Elementary abelian regular coverings of Platonic maps

31 Pages·2015·0.31 MB·English

JAlgebrComb(2015)41:461–491
DOI10.1007/s10801-014-0542-5
Elementary abelian regular coverings of Platonic maps
CaseI:ordinaryrepresentations
GarethA.Jones
Received:22April2012/Accepted:25June2014/Publishedonline:17July2014
©SpringerScience+BusinessMediaNewYork2014
Abstract We classify those orientably regular maps which are elementary abelian
regularbranchedcoveringsofPlatonicmapsM,inthecasewherethecoveringgroup
andtherotationgroup G ofMhavecoprimeorders.Themethodinvolvesstudying
therepresentationsofG oncertainhomologygroupsofthesphere,puncturedatthe
branch-points.Wegiveacompleteclassificationforbranchingoverfaces(or,dually,
vertices)ofM,andweoutlinehowthemethodextendstootherbranchingpatterns.
Keywords Orientablyregularmap·Platonicmap·Elementaryabeliancovering·
Automorphismgroup
MathematicsSubjectClassification 05C10·20B25
1 Introduction
Thispaperaddressestheproblemofclassifyingorientablyregularmaps,thosemapsN
onsurfaces(alwaysassumedtobecompactandorientable)forwhichtheorientation-
preserving automorphism group Aut+N acts transitively on directed edges. Such
mapsareoftenformedasregularcoveringsofasimplerorientablyregularmapM,in
whichcasetheycorrespondtonormalsubgroupsofthefundamentalgroupπ Softhe
1
underlyingsurfaceSofM,puncturedatanybranch-points,whichareinvariantunder
theinducedactionoftheorientation-preservingautomorphismgroupG := Aut+M
ofMonπ S.Inthecaseofabeliancoverings,onecanabelianiseπ S,andinstead
1 1
lookforG-invariantsubmodulesofthefirsthomologygroup H (S;Z):forinstance,
1
B
G.A.Jones( )
SchoolofMathematics,UniversityofSouthampton,SouthamptonSO171BJ,UK
e-mail:[email protected]
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462 JAlgebrComb(2015)41:461–491
SurowskiandJones[13]have usedthismethod toclassifythecyclicregular cover-
ings of the Platonic maps M, branched over the vertices, edges or faces; this work
has recently been extended by Hu [9] to include branching over vertices and faces
(seealso[10]).Whenconsideringcoveringsbyelementaryabelian p-groups,onecan
go further and reduce the homology module mod p, so that G acts on the vector
space H (S;F ): this idea has been used by Kazaz [14–16] to classify the elemen-
1 p
tary abelian unbranched regular coverings of the orientably regular hypermaps of
genus2.
Herewecombinethesetwoapproachestostudyregularbranchedcoveringsofthe
Platonicmaps,theorientablyregularmapsofgenus0,byelementaryabelianp-groups,
asafirststeptowardsamoregeneralclassificationoftheabeliancoveringsofthese
maps (indeed, itishoped thatthetechniques used herecan alsobe applied tomaps
on other surfaces than the sphere). The representation theory of the corresponding
Platonic groups G plays an important role in this work. It is rather easier to apply
when the prime p does not divide |G|, as we shall generally assume here, so that
Maschke’s Theorem applies, and ordinary representation theory can be used. The
remainingcases,where pdivides|G|andmodularrepresentationtheoryisrequired,
willbeconsideredinaseparatepaper.
Theactionofautomorphismgroupsonhomologyhasbeenused,eitherexplicitly
orimplicitly,tostudyabeliancoveringsinothercontexts.Forinstance,itisaclassical
technique for Riemann surfaces, with Farkas and Kra [7, Sect. V.3] describing the
backgroundtheoryandCohen[2]applyingittoKlein’squarticcurve.Inrecentyears
ithasalsobeenapplied,oftenusingthelanguageofvoltageassignments,toabelian
coverings of graphs. For instance Malnicˇ et al. [19] have developed the theory and
applied it to elementary abelian coverings of dipoles and of the Heawood graph,
while Kwak and Oh [17] and Conder and Ma [4,5] have respectively considered
elementaryabeliancoveringsoftheoctahedralgraphandabeliancoveringsofvarious
cubicgraphs.Infact,thepresentworkcouldberestatedintermsofcoveringsofthe
Platonicgraphs,restrictedtothosecoveringswhichrespectthesurface-embeddings.
