Coincidences and colorings of lattices and Z-modules [Elektronische Ressource] / Manuel Joseph C. Loquias. Fakultät für Mathematik. Forschungsschwerpunkt Mathematisierung (FSPM). Sonderforschungsbereich: Spektrale Strukturen und Topologische Methoden in der Mathematik (DFG SFB 701)

Publié par

ZCoincidences and Colorings ofLattices and -modulesManuel Joseph C. LoquiasZCoincidences and Colorings ofLattices and -modulesDissertation zur Erlangung des Doktorgradesan derFakult¨at fu¨r MathematikUniversit¨at Bielefeldvorgelegt vonManuel Joseph C. LoquiasFakult¨at fu¨r MathematikUniversit¨at BielefeldNovember 2010ii1. Berichterstatter: Prof. Dr. Michael Baake und Dr. Peter Zeiner2. Berichterstatter: Prof. Dr. Uwe GrimmDatum der mu¨ndlichen Pru¨fung: 25. Januar 2011Pru¨fungsausschuss: Prof. Dr. M. Baake, Prof. Dr. W.-J. Beyn, Prof. Dr. H. Krause, Prof. Dr. M. Kaßmann,Dr. P. ZeinerGedruckt auf alterungsbest¨andigem Papier ISO 9706ZZZContentsAcknowledgements vIntroduction 1Background 1Outline of the thesis 2Chapter 1. Preliminaries 51.1. Coincidences of lattices and -modules 51.2. Solution of the coincidence problem for certain lattices and -modules 81.3. Lattice colorings 14Chapter 2. Coincidence indices of sublattices and colorings of lattices 152.1. Coincidence index with respect to a sublattice 152.2. Color coincidence 192.3. Further examples 232.4. Application to quasicrystals 28Chapter 3. Coincidences of shifted lattices 313.1. Affine coincidences 313.2. The coincidence problem for a shifted lattice 333.3. The setsH, AC(Γ), and OC(x+Γ) as groupoids 363.4. Coincidences of a shifted square lattice 363.5. Corresponding results for -modules 49Chapter 4. Coincidences of multilattices 534.1.
Publié le : samedi 1 janvier 2011
Lecture(s) : 19
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Source : NBN-RESOLVING.DE/URN:NBN:DE:HBZ:361-18127
Nombre de pages : 90
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Acknowledgements

Iwouldliketoexpressmydeepestgratitudetothefollowingpeopleandinstitu-
tionswhomadethiswork(directlyorindirectly)possible:
ToProf.Dr.MichaelBaake,forgivingmetheopportunitytoworkinhisresearch
group,andforhissupport,guidance,andencouragementduringmyentirestayin
Bielefeld.
ToDr.PeterZeiner,forsharinghisknowledgeandexpertise,forthehelpfuland
valuablediscussions,andforhisinputandthoroughnessinreviewingdraftsofthis
thesis.
Tomysecondreviewer,Prof.Dr.UweGrimm,andtotheothermembersofmy
doctoralcommittee;Prof.Dr.Wolf-Ju¨rgenBeyn,Prof.Dr.HenningKrause,and
Prof.Dr.MoritzKaßmann,fortakingtimeoutoftheirbusyschedulestoreadmy
.krowTotheGermanAcademicExchangeService(DAAD),theResearchCenterfor
MathematicalModeling(FSPM)attheUniversityofBielefeld,andtheCollabora-
tiveResearchCenter(CRC)701:“SpectralStructuresandTopologicalMethodsin
Mathematics”,fornancialsupportofmydoctoralstudies.
TotheInstituteofMathematicsattheUniversityofthePhilippines(UP),Dili-
man,forgrantingmeleaveofabsencetopursuemydoctoralstudies.
Topresentandformermembersoftheresearchgroup“MathematikindenNatur-
wissenschaften”;MichaelBaake,ChristophRichard,FranzGa¨hler,PeterZeiner,
DirkFrettl¨oh,ChristianHuck,JohanNilsson,UweSchwerdtfeger,ManuelaHeuer,
SvenjaGlied,KaiMiguelMatzutt,NataschaNeuma¨rker,MarkusMoll,EnricoPaolo
Bugarin,ClaudiaLu¨tkeho¨lter,OliverSchmidt,NadineEisner,MagnusDu¨mke,and
ThomasBlomenkamp,forprovidingastimulatingandfriendlyworkatmosphere,and
forinterestingandfunnydiscussionsduringlunch.
TomyfamilyhereinGermany,theBayanihanPilipinoe.V.,inparticular,Kuya
Filemon,German,Larry,andAteBabes,Becca,Cris,Edna,Flor,Glecy,Grace,
Larife,Linda,Lisa,Lita,Lulu,Marissa,Nancy,Rosie,Yeny;aswellasAteBelinda,
Josie,Chit,Virgie,andJudith.
TomyfellowFilipinostudentshereinGermany;Andrew,Bugs,Gail,Gori,Karl,
Migs,andSanti,forthefriendship,dinners,andquiznights.
TotheInternationalOceattheUniversityofBielefeldformakingthelifeof
internationalstudentslikemealittlebiteasierandmorefun.
ToallacquaintancesandfriendsImetinmyGermanlanguageclassesinBielefeld
andinMarburg,andintheexcursiontoBrussels.

