The t-core of an s-core

0.15 MB·

by  Matthew Fayers

The t-core of an s-core
Matthew Fayers
QueenMary,UniversityofLondon,MileEndRoad,LondonE14NS,U.K.
[email protected]
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2000Mathematicssubjectclassification: 05E10,05E18
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n Abstract
a We consider the t-core of an s-core partition, when s and t are coprime positive integers.
J
Olssonhasshownthatthet-coreofans-coreisagainans-core,andweexaminecertainactions
5 oftheaffinesymmetricgroupons-coreswhichpreservethet-coreofans-core. Alongtheway,
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we give a new proof of Olsson’s result. We also give a new proof of a result of Vandehey,

] showingthatthereisasimultaneouss-andt-corewhichcontainsallothers.
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2 MatthewFayers
1 Introduction
Inthispaper,wedefineapartition tobeainfiniteweaklydecreasingsequenceofnon-negative
integerswithfinitesum. Ifsisapositiveinteger,thenapartitionisans-coreifithasnorims-hooks,
or equivalently if none of its hook lengthsis divisible by s. The s-core of an arbitrary partition is
obtainedbyremovingasmanyrims-hooksaspossible.
The notion of an s-core was introduced in the representation theory of the symmetric group:
whensisaprime,thes-blocksofallsymmetricgroupsofagivendefectareindexedbythes-cores,
and the relationships between these blocks are controlled by the combinatorics of s-cores. This
can be generalisedto representationsofIwahori–Heckealgebras oftypeA at an sth root ofunity
(wheres > 2nolongerneedstobeprime).
This representation-theoretic work reveals a relationship between the set of s-cores and the
alcovegeometryfortheCoxetergroupoftypeA ,whens> 2. Specifically,s-coresareinbijection
s−1
withalcovesinthedominantregionoftheweightspace,whichinturnareinbijectionwithcosets
ofthefiniteCoxetergroup(oftypeA )initsaffinecounterpart(oftypeA˜ ). Furthermore,the
s−1 s−1
actionofthesegroupsofthesetofalcovescanbeinterpretedintermsoftherelationshipsbetween
s-cores.
A recent trend in the study of s-cores has been to compare s-cores and t-cores, for different
integers s,t. For s > 2 there are infinitely many s-cores, but if s and t are coprime, there are only
finitelymanypartitionswhicharesimultaneouslys-coresandt-cores. Theexactnumberwasfound
byAnderson[A],andthese‘(s,t)-cores’havesincebeenstudiedinmoredetail. Inparticular, itis
knownthatthereisan(s,t)-corewhichislargerthantheothers,havingsize 1 (s2−1)(t2−1),andit
24
wasaskedbyOlssonandStanton[OS]whetherthispartitioncontainsall(s,t)-cores. Thisquestion
ffi
has been answereda rmatively by Vandeheyin an unpublished thesis[V]; in the presentpaper
we give a simpler proof. Another avenue is pursued by Fishel and Vazirani [FV], who examine
(s,t)-coresinconnectionwithalcovegeometryinthecaseswheret ≡ ±1(mod s),exhibitingnatural
bijectionsbetween(s,t)-coresand(bounded)regionsintheextendedShiarrangement.
Anotheraspectofthecomparisonofs-andt-coresisaresultofOlsson[O]whichsaysthatifs
andtarecoprimeandonetakesthet-coreofans-core,thentheresultingpartitionisstillans-core.
Themainfocusofthispaperistoaskwhich(s,t)-coreoneobtainsbytakingthet-coreofans-core.
We explore how the symmetry of the set of s-cores is manifested when one replaces each s-core
with its t-core. One by-productofthis is anew proofofOlsson’s result. Weremark here thatthe
hypothesisthatsandtarecoprimeinOlsson’sresultisunnecessary,asobservedbyNath[N].
Expertsincombinatorialrepresentationtheorywillbeawareofthetheoryofbarpartitionsand
m-bar-cores which control the combinatorics of spin representation of the symmetric group; it is
natural to ask whether analogues of these results in the present paper hold in this context. We
addresstheseissuesinaforthcomingpaper[F].
Wenowsummarisethelayoutofthispaper. InSection2,wegiveabriefaccountofs-coresand
abacusdisplays. InSection3wediscussalcovegeometryandtheaffineWeylgroupintypeA. We
gointomoredetailhere,sincetheconventionsweuseforalcovesareslightlyunusual. InSection
4weconnects-coreswithalcovegeometryandproveourmain resultsonthesymmetryinherent
in taking the t-core of an s-core. Finally in Section 5 we examine the largest (s,t)-core, and prove
thatitcontainsall(s,t)-cores.
Thet-coreofans-core 3
2 Partitions