Themainresultisthefollowingtheorem,whichusesthenotation{n,m}ofCoxeter
andMoser[6]forthem-valentPlatonicmapwithn-gonalfaces:
Theorem1 The orientably regular covering maps N of the Platonic maps M,
branched over the faces, with elementary abelian p-groups of rank c ≥ 1 as cov-
eringgroupsforprimes pnotdividingtheorderofthegroupG = Aut+M,are(up
toorientation-preservingisomorphism)asfollows:
– ifMisthetetrahedron{3,3}thereisonemapN foreach p >3,withc=3;
– ifMisthecube{4,3}therearethreemapsN foreach p >3,withc=2,3and
5;
– if M is the octahedron {3,4} there are seven maps N for each p > 3, with
c=1,3,3,4,4,6and7;
– ifMisthedodecahedron{5,3}therearesevenmapsNforeachp ≡±1 (mod 5),
with c = 3,3,5,6,8,8 and 11, and three when 3 < p ≡ ±2 (mod 5), with
c=5,6and11;
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JAlgebrComb(2015)41:461–491 463
– if M is the icosahedron {3,5} there are 8p + 23 maps N for each p ≡ ±1
(mod 5),withc ∈ {3,4,...,16,19},and4p+11when3 < p ≡ ±2 (mod 5),
withc∈{4,5,6,8,9,10,11,13,14,15,19};
– if M is the dihedron {n,2} there is one map N for each p coprime to 2n, with
c=1;
– ifMisthehosohedron{2,n}thereare2ν−1mapsN foreach pcoprimeto2n,
whereν isthenumberoforbitsΔ(cid:5)={1}ofthegroupgeneratedbytheFrobenius
automorphismandinversiononthesetofn-throotsof1inthealgebraicclosure
F ofthefieldF .
p p
ForeachM={n,m}andprime pthesemapsN havetype{np,m}andgenus
(cid:2) (cid:3)
f f
−1 pc− pc−1+1,
2 2
where f isthenumberoffacesofM;theyareallquotientsofasinglecoveringmap
withc = f −1.ThemapsN areallregular,exceptwhenMistheicosahedron,in
whichcasethereare4(p−1)or2(p−1)chiralpairsas p ≡ ±1or±2 (mod 5),
theother31or15beingregular.InallcasesexceptwhenMistheicosahedron,the
2ν −1mapsN arethejoinsformedfromasetofν irreduciblecoveringsofM.
Comments
(1)Theorientation-preservingautomorphismgroupAut+N ofeachofthesemaps
isanextensionofthecoveringgroup,anelementaryabeliannormalsubgroup K
of order pc, by G. Since G and K have coprime orders, the Schur–Zassenhaus
Theoremimpliesthatthisextensionsplits.AsaG-moduleunderconjugation, K
isisomorphictoaquotientof H (S;F ),identifiedspecificallyinSects.3–9for
1 p
theindividualmapsM.
(2)Anorientablyregularmapiscalledregularorchiralasitisorisnotisomorphic
toitsmirrorimage.WhenMistheicosahedron,thechiralcoveringmapsarise
fromthefactthatitsfullautomorphismgroupAutMhasfewerorbitsthanG on
orderedpairsoffacesofM(seeSects.2.6and7).Thefactthatinthiscasethe
numberofcoveringsisunboundedasafunctionof pisrelatedtothefactthatthe
permutationcharacterofG onfacesofMisnotmultiplicity-free(seeSects.2.5
and7).