v

iv

ACKNOWLEDGEMENTS

Tomyfellowdoctoralstudents:Gino,Carmen,Romar,Tina,andBituin,for
visitingmeinBielefeld,thetravelsinEurope,andmanychatsessions.
Toallmyformermathematicsteachers,forsharingyourknowledgewithme.
Toallmyfriends,forallthethingsthatyouhavedoneformeandforbelieving
.eminTomyfamily,Daddy,Mommy,Joyce,andJandriel,foralwaysbeingthereforme.
Iloveyouall.
Finally,totheAlmightyGod,fortheblessingsyouhavegivenmealltheseyears
despitemyweaknessesandshortcomings.

4INTRODUCTION
obtainedbyshiftingthelattice,withsomevaluesdisappearingortheirmultiplicity
changed.
Similartotheapproachin[59,4],anextensiveanalysisofthecoincidencesofa
shiftedsquarelatticeinChapter3.4isachievedbyidentifyingthelatticewiththe
ringofGaussianintegers.Thecoincidenceproblemiscompletelysolvedwhenthe
shiftconsistsofanirrationalcomponent(Theorem3.26).Fortheremainingcase,
thatis,whentheshiftmaybewrittenasaquotientoftwoGaussianintegersthat
arerelativelyprime,onecomputesforthesetofcoincidencerotationsoftheshifted
squarelatticeusingsomedivisibilityconditioninvolvingthedenominatoroftheshift
(Lemma3.28).Inbothinstances,thesetofcoincidencerotationsofashiftedsquare
latticeformagroup.Anexampleisgivenwherethesetofcoincidenceisometriesofa
shiftedsquarelatticeisnotagroup.Thisshowsthatingeneral,thesetofcoincidence
isometriesofashiftedlatticedoesnotformagroup.Correspondingresultsandan
exampleforplanarmodulesconcludethechapter.
Thenalchapterofthisthesisisconcernedwiththecoincidencesofsetsofpoints
formedbytheunionofalatticewithanitenumberofshiftedcopiesofthelattice.
Suchsetsarereferredtoasmultilattices.Thisideashouldbeusefulinthecontextof
bicrystallography,andingeneral,tocrystalshavingmultipleatomsperprimitiveunit
cell[31,60].Thechapterstartswithananalysisofthecoincidencesofthesimplest
multilattice,thatis,oftheunionofalatticeandashiftedlattice.Thisleadstothe
solutionofthecoincidenceproblemforthediamondpackinggiveninTheorem4.6.
ThemainresultofthechapterisTheorem4.15,whichgivesthesolutionofthe
coincidenceproblemforgeneralmultilattices.Simplyput,the(linear)coincidence
isometriesofamultilatticeareexactlythecoincidenceisometriesofthelatticethat
generatesthemultilattice-onlytheresultingintersectionsandcorrespondingindices
mayvary.
ThemainprobleminChapter2isthenrevisited,wherethereverseconditionis
nowconsidered.Moreaccurately,ifthecoincidenceproblemforasublatticeofa
givenlatticehasalreadybeensolved,thenwhatcanbededucedabouttheCSLsand
correspondingcoincidenceindicesoftheoriginallattice?Thisquestionisresolvedin
Theorem4.17byregardingthelatticeasamultilatticeformedbytheunionofthe
sublatticewiththecosetsofthesublattice.Thisperspectiveestablishesaconnection
amongtherelationshipbetweenthecoincidenceindicesofalatticeandasublattice,
colorcoincidencesofthecoloringofthelatticedeterminedbythesublattice,and
coincidencesofshiftedlattices,whichisencapsulatedinPropositions4.19and4.20.
Thechapterendswithafulldescriptionofthecasewhenasublatticeisofprime
indexinalattice,andthesolutionofthecoincidenceproblemforcertainprimitive
andcenteredrectangularlattices.

