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Combinatorial Complexes, Bruhat Intervals and Reflection Distances

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Combinatorial Complexes, Bruhat Intervals and Reflection Distances Combinatorial complexes, Bruhat intervals and reflection distances Axel Hultman Stockholm 2003 Doctoral Dissertation Royal Institute of Technology Department of Mathematics Akademisk avhandling som med tillst˚andav Kungl Tekniska H¨ogskolan framl¨ag- ges till offentlig granskning f¨oravl¨aggandeav teknisk doktorsexamen tisdagen den 21 oktober 2003 kl 15.15 i sal E2, Huvudbyggnaden, Kungl Tekniska H¨ogskolan, Lindstedtsv¨agen3, Stockholm. ISBN 91-7283-591-5 TRITA-MAT-03-MA-16 ISSN 1401-2278 ISRN KTH/MAT/DA--03/07--SE °c Axel Hultman, September 2003 Universitetsservice US AB, Stockholm 2003 Abstract The various results presented in this thesis are naturally subdivided into three differ- ent topics, namely combinatorial complexes, Bruhat intervals and expected reflection distances. Each topic is made up of one or several of the altogether six papers that constitute the thesis. The following are some of our results, listed by topic: Combinatorial complexes: ² Using a shellability argument, we compute the cohomology groups of the com- plements of polygraph arrangements. These are the subspace arrangements that were exploited by Mark Haiman in his proof of the n! theorem. We also extend these results to Dowling generalizations of polygraph arrangements. ² We consider certain B- and D-analogues of the quotient complex ∆(Πn)=Sn, i.e. the order complex of the partition lattice modulo the symmetric group, and some related complexes. Applying discrete Morse theory and an improved version of known lexicographic shellability techniques, we determine their homotopy types. ² Given a directed graph G, we study the complex of acyclic subgraphs of G as well as the complex of not strongly connected subgraphs of G. Known results in the case of G being the complete graph are generalized. Bruhat intervals: We list the (isomorphism classes of) posets that appear as intervals of length 4 in the Bruhat order on some Weyl group. In the special case of symmetric groups, we list all occuring intervals of lengths 4 and 5. Expected reflection distances: Consider a random walk in the Cayley graph of the complex reflection group G(r; 1; n) with respect to the generating set of reflections. We determine the ex- pected distance from the starting point after t steps. The symmetric group case (r = 1) has bearing on the biologist’s problem of computing evolutionary distances between different genomes. More precisely, it is a good approximation of the ex- pected reversal distance between a genome and the genome with t random reversals applied to it. ISBN 91-7283-591-5 ² TRITA-MAT-03-MA-16 ² ISSN 1401-2278 ² ISRN KTH/MAT/DA--03/07--SE iii iv Preface If you are a mathematician, then you know all too well the following question: “But, what does a mathematician really do?”. Luckily, thanks to P´alErd˝os,we all know the proper answer: mathematicians convert coffee into theorems. However, some people are rather persistent and keep going: “But, what is a theorem anyway?”. The answer is, of course, that a theorem is a tautology, a way to rephrase something. It doesn’t say anything new! Thus, mathematics is basically the art of converting coffee into nothing, which is, clearly, to violate the first law of thermodynamics. This, of course, must never be told to the people that fund us. During my graduate studies, a range of helpful people have guided and aided my attempts to violate the first law of thermodynamics. Among them are the following: ² Anders Bj¨ornerhas been a superb advisor. Being an enormously rich source of advice and hints, of answers and questions, of ideas and suggestions, he has provided the best possible support a graduate student can hope for. ² Kimmo Eriksson helped me excellently through the first part of my graduate studies, while Anders was ill. He also carefully read the article which has become Paper 5 of this thesis. ² After Kimmo had left for M¨alardalenUniversity, Dmitry Kozlov was kind enough to take his place. Apart from being ever helpful, he guided me through writing my first paper, which forms Paper 2 of this thesis. ² Through a large portion of my time as a graduate student, Niklas Eriksen was the only other combinatorics student at the department. We had many interesting discussions, and we wrote two articles together. They form Papers 5 and 6 of this thesis. Niklas also filled the important position as organizer of the department’s weekly floorball matches. ² Niklas and I tried for some time to prove a crucial result (Theorem 5.5.1) without success. We then asked Richard Stanley of MIT for advice. He replied with the solution virtually before we had even asked the question. v vi Preface ² The entire staff of the department has provided a helpful and friendly envi- ronment suitable to work in. ² Finally, the people who have helped me the most (although the majority couldn’t care less about mathematics!), and without whom I would never have had the opportunity nor the energy to work on this thesis, are my ever- supportive family and friends. You know who you are. Contents Introduction 1 0.1 Combinatorial complexes . 