Computational and combinatorial aspects of Coxeter groups [PhD thesis]

104 Pages·2008·0.89 MB·English

by  Henrik Eriksson

COMPUTATIONAL AND
COMBINATORIAL ASPECTS
OF COXETER GROUPS
HENRIKERIKSSON
NADA,KTH
S-100 44 STOCKHOLM, SWEDEN
[email protected]
DISSERTATION,SEPTEMBER 1994
i
COMPUTATIONAL AND COMBINATORIAL ASPECTS OF COXETER GROUPS HENRIK ERIKSSON NADA, KTH S-100 44 STOCK
Akademisk avhandling för teknisk doktorsexamen vid
Kungl. Tekniska Högskolan, Stockholm
September 1994
(cid:1)c Henrik Eriksson 1994
ISBN 91-7170-896-0
KTH Tryck & Kopiering, Stockholm
mcmxciv
COMPUTATIONAL ANDCOMBINATORIAL ASPECTSOF COXETER GROUPS HENRIK ERIKSSON NADA, KTHS-100 44 STO
Abstract
This thesis deals with combinatorics in connection with Coxeter groups, finitely generated but
not necessarily finite. The representation theory of groups as nonsingular matrices over a field
is of immense theoretical importance, but also basic for computational group theory, where the
group elements are data structures in a computer. Matrices are unnecessarily large structures,
and part of this thesis is concerned with small and efficient representations of a large class of
Coxeter groups (including most Coxeter groups that anyone ever payed any attention to.)
The main contents of the thesis can be summarized as follows.
• We prove that for all Coxeter graphs constructed from an n-path of unlabelled edges by
adding a new labelled edge and a new vertex (sometimes two new edges and vertices),
there is a permutational representation of the corresponding group. Group elements
correspondtointegern-sequencesandthenodesinthepathgeneratealln!permutations.
The extra node has a more complicated action, adding a certain quantity to some of the
numbers.
• We derive formulas and interpretations of many important group concepts in the permu-
tational representations, such as length and Bruhat order.
• Our computer implementation of the permutational representations have been used suc-
cessfully for various investigations. We obtain interesting variants of Peter Ungar’s the-
orem about the minimal number of partial flip moves required to accomplish a total
reversal of a permutation. The theorems give sharp bounds for the complexity of stack-
based sorting - a well-known problem in computer science.
• Is the language of reduced Coxeter group expressions regular? After much research,
giving only partial answers, the question was recently settled in the affirmative. We are
now able to construct simple finite automata for all Coxeter groups.
• The original pebbling game startswith onepebbleonthelower leftsquareofapotentially
infinite chessboard. Any pebble with empty squares above and to the right may be split
into two and moved to these two squares. We are able to enumerate and characterize
legal positions, not only in the plane, but in higher dimension and on general posets. In
particular, if played on a Coxeter group, the legal positions give unexpected connections
between the two poset structures, weak order and Bruhat order.
• The chip-firing game has been a rich source of combinatorical results in recent years. We
find a surprising connection between minimal recurrent games and conjugacy classes of
Coxeter elements.
Keywords: permutational representation, game, pebbling, polygon property, allowable se-
quence, essential chain, reachability, Coxeter groups, weak order, Bruhat order, polygon poset,
chip-firing, Coxeter element, conjugacy class
iiCOMPUTATIONAL AND COMBINATORIAL ASPECTS OF COXETER GROUPS HENRIK ERIKSSON NADA, KTH S-100 44 STOCK
Sammanfattning
Ämnet för denna avhandling är kombinatorik i samband med coxetergrupper, ändligt gener-
erademenintenödvändigtvisändliga. Teorinförgrupprepresentation medickesinguläramatriser
över en kropp är av högsta teoretiska betydelse, men också grundläggande för datorräkningar
med grupper, där gruppelementen är datastrukturer i datorns minne. Matriser är här onödigt
utrymmeskrävandeochendelavavhandlingenbehandlarmindreocheffektivarerepresentationer
av en stor klass av coxetergrupper (de flesta grupper som någon intresserat sig för torde ingå).
Avhandlingen har i stora drag följande innehåll.
• Varjecoxetergrafsommankanbildaavenn-stigavomärktakantergenomattläggatillen
ny märkt kant (ibland två nya kanter) och en ny nod har en permutationsrepresentation.
Gruppelementen motsvaras av n-vektorer av heltal och stigens noder genererar alla n!
permutationer av komponenterna. Den extra noden har mer komplicerad verkan och
adderar en viss storhet till vissa av talen.
• Vi härleder ett otal formler och tolkningar av viktiga gruppbegrepp, såsom längd och
bruhatordningen.
• Vår datorimplementation av permutationsrepresentationerna har framgångsrikt använts
för olika undersökningar. Vi har erhållit intressanta varianter av Peter Ungars sats om
minimala antalet partiella omvändningar som behövs för att vända hela permutationen
bak och fram. Manfårdärmed skarpagränser förkomplexiteten hos sortering med stack,
ett känt problem inom datalogin.