162.COINCIDENCEINDICESOFSUBLATTICESANDCOLORINGSOFLATTICES
relationisneverempty.Infact,(c0,c0)2becausethelatticesR[2∩1(R1)]=
R2∩1(R)and2∩1(R)arecommensurate.
Remark2.1:
(i)Fromeachcj+22J,wecanalwayschooseasuitablecosetrepresentative
c˜jsuchthatc˜j+2=cj+2withc˜j21(R1).Similarly,forevery
ck+22K,thereexistscksatisfyingck+2=ck+2withck21(R).
(ii)Givenacosetc`+26=2thatisbothinJandK,itmayhappenthat
(c`+2)∩1(R1)∩1(R)=?.Insuchacase,itisnotpossibletond
acosetrepresentativec`0ofc`+2havingthepropertyc`0+2=c`+2
withc`021(R1)∩1(R).
Thefollowinglemmatellsusthat2(R)consistsofthosepointscoloredc0inthe
coloringof1(R)whosepreimagesunderRarealsopointscoloredc0inthecoloring
of1(R1).
Lemma2.2:Let2beasublatticeof1andR2OC(1).Thenthelattices
2∩1(R)andR2∩1(R)arecommensuratewithintersection2(R).Inparticular,
ifR2∩1(R)=2∩1(R)then2(R)=R2∩1(R)=2∩1(R).
Proof:Onehas[2∩1(R)]∩[R2∩1(R)]=(2∩R2)∩1(R)=2(R).
R111(R)1(R)
m1(R)m

2tsR2

2∩1(R)R2∩1(R)
vu2(R)2(R)
)R(2Figure2.Latticediagramofthelattices1,R1,2,R2,1(R),
2(R),2∩1(R),andR2∩1(R)(asgroups)andcorresponding
seicdinFigure2exhibitstherelationshipsamongthevariouslattices.Thefollowing
notationsshallbeusedtoindicatethecorrespondinglatticeindices(seeFigure2):
s:=[1(R):R2∩1(R)],u:=[2∩1(R):2(R)]
).52(t:=[1(R):2∩1(R)],v:=[R2∩1(R):2(R)]
Thenextlemmaisaconsequenceofthesecondisomorphismtheoremandwillbe
usedrepeatedlyintheproofofthesucceedingtheorem.