(3)Inthecaseofthehosohedron,themethodforfindingtherankscofthecovering
groups K,toocomplicatedtostatehere,isdescribedinSect.9.2.
(4)TheconclusionsofTheorem1alsoapplyinafewcaseswhere pdivides|G|,
such as the tetrahedron and octahedron with p = 3, and the hosohedron with n
oddand p =2;thesecaseswillbestudiedinalaterpaper.
Thepaperisorganisedasfollows.InSect.2someusefultechniquesareoutlined.
These are applied in Sects. 3–9 to enumerate and describe the coverings in Theo-
rem 1. The individual cases are fairly straightforward, except for that involving the
hosohedronM={n,2},whichdependsonthereductionmod pofcyclotomicpoly-
nomials.Branchingoververticesiseasilydealtwithbyduality.InSect.10weconsider
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464 JAlgebrComb(2015)41:461–491
simultaneousbranchingoververticesandfaces,concentratingonthetetrahedronasa
typicalexample.InSects.11and12webrieflyconsiderthehypermapsarisingfrom
branchingoveredges,andoververtices,edgesandfaces.Theresultsforthesedifferent
branchingpatternsaresimilartothoseinTheorem1.
2 Techniques
Inthissection,severaltechniquesfromalgebraicmaptheory,topologyandrepresen-
tation theory are used to reduce the classification problem for maps to a number of
mainly group- and number-theoretic calculations. No originality is claimed for the
resultsquoted,whichareallstandardintheirareas.Tohelpthereaderthroughwhat
isaratherlongnarrative,someimportantobservations,oftenusedimplicitlyinlater
calculations,areitalicised.
2.1 Maps
First,webrieflysketchtheconnectionsbetweenmapsandtrianglegroups;forback-
ground,see[11].Mapsarealwaysassumedtobeoriented,thatis,onorientablesurfaces
withachosenorientation.Theaimistoclassifythemuptoorientation-preservingiso-
morphism.
Wedefineatrianglegrouptobeagroupoftheform
Δ(p,q,r)=(cid:6)x,y,z | xp = yq =zr = xyz =1(cid:7)
where p,q,r ∈N∪{∞}andweignoreanyrelationg∞ =1.Anym-valentmapN
correspondstoasubgroup N ofthetrianglegroup
Δ:=Δ(m,2,∞)=(cid:6)x,y,z | xm = y2 = xyz =1(cid:7),
withvertices,edgesandfacescorrespondingtothecyclesofx,y andzonthecosets
of N.
ThemapN isorientablyregularifandonlyifN isnormalinΔ,inwhichcasethe
orientation-preservingautomorphismgroupAut+N isisomorphictoΔ/N;wewill
assumethisthroughout.
Inparticular,thePlatonicmapM = {n,m}correspondstothenormalclosure M
ofzn inΔ,and
Aut+M∼=Δ/M ∼=Δ(m,2,n).
AmapN isad-sheetedcoveringofMifandonlyifN isasubgroupofindexdinM,
inwhichcaseN →Misaregularcoveringandthegroupofcoveringtransformations
is M/N.
Thustheorientablyregularm-valentmapswhichared-sheetedabeliancoverings
ofMare(uptoorientation-preservingisomorphism)inbijectivecorrespondencewith
thesubgroups N of M whicharenormalinΔ,with M/N abelianoforderd.
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JAlgebrComb(2015)41:461–491 465
If M/N hasexponente,suchsubgroups N containthecommutatorsubgroup M(cid:12)
and the subgroup Me generated by the e-th powers in M, so they correspond to
subgroups N = N/M(cid:12)Me of M = M/M(cid:12)Me,with M/N ∼= M/N.TheactionofΔ
(cid:12)
byconjugationonthenormalsubgroup M preservesitscharacteristicsubgroups M
and Me,givinganinducedactionofΔon M;since M actstriviallyon M,wehave
anactionofthegroup
G :=Aut+M∼=Δ/M
on M,whichisamoduleforG overtheringZ :=Z/eZ.
e
Itfollowsthatasubgroup N ofM,containingM(cid:12)Me,isnormalinΔifandonlyif
N isaG-submoduleof M.