• Utgör de reducerade uttrycken i en coxetergrupp ett reguljärt språk? Många har sysslat
medfråganunderdesenasteårenochmångapartiellaresultatharernåtts. Vikanhärför
godtycklig coxetergrupp beskrivaenändlig automat som känner igen reducerade uttryck.
• Detursprungligapebblingspelet börjarmedenstenpånedrevänstrarutaniettpotentiellt
oändligt schackbräde. Om rutorna norr och öster om en sten är tomma får stenen klyvas
och flyttas till dessas rutor. Vi lyckas räkna och karakterisera legala ställningar, inte
bara i planet utan i högre dimension och på allmänna pomängder. Om spelet utförs
på en coxetergrupp ger legala ställningar oväntade samband mellan svaga ordningen och
bruhatordningen.
• Chipsspelet har på senare år varit en rik källa till kombinatoriska resultat. Vi finner
ett överraskande samband mellan minimala rekurrenta spel och konjugatklasser för cox-
eterelement.
Nyckelord: permutationsrepresentation, spel, polygonegenskapen, coxetergrupper, nåbarhet,
svaga ordningen, bruhatordningen, polygonpomängd, chipsspel, konjugatklass
COMPUTATIONAL ANDCOMBINATORIAL ASPECTSOF COXETER GROUPS HENRIK ERIKSSON NADA, KTHS-100 44 STO
Contents
Preface vi
Part 1. Combinatorial representations of Coxeter groups 1
1. Computational group theory 2
1.1. Presentation and representation of groups 2
1.2. Coset enumeration and other algoritms 3
1.3. Computability and complexity 4
2. Coxeter groups 4
2.1. Reflections and symmetries 5
2.2. The standard geometric representation 5
2.3. Root systems and hyperplane arrangements 6
3. The numbers game 7
4. Length, order and permutations 8
5. Introduction and outline of results 11
6. Representations of Dn, E6, E7 and E8 and friends 12
6.1. Analysis of permutational representations 12
6.2. Representations of Dn,E6,E7,E8 15
6.3. Interpretation of Dn, E6, E7, E8 as subgroups of Sk 19
6.4. The length function for Dn, E6, E7, E8 20
6.5. Geometric interpretation of the formulas 21
6.6. Representations of E˜7 and E˜8 22
7. Representations of D˜n, E˜6 and friends 25
7.1. Analysis of possible permutational representations 25
7.2. Representations of D˜n and E˜6 26
8. Representations of Bn, B˜n, C˜n and friends 26
8.1. Analysis of possible representations 27
8.2. Representations of Bn,H3,H4 and I2(p) 28
8.3. A special study of G˜2 30
8.4. Two extra edges, B˜n, C˜n and D˜n 31
9. Representations of F4, F˜4, A˜n and friends 32
9.1. Possible representations of F–type groups 33
9.2. Representations of F4 and F˜4 35
9.3. The length function for F4 and F˜4. 36
9.4. Other edge labels or extra edges 37
9.5. A special study of A˜n−1 38
10. The affine groups as infinite permutations 39
11. The Cox Box for building permutation representations 44
11.1. The Cox Box blocks 45
11.2. Examples of Cox Box constructions 45
12. The Bruhat order for permutation representations 48
13. Introduction 50
14. Language automata 51
15. A finite automaton for reduced expressions 53
15.1. An automaton for A˜2 54
15.2. Automata for affine groups 54
15.3. Automata for nonaffine infinite groups 55
16. Normal forms for reduced words 58
16.1. Orderings of words in Coxeter groups 58
16.2. The normal form for finite Coxeter groups 59
ivCOMPUTATIONAL AND COMBINATORIAL ASPECTS OF COXETER GROUPS HENRIK ERIKSSON NADA, KTH S-100 44 STOCK
16.3. The normal form for infinite Coxeter groups 60
Part 2. Games and Coxeter groups 62
17. The pebbling game 63
18. Pebbling in Zn 63
19. Pebbling a poset 69
20. Pebbling a Coxeter group 72
21. Conclusions 77
22. Introduction 79
23. Edge orientations and chip-firing 79
24. Coxeter elements 80
25. The flip game and its applications 83
25.1. Slopes of n points in plane geometry 83
25.2. Stack sorting in computer science 84
25.3. Flip mutations in genetics 85
26. Analysis of the flip game 86
26.1. Flip sequences and the Möbius function 87
26.2. Ungarian theorems for finite Coxeter groups 87
Bibliography 91
References 91
Index 93
Index 94
COMPUTATIONAL ANDCOMBINATORIAL ASPECTSOF COXETER GROUPS HENRIK ERIKSSON NADA, KTHS-100 44 STO
*Preface
I am a victim of the enigmatic ADE-phenomenon, so named by the famous Russian mathe-
matician V.I.Arnol’d [2]. Even a casual reader of this thesis must notice the ubiquitous pictures
of small circles connected with straight lines. The most common types are named A, D and E,
and the challenge is to explain why these little pictures pop up all over the place, not only in
combinatorics. For some years, I tried to ignore them, but as you can see, they prevailed.