2.1.COINCIDENCEINDEXWITHRESPECTTOASUBLATTICE17
Lemma2.3:Let2and20besublatticesofthelattice1.Thenthefollowingholds.
(i)[20:2∩20]=|{`+221/2:(`+2)∩206=?}|,
(ii)[20:2∩20]divides[1:2],
(iii)If(`+2)∩206=?then(`+2)∩20isthecoset`_+(2∩20)of2∩20
in20whenever`_2(`+2)∩20.
Proof:Fromthesecondisomorphismtheorem,
20/(2∩20)=(2+20)/2={`+221/2:(`+2)∩206=?},
andthisproves(i).
Since2+20isasublatticeof1,(2+20)/2isasubgroupof1/2.Thus,
[2+20:2]=[20:2∩02]divides[1:2]byLagrange’sTheorem.
Thelaststatementisclearbyreplacing`+2by`_+2.
UsingLemma2.3,onecannowgiverestrictionsonthevaluesofs,t,u,andv,
aswellasinterpretationsofthesevaluesinrelationtothecoloringsof1(R1)and
1(R)determinedby2.Theseresultsareexplicitlystatedinthefollowingtheorem.
Theorem2.4:Considerthecoloringofalattice1determinedbyasublattice2of
1ofindexmwhereeachcosetcj+2isassignedthecolorcjfor0jm1,
withc0=0.IfR2OC(1),then
2(R)=tu1(R)=sv1(R),(2.6)
mmwheresandtarethenumberofcolorsinthecoloringof1(R1)and1(R),respec-
tively,determinedby2;uisthenumberofcolorscjwiththepropertythatsome
pointsof1(R1)coloredcjaremappedbyRtopointscoloredc0inthecoloringof
1(R);andvisthenumberofcolorsinthecoloringof1(R)thatisintersectedby
theimagesunderRofthosepointsof1(R1)coloredc0.Moreover,s|m,t|m,
u|s,andv|t.
Proof:ComparingindicesinFigure2givestheformulafor2(R)intermsof1(R)
in(2.6).
Takethesublattices2and1(R)of1.ApplyingLemma2.3,onereadilyobtains
thatt=|K|=|CR|(see(2.1)and(2.3))andt|m.Correspondingstatementsfors
aresimilarlyprovedbylookingatthesublatticesR2and1(R)ofR1.Lemma2.3
alsoimpliesthatforallcj+22Jandck+22K,
R[(cj+2)∩1(R1)]=Rc˜j+[R2∩1(R)]and
).72((ck+2)∩1(R)=ck+[2∩1(R)],
forsomec˜j2(cj+2)∩1(R1)andck2(ck+2)∩1(R)(seeRemark2.1).
Tocompletetheproof,considerthefollowingsets:
D:={Rc˜j+[R2∩1(R)]:cj+22Jwith(Rc˜j+[R2∩1(R)])∩[2∩1(R)]6=?}
E:=ck+[2∩1(R)]:ck+22Kwithck+[2∩1(R)]∩[R2∩1(R)]6=?.

)8.2(

202.COINCIDENCEINDICESOFSUBLATTICESANDCOLORINGSOFLATTICES

(a)Coloringof1(R1)(whitedotscorre-
spondtopointsof1\1(R1))

(b)Coloringof1(R)

(c)2(R)(=blackdots)asasublatticeof1.
Figure4.Coloringsof1(R1)and1(R)determinedbythesublat-
tice2ofFigure3,whereRisthecounterclockwiserotationaboutthe
originbytan1(43)37.TheCSL2(R)ofindex10in2isobtained
bytakingallblackpointsof1(R)whosepreimagesunderRarealso
coloredblack.
Proof:LetRbeacolorcoincidenceofthecoloringof1.Thecolorc0appearsin
thecoloringof1(R1)andRsendscolorc0toexactlyonecolorckinthecoloring
of1(R).Hence,R[2∩1(R1)]=(ck+2)∩1(R).Since022∩1(R1)and
R(0)=0,02(ck+2)∩1(R)whichimpliesthatck=c0.Thus,Rxescolorc0.
Intheotherdirection,supposeRxescolorc0,thatis,R[2∩1(R1)]=
2∩1(R).Thismeansthat2(R)=R2∩1(R)=2∩1(R)byLemma2.2.
Hence,u=v=1ands=t(referto(2.5)).FromTheorem2.4,thecoloringsof
1(R)and1(R1)musthavethesamenumberofcolors.Now,foreachcj+22J,
choosec˜j2(cj+2)∩1(R1).ThenRc˜j+22KwithRc˜j21(R),and
R[(cj+2)∩1(R1)]=Rc˜j+[R2∩1(R)]=Rc˜j+[2∩1(R)]=(Rc˜j+2)∩1(R).
(2.7)(2.7)