Theorientation-preservingautomorphismgroupG˜ := Aut+N ∼= Δ/N ofN has
an abelian normal subgroup K,corresponding to M/N,with G˜/K ∼= G;under the
inducedactionofGonK byconjugation,K isisomorphicasaG-moduletoM/N.If
eiscoprimeto|G|,thentheSchur–ZassenhausTheoremimpliesthatG˜ isasemidirect
productof K byG.
2.2 Homologymodules
Let S = S2 \{c ,...,c }, where c ,...,c are the centres of the f faces of M.
1 f 1 f
ThenMcanbeidentifiedwiththefundamentalgroupπ (S)ofS,afreegroupofrank
1
f −1generated bythehomotopy classes g ofloopsaround thepunctures c ,with
i i
asingledefiningrelation g ...g = 1.Itfollowsthatthegroup Mab = M/M(cid:12) can
1 f
beidentifiedwiththefirstintegerhomologygroup H (S;Z) = π (S)ab of S,afree
1 1
abelian group of rank f −1 generated by the homology classes [g ], with defining
i
relation[g ]+···+[g ]=0.BytheUniversalCoefficientTheorem,M = M/M(cid:12)Me
1 f
isidentifiedwiththemodehomologygroup H1(S;Ze)= H1(S;Z)⊗ZZe ∼=Zef−1.
Undertheseidentifications,theactionsofG inducedbyconjugationinΔandby
homeomorphisms of S are the same, so our problem is to find the G-submodules of
H1(S;Ze). (cid:4)
Theprimepowerfactorisation p pip (pprime,ip ≥1)ofeinducesaG-invariant
decompositionof H (S;Z )asadirectsumofitsSylow p-subgroups H (S;Z ),
1 e 1 pip
soitissufficienttorestrictattentiontothecasewhereeisapowerofaprime p.For
eachprime pwehaveaninfinitedescendingseries
Mab = M/M(cid:12) > M(cid:12)Mp/M(cid:12) > M(cid:12)Mp2/M(cid:12) >···> M(cid:12)Mpi/M(cid:12) >··· ,
ofcharacteristicsubgroupsof Mab,correspondingtoadescendingseries
H (S;Z)= H0 > H1 >···> Hi >··· ,
1
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466 JAlgebrComb(2015)41:461–491
of G-submodules of H (S;Z), where each Hi is the kernel of the reduction
1
(mod pi): H (S;Z)→ H (S;Z ).Thisinducesafinitedescendingseries
1 1 pi
H (S;Z )= H0/Hi > H1/Hi >···> Hi/Hi =0
1 pi
ofG-submodulesof H (S;Z ).Successivequotients
1 pi
(Hj/Hi)/(Hj+1/Hi)∼= Hj/Hj+1 ∼= M(cid:12)Mpj/M(cid:12)Mpj+1
inthisseriesareG-isomorphictothemodule
H (S;F )= H (S;Z )= H0/H1 ∼= M := M/M(cid:12)Mp,
1 p 1 p p
where F is the field of p elements. This G-module H (S;F ), along with all the
p 1 p
otherquotientsintheseries,isavectorspaceoverF ofdimension f −1affording
p
arepresentationρp :G →Aut(H1(S;Fp))∼=GLf−1(Fp)ofG.
Our main problem therefore is to determine the structure of this G-module
H (S;F ) for each prime p; in particular, this will immediately determine the ele-
1 p
mentaryabelianregularcoveringsofM.