The ADE-diagrams symbolise Coxeter groups, named after Harold Scott Macdonald Coxeter,
one of my all-time favourite mathematicians. His name is always associated with beautiful math
and math that is fun. I thoroughly enjoyed his work in recreational and aesthetic mathematics
(in particular [33] and [31]) long before I could imagine that I would one day be writing a thesis
with his name in the title. Being an embarrassingly over-age Ph.D. candidate, I find it especially
encouraging that professor Coxeter at ninety is still active and top-ranking in his field.
Mathematical research is, in principle, a very lonely occupation. However, to be successful
you need fellow–creatures, for advice, criticism and encouragement. I would like to express my
gratitude to all those who have helped me in any of these ways.
I count myself fortunate to have had Anders Björner as my supervisor. He taught me every-
thing I know in combinatorics and he introduced me to the world of international mathematics.
I also want to thank my younger son Kimmo Eriksson for countless discussions of mathematics
(mainly atnight) eversincehis childhood, andmy olderson Viggo Kannforsolvingallmy LATEX
problems.
Henrik Eriksson
Stockholm
October 8, 2004.

COMPUTATIONAL ANDCOMBINATORIAL ASPECTSOF COXETER GROUPS HENRIK ERIKSSON NADA, KTHS-100 44 STO
Part 1. Combinatorial representations of Coxeter groups
2COMPUTATIONAL AND COMBINATORIAL ASPECTS OF COXETER GROUPS HENRIK ERIKSSON NADA, KTH S-100 44 STOCK
Computational Coxeter group theory
1. Computational group theory
A modern encyclopedia (Nationalencyklopedin, 1992) defines computer as an automatic ma-
chine for numeric calculation and symbolic manipulation. It is a strange fact that in computa-
tional group theory, computers are used for symbolic calculation and the methods that we are
going to introduce could well be called numeric manipulation!
The first mathematician to think of using computers for group-theoretical calculations was
Alan Turing. During World War II, Turing and other British mathematicians were engaged in
cryptographic activities, mainly in breaking the German secret codes. At the end of the war,
Turingstartedtoworkonthedesignofanautomaticcomputingengine. Inadocumentcirculated
at the end of 1945, Turing suggested several tasks that his engine would be able to take on. One
of these was the enumeration of all groups of order 720.
However, thefundamentalalgorithms ofcomputational grouptheorypredatetheadventofthe
computer by almost a century. In fact, calculation with groups was an established activity from
the middle of thenineteenth century, although the definition of anabstract group did notappear
until1893(Weber[81]). Cayleynevergavethedefinition, buthecertainlyknewsomethingabout
groups. The famous theorems of Sylow date from 1872 and were originally results about certain
sets of permutations. Paradoxically stated, computational group theory is the mother of group
theory. The father? There is always some doubt, but a candidate is the surge of axiomatising,
abstraction and striving for purity and clarity, so evident in Hilbert’s talk in Paris at the turn of
the century. This emphasis on structure with some disregard for substance had its climax in the
nineteen-sixties, when many areas of mathematics were rewritten in the categorical language of
morphisms.
Since then, a remarkable change of attitude is evident. Much more attention is now paid
to concrete representations of mathematical objects. Chronologically, this change of attitude
coincides with the entry of computers into the academic world. It could not be termed far-
fetched to suggest a causal connection.
Conjecture 1. The current upswing in interest for concrete representations of mathematical
abstractions is a mental side-effect, when the computers go marching in.
(It should be noted that the author in an earlier paper, [38], 1985, has denied any impact
whatsoever of computers on university mathematics.)
1.1. Presentation and representation of groups. Atypicalproblemincomputationalgroup
theory begins something like this: “Given a group G, determine ...”. This problem may be easy
or difficult depending on how G is specified. The recent survey by Sims [72] states that there are
really only three methods in common use. We add one here, the game method, that we believe
to be of increasing importance.
(1) The simplest way is to start with a list of concrete objects, that we already know how to
multiply, anddefine Gasthegroupgeneratedby theseelements. Themostcommonsuch
objects are permutations and square matrices. Note that the matrix elements can belong
to any ring, for examples be polynomials in x and y with integer coefficients modulo 15.
One may be led to think that all problems about representation are solved with this
concrete specification, but that is not always true. The concrete objects may be unnec-
essarily large, making the representation unefficient. In some cases, Coxeter groups are
naturallydefinedbyn×n-matricesandtheseareoftenlargeobjects. Acommonsituation
is the specification of n such reflection matrices and, most often, these reflections will
generate super-exponentially many other matrices, say ∼ nn. Even though n is seldom
greater than 10, memory space will soon be exhausted. That is one of the reasons why
we are going to derive permutational representations, where each matrix can be replaced
by an n-vector, resulting in space savings by a factor n.