Table Of ContentFoundations for Undergraduate Research in Mathematics
Aaron Wootton
Valerie Peterson
Christopher Lee
Editors
A Primer for
Undergraduate
Research
From Groups and Tiles to Frames
and Vaccines
Foundations for Undergraduate Research
in Mathematics
Serieseditor
AaronWootton
DepartmentofMathematics,UniversityofPortland,Portland,USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/15561
Aaron Wootton • Valerie Peterson
Christopher Lee
Editors
A Primer for
Undergraduate Research
From Groups and Tiles to Frames
and Vaccines
Editors
AaronWootton ValeriePeterson
DepartmentofMathematics DepartmentofMathematics
UniversityofPortland UniversityofPortland
Portland,OR,USA Portland,OR,USA
ChristopherLee
DepartmentofMathematics
UniversityofPortland
Portland,OR,USA
ISSN2520-1212 ISSN2520-1220 (electronic)
FoundationsforUndergraduateResearchinMathematics
ISBN978-3-319-66064-6 ISBN978-3-319-66065-3 (eBook)
DOI10.1007/978-3-319-66065-3
LibraryofCongressControlNumber:2017959749
MathematicsSubjectClassification:00A05,00A07,00A08,00B10
©SpringerInternationalPublishingAG2017
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisbookispublishedunderthetradenameBirkhäuser,www.birkhauser-science.com
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Contents
CoxeterGroupsandtheDavisComplex ...................................... 1
TimothyA.Schroeder
ATaleofTwoSymmetries:Embeddable andNon-embeddable
GroupActionsonSurfaces...................................................... 35
ValeriePetersonandAaronWootton
TileInvariantsforTacklingTilingQuestions................................. 61
MichaelP.Hitchman
ForbiddenMinors:FindingtheFiniteFew ................................... 85
ThomasW.Mattman
IntroductiontoCompetitiveGraphColoring................................. 99
C.Dunn,V.Larsen,andJ.F.Nordstrom
Matroids........................................................................... 127
ErinMcNicholas,NancyAnnNeudauer,andColinStarr
FiniteFrameTheory............................................................. 145
SomantikaDattaandJesseOldroyd
MathematicalDecision-MakingwithLinearandConvexProgramming.. 171
JakobKotas
ComputingWeightMultiplicities............................................... 193
PamelaE.Harris
VaccinationStrategiesforSmallWorlds ...................................... 223
WinfriedJustandHannahCallenderHighlander
SteadyandStable:NumericalInvestigationsofNonlinearPartial
DifferentialEquations........................................................... 265
R.CorbanHarwood
Index............................................................................... 305
vii
Coxeter Groups and the Davis Complex
TimothyA.Schroeder
SuggestedPrerequisites. Grouptheory,Graphtheory,Combinatorialtopology.
1 Introduction
Thischapterpresentstopicsinanareaofmathematicsattheintersectionofgeome-
try,topology,andalgebracalledGeometricGroupTheory.Itislikelythatstudents
havebeenexposedtogeometryandabstractalgebratopicsasundergraduates.Some
readingthismayhavealsobeenintroducedtotopology,aswell.Inthischapter,we
willbeusingtermsandconceptsfromeachoftheseareas,thepointbeingtodevelop
aworkingknowledgeofthesetermsandconceptsthatallowsustoprogresstoward
ourgoal:Areflectiongroupactingonatopologicalspace.Wewillnotspendmuch
timeontopologicaldetails(studentsarecertainlyencouragedtopursuethatsubject
formally).Geometricdetails,referencedthroughout,butspecificallyinSection3.2,
are suggested as a project in Section 6. That leaves algebra. Perhaps the best, and
mostfamiliarplacetobegin.WerecallthedefinitionofagroupG:
Definition1. A group G is a set G, together with a closed binary operation,
denoted·,suchthatthefollowinghold:
(cid:129) Associativity:Foralla,b,c∈G,(a·b)·c=a·(b·c).
(cid:129) Identity:ThereisanelementeinGsuchthatforallx∈G,e·x=x·e=x.
(cid:129) Inverse:Foreverya∈G,thereisanelementa(cid:2) ∈Gsuchthata·a(cid:2) =a(cid:2)·a=e.
Suchanelementa(cid:2)isunique,andisdenoteda−1.
T.A.Schroeder((cid:2))
MurrayStateUniversity,Murray,KY42071,USA
e-mail:[email protected]
©SpringerInternationalPublishingAG2017 1
A.Woottonetal.(eds.),APrimerforUndergraduateResearch,Foundationsfor
UndergraduateResearchinMathematics,DOI10.1007/978-3-319-66065-3_1
2 T.A.Schroeder
Students in an abstract algebra course begin with this definition, and likely
embark on a journey through group theory: element orders, subgroups, cosets,
homomorphisms,isomorphisms,cyclicgroups,generators,etc.Eventually,students
maybeintroducedtogroupactions,alongwiththeirstabilizersandorbits.Allthe
while,mostlikely,usingthefavoritedihedralgroupsasprototypes.