AG-submoduleLofH (S;F )ofcodimensionccorrespondstoanormalsubgroup
1 p
NofΔcontainedinM,withthegroupK := M/Nelementaryabelianoforderpc,and
hencetoaregular pc-sheetedcoveringN ofM,branchedoveritsface-centres,with
coveringgroupK.TheimageinK ofageneratorg ofMinK iscalledamonodromy
i
generatorofK,representingthepermutationofthesheetsofthecoveringinducedby
traversingasmallloopinSaroundthecorrespondingbranch-point.
ThiscoveringN isanorientablyregularmapoftype{np,m}.Therearepc−1points
aboveeachofthe f face-centresofM,sothetotalorderofbranchingofthecovering
N →Mis(pc−pc−1)f.BytheRiemann-Hurwitzformula,N thereforehasgenus
(cid:2) (cid:3)
1 f f
g =1− pc+ pc−1(p−1)f = −1 pc− pc−1+1.
2 2 2
Theorientation-preservingautomorphismgroupG˜ =Aut+N ofN hasanelementary
abeliannormalsubgroup K oforder pc,isomorphicto H (S;F )/L asaG-module
1 p
overF .
p
Fornotationalconvenience,weletE (M)denotethesetof(orientation-preserving
p
isomorphism classes of) such regular covers N of M with elementary abelian p-
groupsascoveringgroups,andforeachN ∈ E (M)weletdim(N)=cwherethe
p
coveringN →Mhas pcsheets.Thusdim(N)issimplythedimensionofthecorre-
spondingG-moduleH (S;F )/L,orequivalentlythecodimensionofLinH (S;F ).
1 p 1 p
WewilldenotetherepresentationofG on H (S;F )/L andthecorrespondingchar-
1 p
acterbyρN andχN,andwewillsaythattheyarebothaffordedbyN;thisisbecause
H (S;F )/L isisomorphic,asaG-module,to M/N andhenceto K.
1 p
The representation ρN can therefore be regarded as the induced action of G, by
conjugation,onthecoveringgroup K associatedwithN.
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JAlgebrComb(2015)41:461–491 467
Ifpiscoprimeto|G|thentherepresentationtheoryofGoverfieldsofcharacteristic
pisessentiallythesameasthatoverfieldsofcharacteristic0(ordinaryrepresentation
theory):forinstanceMaschke’sTheoremappliesinbothcases.Inthissituation,the
decompositionof H (S;F )asa G-modulecanbeobtainedfromthatofthecorre-
1 p
sponding homomology module over C (or indeed any algebraically closed field of
characteristic0,suchasthefieldQofalgebraicnumbers).Moreover,if piscoprime
to|G|thenG˜ isasemidirectproductofK byG,bytheSchur–ZassenhausTheorem.
If,ontheotherhand, pdivides|G|thenthisextensionsplitsifandonlyifitsSylow
p-subgroupssplitover K.Moreover,inthissituation,Maschke’sTheoremdoesnot
apply,andthesubmodulestructureofH (S;F )islesstransparent,requiringmodular
1 p
ratherthanordinaryrepresentationtheory.Inthispaperwethereforeconcentrateon
the ordinary case, avoiding the finitely many primes p dividing |G|; they will be
consideredinaseparatepaper.
2.3 Permutationmodules
LetΦbethesetoffacesofM,andletCΦbethecorrespondingpermutationmodule
forG;thisisan f-dimensionalcomplexvectorspacewithbasisΦpermutednaturally
by G. The homology module H1(S;C) is isomorphic(cid:5)to the quotient of CΦ by the
1-dimensionalG-submodulespannedbytheelement φ.
φ∈Φ
The G-module decomposition of CΦ corresponds to that of the corresponding
permutationcharacterπ,whereπ(g)isthenumberoffacesφ ∈ Φ invariantunder
eachg ∈G.
IfH denotesthesubgroup(cid:6)z(cid:7)∼=C ofGleavinginvariantaface,thensinceGacts
n
transitivelyonΦ wehaveπ = 1G,thecharacterobtainedbyinducingtheprincipal
H
character1 of H uptoG.ByFrobeniusreciprocity,themultiplicity(χ ,π)ofany
H i
irreduciblecomplexcharacterχ ofG inπ isgivenby
i
(cid:6)
(χ ,π)=(χ | ,1 )=|H|−1 χ (g),
i i H H i
g∈H
themultiplicityof1 intherestrictionofχ to H.