2.3.FURTHEREXAMPLES23
Thus,R2R1xescolorc0+2=2,whichmeansthatR2R12HbyTheorem2.8.
2.3.Furtherexamples
Wearenowgoingtoimplementthetheorydevelopedintheprevioustwosections
tosomespecialcases.Examplesinvolvingthecubicandhypercubiclatticesarealso
presentedinthissection.
ThenextlemmaconsidersthesituationwhereatleastoneoftheCSLs1(R)and
1(R1)liesinthesublattice2.
Lemma2.14:Let2beasublatticeof1with[1:2]=m,andR2OC(1).If
1(R)or1(R1)isasublatticeof2,then2(R)|1(R).Inparticular,both1(R)
and1(R1)aresublatticesof2ifandonlyif2(R)=m11(R).
Proof:If1(R)isasublatticeof2then2∩1(R)=1(R),thatis,t=1in(2.5).
Sincev|tbyTheorem2.4,v=1.Equation(2.6)yields1(R)=sm2(R),andthus,
2(R)|1(R)becauses|m.
Similarly,if2contains1(R1)thens=u=1.This,with(2.6),impliesthat
2(R)|1(R).
Finally,both1(R)and1(R1)aresublatticesof2ifandonlyifs=t=u=
v=1.Applying(2.6)completestheproof.
Thepossibilitiesarequitelimitedwhenthesublattice2isofprimeindexin1,
ascanbeseeninthenextproposition.
Proposition2.15:Suppose2isasublatticeof1ofindexp,wherepisprime,and
R2OC(1).
(i)Ifboth1(R)and1(R1)aresublatticesof2then2(R)=p11(R).
(ii)Ifeither1(R)or1(R1)isasublatticeof2then2(R)=1(R).
(iii)Ifneither1(R)nor1(R1)isasublatticeof2,then2(R)=1(R)
wheneverRisacolorcoincidenceofthecoloringof1inducedby2,and
2(R)=p1(R)otherwise.
Proof:Statements(i)and(ii)areimmediatefromLemma2.14anditsproof.
Ifneither1(R)nor1(R1)liein2thens,t>1(see(2.5)).Then,s=t=p
becausebothsandtdividetheprimepbyTheorem2.4.NotethatRisacolor
coincidenceofthecoloringof1ifandonlyifu=v=1byTheorem2.8and
Lemma2.2.Thus,2(R)=1(R)wheneverRisacolorcoincidenceofthecoloring
of1by(2.6).Otherwise,u=v=pbecauseu|sandv|t,anditfollowsfrom(2.6)
that2(R)=p1(R).
Thenextpropositionlooksattheinstancewhenthecoincidenceindexofacoin-
cidenceisometryofalattice1isrelativelyprimetotheindexofthesublattice2
.in1Proposition2.16:Let2beasublatticeof1with[1:2]=m,andR2OC(1).
If1(R)andmarerelativelyprime,thenallcolorsinthecoloringof1determined
by2appearinbothcoloringsof1(R)and1(R1),thatis,s=t=min(2.5).

302.COINCIDENCEINDICESOFSUBLATTICESANDCOLORINGSOFLATTICES
Risacolorcoincidenceofthecoloring(withallcolorsbeingxed),andthesetof
coincidingpoints,P2(R),includesallpointsofP2inthecoloringofP1(R).

(a)ColoringofP1determinedbyP2
0D0EE0CABB0A0
DC

EBADC

(b)ColoringofP1(R1)(c)ColoringofP1(R)
Figure7.ColoringsofP1(R1)andP1(R)determinedbyP2(see
abouttheoriginbytan122109.5inthecounterclockwise
Figure6),wherethecoincidencperotationRcorrespondstoarotation
direction.ThepointslabeledA,B,C,D,andEaremappedby
RtothepointslabeledA0,B0,C0,D0,andE0,respectively.Risa
colorcoincidenceofthecoloringofP1thatxesallthecolors.The
intersectionofP2andRP2,P2(R),ismadeupofallblackdotsinthe
coloringofP1(R).

25

3.COINCIDENCESOFSHIFTEDLATTICES

andisasubgroupofSOC(M)ofindex2.Furthermore,xx=12M.
ThismeansthatthereectionsymmetryT1,12P(M)isacoincidencereection
ofx+MbyLemma3.45.ItfollowsthenfromTheorem3.46(iii)thatOC(x+M)
formsasubgroupofOC(M)ofindex2andisgivenby

ofOC(M)ofindex2andisgivenby
OC(x+M)=SOC(x+M)ohT1,1i.

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