1 0.1.1 Subspace arrangements and shellability . 2 0.1.2 Boolean cell complexes . 3 0.1.3 Graph complexes . 4 0.2 Bruhat intervals . 4 0.3 Expected reflection distances . 5 0.4 Overview of the papers . 6 I Combinatorial complexes 9 Paper 1. Polygraph arrangements. 11 1.1 Introduction . 11 1.2 Tools for investigation of subspace arrangements . 12 1.2.1 The Goresky-MacPherson formula . 12 1.2.2 Lexicographic shellings . 13 1.3 Polygraph arrangements . 13 1.4 An EL-labelling of L(Q; P ) ...................... 14 1.5 A Dowling generalization . 18 1.5.1 Dowling lattices . 18 1.5.2 Dowling analogues of L(Q; P ) . 19 Paper 2. Quotient complexes and lexicographic shellability. 25 2.1 Introduction . 25 2.2 Basic definitions and notation . 26 2.2.1 Shelling cell complexes of boolean type . 26 2.2.2 Discrete Morse theory . 27 2.2.3 Quotient complexes . 27 2.3 Lexicographic shellability of balanced boolean cell complexes . 28 2.4 The objects of study . 31 2.5 Describing ∆n;k;h using trees . 32 vii viii Contents 2.5.1 The trees . 33 2.5.2 The deletion operator . 34 2.5.3 A description of ∆n;k;h ..................... 34 2.6 The homotopy type of ∆n;k;h ..................... 34 2.6.1 The Dn;k case . 35 2.6.2 The Bn;k;h case . 36 Paper 3. Directed subgraph complexes. 41 3.1 Introduction . 41 3.2 Basic topological combinatorics . 42 3.3 Acyclic subgraphs . 43 3.4 Not strongly connected subgraphs . 44 II Bruhat intervals 49 Paper 4. Bruhat intervals of length 4 in Weyl groups. 51 4.1 Introduction . 51 4.2 Foundations . 52 4.3 Classifying the intervals of length 4 . 55 4.3.1 Crown indices . 55 4.3.2 Cell decompositions of the 2-sphere . 56 4.3.3 The classification . 58 4.4 A completely computerized approach . 65 Complement. Bruhat intervals of length 5 in symmetric groups 67 4.5 More computer calculations . 67 III Expected reflection distances 73 Paper 5. Estimating the expected reversal distance after t random reversals. 75 5.1 Introduction . 75 5.2 Preliminaries . 76 5.2.1 The cycle graph . 77 5.3 The analogy . 79 5.4 The Markov chain approach . 80 5.5 Decomposing the length function . 83 5.5.1 Proof of Theorem 5.4.3 . 85 5.5.2 Computing the variance . 85 5.6 Experimental results . 86 5.6.1 Predicting the true reversal distance . 86 5.7 Conclusions . 87 Contents ix Paper 6. Expected reflection distance in G(r; 1; n) after t random reflections. 89 6.1 Introduction . 89 6.2 The group G(r; 1; n) and its reflection graph . 90 6.3 Irreducible characters and “symmetric functions” . 91 6.4 The Markov chain . 92 6.5 The eigenvalues of the adjacency matrices . 95 6.6 Decomposing the distance function . 97 6.7 Proof of Theorem 6.4.2 . 101 x Introduction This thesis consists of six different papers. There are ways to tie all of them together. For instance, the symmetric group and its relatives play important roles in all of the papers. However, a less artificial and more natural point of view is probably to divide the papers into three different categories: ² Combinatorial complexes. This category includes the first three papers, which all deal with topological properties of combinatorially defined cell com- plexes. ² Bruhat intervals. Paper 4 and a complement to it are the only members of this category. They address the problem of characterizing those posets that occur as intervals in Bruhat orders of Weyl groups. ² Expected reflection distances. The remainder of the thesis, Papers 5 and 6, concern expected distances after random walks on Cayley graphs of reflection groups. Any of the above categories is mostly independent of the others. Below, we thus give an independent introduction to each. We will assume some knowledge of combinatorics, topology, Coxeter groups and complex reflection groups. We refer to [69], [62], [50] and [67], respectively, for more on these subjects. 0.1 Combinatorial complexes Among the various classes of cell complexes, the most fundamental and important is probably the class of simplicial complexes. Viewing a simplicial complex as an abstract complex, i.e. as a collection of sets which is closed under taking subsets, it is clear that simplicial complexes are indeed very combinatorial by nature. On the other hand, viewing a complex as its geometric realization, it has a distinct topo- logical flavour. Exploiting this connection between topology and combinatorics, the following question has often turned out to be highly interesting: “What are the topological properties of this combinatorially defined simplicial complex?”.
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