H i
Equivalently,themultiplicity(χ ,π)ofanirreduciblecharacterχ ofGinπ isthe
i i
multiplicityof1asaneigenvalueofz intherepresentationρ ofG correspondingto
i
χ .
i
Giventhesemultiplicities,weknowthedirectsumdecompositionofCΦ,andhence
thatof H (S;C).Inparticular,thismoduleaffordsthecharacter
1
χ =π −χ ,
1
whereχ istheprincipalcharacter1 ofG,givenbyχ (g)=1forallg ∈G.
1 G 1
ThesamedecompositionsapplyifCisreplacedwiththealgebraicclosureF ofF
p p
foranyprime pnotdividing|G|.TofindthecorrespondingdecompositionsoverF ,
p
variousalgebraicallyconjugatesummandsmustbemergedtogivesummandsrealised
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468 JAlgebrComb(2015)41:461–491
overF .ForthevariousPlatonicmapsM,howthishappensdependsonwhether p
p
satisfiescertaincongruences.
2.4 Findingsubmodules
SomesubmodulesLofH (S;F )arisenaturallyfromtheactionofGonΦ,whether
1 p
or not p divides |G|.Let P denote thepermutation module(cid:5)FpΦ,a G-module over
F . Given any subset Ψ ⊆ Φ, let Ψ denote the element φ of P. The G-
p φ∈Ψ
module H (S;F )canthenbeidentifiedwiththequotient Q := P/P of P bythe
1 p 1
1-dimensionalsubmodule P spannedbyΦ.
1
2.4.1 Thesubmodule L =0of Q
ThereisaregularcoveringM ∈ E (M),withdim(M ) = f −1,corresponding
0 p 0
tothesubmodule L = 0of Q = H (S;F ),i.e.tothesubmodule P of P;allother
1 p 1
regular coverings N of M in E (M) are proper quotients of M , so they satisfy
p 0
dim(N)≤ f −2.
2.4.2 Thesubmodule L = Q1of Q
T(cid:5)here is a G-submodule P1 of(cid:5)codimension 1 in P, consisting of the elements
φ∈Φaφφwithcoordinate-sum φ∈Φaφ =0.Ifpdivides f thenP1 ≤ P1,givinga
G-submoduleL = Q1 = P1/P ofcodimension1inQ,andhencearegularcovering
1
M1 ∈ E (M) with c = dim(M1) = 1. This is therefore a cyclic covering of M
p
(meaningthatthecoveringgroup K iscyclic),andsince L/Q1 ∼= P/P1 affordsthe
principalrepresentationofG itisalsoacentralcovering,inthesensethat K isinthe
˜
centre of G. The cyclic coverings of the Platonic hypermaps were classified by the
˜
authorandSurowskiin[13](seealso[20]),whereProposition3impliesthat G isa
splitextensionofGbyK,givingG˜ =G×K ∼=G×C ,ifandonlyif pdivides2m
p
butnotn.Inthecaseswhere pdoesnotdivide f wehave P = P ⊕P1,so Q ∼= P1
1
andhence P1doesnotcorrespondtoapropercoveringofM.
