?s problem ofcomputing evolutionary distances between different genomes. Moreprecisely, it is a good approximation of the expected reversaldistance between a genome and the genome with t random reversalsapplied to it.