As useful and correct and edifying as that exposure is, it may strike students
as somewhat sterile. The groups are axiomatically presented, they are thought of
abstractly,andpictures(asidefromregularpolygonsortheoccasionalCayleygraph
ofafinitegroup)maybefewandfarbetween.Itisourintentionthatthegroupswe
presentinthiscontextwillhaveadecidedlyconstructive,andevengeometric,flavor
tothem.
In this chapter, we will study finitely presented groups, specifically Coxeter
groups, and we will present a constructive view of a topological space on which
these Coxeter groups act. We begin with an overview of group presentations and
graph theory, then define Coxeter groups and the associated spaces. The reader
should note that many technical terms are italicized, and can be referenced in the
includedcitationsorelsewhere.
2 GroupPresentationsandGraphs
Let A be a set and define the set A−1 = {a−1 | a ∈ A}. Define W to be the
A
collection of all finite length words in A∪A−1. A “word” being a finite string of
elementsfromthesetA∪A−1.(WearethinkingofthesetAasour“alphabet.”)For
example,ifA={a,b},thena,ab,ba,aba−1,bb−1abareallelementsofthesetof
wordsW .Perhapsyousensethatthislistisredundant.Thatis,youmayalreadybe
A
thinking that two of the words are actually equivalent. You’re right, of course, for
asthenotationsuggests,elementsofthesetA−1 aretoplaytheroleofinverses.To
makethisclear,let’sgetalittlemoreformal.
First,tothesetW ,weincludethewordcomprisedofnoelements,theso-called
A
emptyword,denotedbythesymbol1.
Now,tohaveagroup,wemusthaveanassociativeoperation,usuallydescribed
and understood as some sort of multiplication. So, on the set W , we define
A
multiplication by concatenation. (In other words, “put next to each other.”) In the
example above, to multiply ba and aba−1, we have ba · aba−1 = baaba−1 =
ba2ba−1.Youshouldverifythat,ingeneral,concatenationisinfactassociative.
Next, we say w ∈ W is obtained from w ∈ W by an elementary reduction
2 A 1 A
(or expansion) if w is obtained from w by deleting (or inserting) a sub-word of
2 1
the form aa−1 or a−1a, for some element a ∈ A. We say that two words w and
w(cid:2) are equivalent if we may pass from w to w(cid:2) by a finite sequence of elementary
reductions or expansions (both are allowed), and we write w ∼ w(cid:2). (As you may
have guessed, this means that in our list above, ab ∼ bb−1ab.) The relation ∼
definesanequivalencerelationonthesetW .(SeeExercise1.)
A
CoxeterGroupsandtheDavisComplex 3
Finally,letF bethecollectionofequivalenceclassesof∼,whereforw ∈ W ,
A A
[w]denotestheequivalenceclasscontainingw.ConcatenationonW inducesawell-
A
definedoperationonF :Foru,v ∈W ,[u]·[v]=[uv].(SeeExercise2.)Infact,F
A A A
isagroupwithidentitytheequivalenceclasscontaining1.Itiscalledthefreegroup
onA.
2.1 GroupPresentations
It could be the case that within a set of words, or within a group, there could be
manyways,besidesusingelementaryreductionsorexpansions,torepresentagiven
element.Tohandlethismorecomplicatedsituation,weaddwhatarecalledrelators
toourgroup,andstudypresentationsofgroups.
LetAbeasetandconsideragainF ,thefreegrouponA.LetRbeasetofwords
A
inthealphabetA∪A−1,anddefineN(R)tobethesmallestnormalsubgroupofF
A
containing the equivalence classes of the elements of R. (This normal subgroup is
formedbytakingthecollectionofallfiniteproductsofconjugatesofelementsofR
andtheirinversesinF .)Wehavethefollowingdefinition.
A
Definition2. LetA,RandN(R)beasabove.Thegroupdefinedbythepresentation
(cid:5)A|R(cid:6)isthequotientgroupF /N(R).
A
That is, a group defined by a presentation is the quotient of a free group by the
normalsubgroupgeneratedbythewordsinsomesetR.Weequateagroupwithits
presentation, writing F /N(R) = (cid:5)A|R(cid:6), but note that it is the case that a given
A
groupcanhavemultiplepresentations.
If A is a finite set (for it could be the case that A is an infinite set), we say the
correspondinggroupisfinitelygenerated.IfRisalsofinite,wesaythatthegroupis
finitelypresented.Forreferenceongrouppresentations,see [6]or [10].