2.4.3 Thesubmodules L = Q∼of Q
IfGactsimprimitivelyonΦ,preservinganon-trivialequivalencerelation∼(equiva-
lently,ifthereisasubgroupH∼ofGsuchthatH < H∼ <G),thenasΨ rangesover
thek =|G : H∼|equivalenceclasses,theelementsΨ formabasisforak-dimensional
G-submodule P∼ of P,containing P1 (andcontainedin P1 ifandonlyif p divides
the size |H∼ : H| = f/k of each class). The G-submodule L = Q∼ = P∼/P1
of Q therefore corresponds to a regular covering M∼ ∈ Ep(M) of dimension
dim(M∼) = f − k, affording the character χM = π − π∼ of G where π∼ is
∼
thepermutationcharacterofG ontheequivalenceclassesof∼,i.e.onthecosetsof
H∼ in G. If ≈ is a G-invariant equivalence relation which refines ∼ (i.e. φ1 ≈ φ2
impliesφ1 ∼φ2),thentheinclusion Q∼ ≤ Q≈inducesacoveringM∼ →M≈.
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JAlgebrComb(2015)41:461–491 469
For instance, suppose that M has an antipodal automorphism, so that f is even.
Then P has a G-submodule P ≥ P of dimension k = f/2 with basis elements
a 1
Ψ corresponding to the antipodal pairs Ψ ⊆ Φ. This gives a G-submodule L =
Q = P /P of codimension f −k = f/2 in Q, so we obtain a regular covering
a a 1
Ma ∈ Ep(M)withdim(Ma)= f/2.If p >2thereisasecondG-submodulePa(cid:12)of
dimension f/2inP,withabasiselementφ−φ(cid:12)foreachantipodalpair{φ,φ(cid:12)}⊆Φ(so
Pa(cid:12) ≤ P1).TheG-submodule L = Qa(cid:12) =(Pa(cid:12) ⊕ P1)/P1ofcodimension(f −2)/2
inQcorrespondstoaregularcoveringMa(cid:12) ∈ Ep(M)withdim(Ma(cid:12))=(f −2)/2.
Asimplecalculationshowsthat P = Pa ⊕ Pa(cid:12),so Q = Qa ⊕Qa(cid:12).Since Pa affords
thepermutationcharacterπ ofG ontheantipodalpairsoffaces,itfollowsthat Q
a a
andhence Q/Qa(cid:12) affordπa −χ1,while Qa(cid:12) and Q/Qa affordχ −πa.
2.4.4 Thecase p =2
If p =2onecanidentify P withthepowersetP(Φ)ofΦ,agroupundersymmetric
difference,withthenaturalinducedactionofG,sothat P1 ={Ψ ⊆Φ ||Ψ|iseven}
andP ={∅,Φ}.Inthiscase,Qcanbeidentifiedwiththesetofcomplementarypairs
1
{Ψ,Φ\Ψ}ofsubsetsofΦ.
2.4.5 Thenaturalrepresentationρ
n
For each M, the group G has a natural 3-dimensional complex representation ρ ,
n
obtainedbyextendingitsnaturalrealrepresentation,astherotationgroupofM,by
linearitytoC.Thisrepresentationisirreducible,exceptwhenMisthedihedronor
hosohedron,inwhichcaseitsplitsasadirectsumof1-and2-dimensionalrepresen-
tations. When M is one of the five Platonic solids, by sending each basis element
φ (φ ∈ Φ)ofCΦ totheposition-vectorinR3 ⊂ C3 oftheface-centreofφ,wesee
thatρ isaquotient,andhenceadirectsummand,oftherepresentationofG onCΦ.
n
Frobeniusreciprocityimpliesthatρ hasmultiplicity1inthisrepresentation,andin
n
thecasesofthedodecahedronandicosahedron,thesameappliestotheGaloiscon-
jugaterepresentationρn∗.Byrealisingρn overtheringofintegersofanappropr√iate
algebraic number field (namely Q for the tetrahedron, cube or octahedron, Q( 5)
forthedodecahedronoricosahedron,andacyclotomicfieldQ(ζ )forthedihedron
n
or hosohedron, where ζ = e2πi/n), and taking quotients modulo some prime ideal
n
containing p,wefindthatthemod preductionofρn orofρn⊕ρn∗isasummandof
therepresentationofG on P,andhenceon Q.
