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Combinatorial complexes, Bruhat intervals and reflection distancesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2003 (English)Doctoral thesis, monograph (Other scientific)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Stockholm: KTH , 2003. , p. ix, 108
##### Series

Trita-MAT. MA, ISSN 1401-2278 ; 2003:07
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:kth:diva-3620ISBN: 91-7283-591-5 (print)OAI: oai:DiVA.org:kth-3620DiVA, id: diva2:9451
##### Public defence

2003-10-21, 00:00
#####

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##### Note

QC 20100616Available from: 2003-10-21 Created: 2003-10-21 Last updated: 2010-06-16Bibliographically approved

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 certain*B*- and*D*-analogues of the quotient complex Δ(Π_{n})=*S*_{n}, 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 graph*G*, we study the complex of acyclic subgraphs of*G*as well as the complex of not strongly connectedsubgraphs of*G*. Known results in the case of*G*being 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 group*G*(*r*, 1,*n*) with respect to the generating set of reflections. Wedetermine the expected distance from the starting point after*t*steps. The symmetric group case (*r*= 1) has bearing on the biologists 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.

isbn
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