In practice we often take a slightly more constructive approach to defining
and working with the elements of a finitely presented group. We highlight this
perspectivenext.
2.1.1 AConstructiveApproach
Resettingthetable,considerthesetW ofwordsinthealphabetA∪A−1,whereA
A
issomefiniteset,andletRbeafinitesetofwordscontainedinW .Theelementsof
A
R,againcalled“relators,”willservetoidentifycertainwordsinW ,besidesthose
A
identified in the free group F . Indeed, we say a word w in W is obtained from
A 2 A
w ∈ W by a simple R-reduction (or R-expansion) if w is obtained from w by
1 A 2 1
deleting(orinserting)asub-wordr,wherer ∈R.Wethensaytwowordswandw(cid:2)
are R-equivalent, and write w ∼ w(cid:2), if there exist a finite sequence of simple R-
R
reductions,simpleR-expansions,elementaryreductions,andelementaryexpansions
leading from w to w(cid:2). As before, concatenation induces an operation on the set of
equivalence classes of R-equivalent words of W , and we have the structure of a
A
groupGwithpresentation(cid:5)A|R(cid:6).Weagainequatethegroupwithitspresentation
andwriteG=(cid:5)A|R(cid:6).
4 T.A.Schroeder
Now, for an element w ∈ W , we think of w as representing its entire
A
equivalence class of R-equivalent words and drop the equivalence class notation
(or the associated coset), understanding that other words may be equivalent to w.
Thatis,wethinkofthewordsthemselvesaselementsofthegroup;butinthatset,
thereismuchredundancy.Inparticular,wewritew = w(cid:2) asgroupelementswhen
w∼ w(cid:2)aswords(orwhenwandw(cid:2)areinthesamecosetofN(R)).Multiplication
R
isstillrepresentedbyconcatenation,andtheidentityelementstillrepresentedby1.
ThismeansthatthegeneratingsetAisconsideredasubsetofthegroupitself.Also,
R-equivalencemeansthatweareabletoinsertordeleteeachr ∈Rintoanypartofa
wordwwithoutchangingthegroupelement.Thisamountstoequatingeachrelator
r with the identity element 1, a perspective often indicated in the presentation as
each r ∈ R will often be equated to 1 in the right hand side of the presentation.
Finally, note that if A = {a ,a ,...,a } and R = {r ,r ,...,r }, then we may
1 2 n 1 2 k
write(cid:5)a ,a ,...,a |r ,r ,...,r (cid:6)todenote(cid:5)A|R(cid:6).
1 2 n 1 2 k
2.2 SomeBasicGraphTheory
In order to fully explore group presentations, and the structures of the resulting
groups,itdoesuswelltoreview(orintroduce)somebasicgraphtheory.
LetV beaset,andletE beacollectionoftwoelementsubsetsofV,wherean
individual set can be repeated in the collection, and an individual element can be
repeatedtoformatwo-elementsubset.SuchsetsdefineagraphΓ = (V,E)where
theelementsofV aretheverticesofthegraphandtheelementsofE theedges.A
set{v,w} ∈ Eindicatesanedgebetweenverticesv andw,and{v,v} ∈ Edefines
anedgeinΓ fromvtoitself,i.e.aloopinΓ.IfasubsetinEhasmultiplicityn,then
we include n edges between the associated vertices in Γ. Here, Γ is said to have
multi-edges.
WedefineadirectedgraphbytakingE ⊆ V ×V wheretheorderedpair(v,w)
indicatesadirectededgebetweentheverticesv andw.Weindicatethispictorially
by placing an arrow on the associated edge. Of course, this sort of structure can
indicate an orientation to the edges in the graph, where traversing the associated
edgeindifferentdirectionshasdifferentimplications.Forourpurposes,itispossible
to have a graph with both directed edges and undirected edges. In this case, the
edgesetEwilldenoteundirectededgesbytwoelementsets,anddirectededgesby
orderedpairs.
AgraphordirectedgraphΓ =(V,E)issaidtobelabeled,orweighted,ifthere
isafunctionfromthesetofedgesEtoasetoflabels.
AgraphΓ issaidtobefiniteifbothV andEarefinitesets.Γ issaidtobesimple
ifΓ includesnoloopsnormulti-edges.
Two graphs Γ − (V ,E ) and Γ = (V ,E ) are isomorphic if there exists
1 1 1 2 2 2
a bijection f : V → V such that if {u,v} ∈ E , then {f(u),f(v)} ∈ E . A
1 2 1 2
similardefinitionexistsfordirectededges,thatisif(u,v)∈E ,then(f(u),f(v))∈
1
E . If the bijection f maps V to itself, and E = E , then the map f defines an
2 1 1 2
automorphismofthegraphΓ (V ,E ).
1 1 1