2.5 Diagonalsubmodules
Weneedtodealwiththosecaseswhere Qhassummandsofmultiplicitygreaterthan
one.If A and A areisomorphic FG-modulesforsomefield F andgroup G,then
1 2
adiagonalsubmoduleof A ⊕ A isasubmoduleoftheform{(a,a(cid:12)) | a ∈ A }for
1 2 1
someisomorphism A → A ,a (cid:22)→a(cid:12).
1 2
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470 JAlgebrComb(2015)41:461–491
Lemma1 LetU beasubmoduleofamoduleV = V ⊕V ,letU =U∩V ,andlet
1 2 i i
V = V /U fori =1,2.ThenU istheinverseimageinV ofadiagonalsubmodule
i i i
ofW ⊕W forsomesubmodulesW suchthatU ≤ W ≤ V fori =1,2.
1 2 i i i i
Proof The image of U in V ⊕ V intersects each summand V trivially, so it is
1 2 i
isomorphictoitsprojectionW ineachV ,andisthereforeadiagonalsubmoduleof
i i
W ⊕W . (cid:24)(cid:25)
1 2
Thus,ifthesubmodulestructuresofeachV areknown,soisthatofV.
i
Lemma2 IfV andV areisomorphicirreducibleFG-modules,thentheonlyproper
1 2
submodulesofV ⊕V are0,V ,V andthediagonalsubmodules;givenanyisomor-
1 2 1 2
phismV → V ,v (cid:22)→v(cid:12),eachofthelatterhastheform{(v,λv(cid:12))|v ∈ V }forsome
1 2 1
λ∈ F∗.
Proof LetU beasubmoduleofV = V ⊕V .InthenotationofLemma1,foreach
1 2
i = 1,2 the irreducibility of V implies that U = 0 or V . If U = V for some i
i i i i i
then U ≥ V , so the irreducibility of V/V implies that U = V or V. Otherwise,
i i i
U = U = 0,soinLemma1wehaveeither W = 0forsomei,givingU = 0,or
1 2 i
W = V foreachi,inwhichcaseU isadiagonalsubmoduleofV.Inthelattercase,
i i
Schur’sLemmaimpliesthattheisomorphisms V → V arethefunctionsv (cid:22)→ λv(cid:12)
1 2
forλ∈ F∗. (cid:24)(cid:25)
ThisshowsthatifV = V isirreducibleandF =F ,thenV = V ⊕V = V ⊕V
1 2 q 1 2 1 1
hasq+1non-trivialpropersubmodules,allisomorphictoV .Wecanwritethemas
1
V(λ)={(αv,βv)|v ∈ V , α/β =λ}
1
for λ ∈ P1(F ) = F ∪{∞}; here V(∞) and V(0) are the first and second direct
q q
summands,withβ =0andα =0,respectively.
2.6 Regularityandchirality
Each of the covering maps N constructed here is orientably regular, since it corre-
spondstoanormalsubgroup N ofΔ.Itisregular(i.e.italsoadmitsanorientation-
reversing automorphism) if and only if N is normal in the extended triangle group
Γ = Δ[m,2,∞], which contains Δ as a subgroup of index 2. This is equivalent
to the corresponding submodule L of H (S;Z ) being invariant, not just under the
1 e
orientation-preservingautomorphismgroupG = Aut+M ∼= Δ/N ofM,butunder
itsfullautomorphismgroup A=AutM∼=Γ/N.Whene= p,itisclearfromtheir
constructionthatthemapsM0,M1,Ma andMa(cid:12) (wheretheyexist)areallregular.
OnecanfindthestructureofH (S;F )asanA-modulebytreatingitasthequotient
1 p
Q = P/P of P,where P isnowregardedasthepermutationmodulefor AonΦ.
1
Inparticular,thepermutationcharacterπ| of AonPcanbefoundasforπ| (see
A G
Sect.2.3),taking H tobethestabiliser Aφ ∼= Dn ofafaceφ ∈Φ in Ainsteadofits
∼
stabiliserGφ =Cn inG.
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