2.1 Partitions ands-cores
A partition is a sequenceλ = (λ ,λ ,...) ofnon-negative integerssuch that λ > λ > ··· and
1 2 1 2
+ +
the sum λ λ ... is finite. When writing a partition, we usually omit the trailing zeroes. A
1 2
partitionisoftenrepresentedbyitsYoungdiagram,whichistheset
[λ] = (i,j) ∈N2 j6 λ .
i
n (cid:12) o
(cid:12)
WedrawtheYoungdiagramasanarrayofboxesint(cid:12)heplane;forexample,thearray
represents the partition (6,6,2,1). (It is usual to use a symbol such as ∅ in place of the Young
diagramforthepartition(0,0,...),butinthispaperweshalljustuseanemptydiagram.) Therim
ofλisthesetofnodes(i,j) ∈[λ]forwhich(i+1,j+1) < [λ].
Nowfixapositiveintegers. Ifλisapartition,arims-hookofλisaconnectedportionoftherim,
consisting of exactly s boxes, which can be removed from [λ] to leave a new Young diagram. A
partitionisans-coreifithasnorims-hooks. Startingfromanypartitionλandrepeatedlyremoving
rim s-hooks, one obtains an s-core, which is independent of the choice of rim hook removed at
eachstage;thiss-coreisreferredtoasthes-coreofλ.
Forexample, themarkedboxesin thefollowing Youngdiagramfor (6,6,2,1) constitutearim
5-hook. Whenthisisremoved,theremainingpartitionis(5,2,2,1),whichhasnorim5-hooks,and
soisthe5-coreof(6,6,2,1).
•
• • • •
Thenotionofans-corederivesfromtherepresentationtheoryofthesymmetricgroup: ifsisa
primeandλ,µaretwopartitionsofsizen,thecorrespondingordinaryirreduciblerepresentations
of S lie in the same s-block of S if and only if λ and µ have the same s-core. So the results in
n n
this paper can be interpretedas comparing the representationtheoryof the symmetricgroup for
twodifferentprimes. Butfromacombinatorial pointofview, thereis noneedtoassumethat sis
prime.
2.2 Beta-numbers
Now we define beta-numbers and the abacus; these were introduced by James [J]. Given a
partitionλ,define
β = λ −i
i i
for i ∈ N. Then the sequence β ,β ,... is a strictly decreasing sequence such that β = −i for
1 2 i
sufficiently large i. Conversely, any such sequence is the sequence of beta-numbers of some
partition. Werefertotheset{β ,β ,...}asthebeta-setofλ. Givenapositiveintegers,thes-runner
1 2
abacusisanabacuswithsinfiniteverticalrunners,numbered0,...,s−1fromlefttoright;foreach
4 MatthewFayers
j,runner jhasmarkedpositionslabelledbytheintegerscongruentto jmodulosincreasingdown
therunner. Forexample,the4-runnerabacusisasfollows.
0 1 2 3
p p p p
p p p p
p p p p
−4 −3 −2 −1
0 1 2 3
4 5 6 7
p p p p
p p p p
p p p p
Thes-runnerabacusdisplayforλisobtainedbyplacingabeadontheabacusatpositionβ foreachi.
i
Theabacusdisplaymakesitveryeasytoseewhetherapartitionisans-core,sinceremovingarim
s-hookcorrespondstomovingabeadupintoanemptyspaceimmediatelyaboveit. Henceλisan
s-coreifandonlyifinthes-runnerabacusdisplayforλeverybeadhasabeadimmediatelyabove
it. Moreover,itiseasytoobtaintheabacusdisplayforthes-coreofλfromtheabacusdisplayfor
λ: onejustslidesbeadsupuntilthereisnobeadwithanemptyspaceaboveit.
Example.Supposeλ = (6,6,2,1). Thenthebeta-setforλis
{5,4,−1,−3,−5,−6,−7,...}.
Sothe5-runnerabacusdisplayforλisasfollows.
0 1 2 3 4
p p p p p
p p p p p
p p p p p
v v v v v
v v v v v
v v v
v
v
p p p p p
p p p p p
p p p p p
Theabacusdisplayofthe5-core(5,2,2,1)ofλisobtainedbymovingthelowestbeadonrunner0
uponeposition.
ffi
3 Alcoves and the a ne symmetric group
ffi
Inthissectionweintroducealcovesandthea nesymmetricgroup. Thismaterialwillbevery
familiar tomanyreaders,butwegiveadetailedaccountherebecausetheconventionsweuseare
slightlyunusual.
3.1 Alcoves ands-points
Asbefore,weassumesisapositiveinteger. Infact,fromnowonweassumethats > 2. LetPs
ffi
denotethea nespace
Ps = p∈ Rs p +···+p = s .
1 s
2
(cid:26) (cid:12)(cid:12) (cid:16) (cid:17)(cid:27)
We definethe dominant region to be thesubse(cid:12)(cid:12)tofPs consistingof pointsp for which p1 6 ··· 6 ps.
(Notethatthisisunconventional–theinequalitiesareusuallytakentheotherwayround.)
Thet-coreofans-core 5
Foreach1 6 i,j 6 swithi , jandforeachintegerk,wedefinethehyperplane
Hk = p ∈ Ps p −p = ks ,
ij j i
n (cid:12) o
(cid:12)
(cid:12)
and we let H = Hk 16 i < j6 s, k ∈Z . The connected components of the complement in Ps
ij
(cid:26) (cid:12) (cid:27)
oftheunionofthehy(cid:12)perplanesinH arecalledalcoves. Wewillabuseterminologybyreferringto
(cid:12)
(cid:12)
thepoint(0,1,...,s−1)astheorigin,anddenotingit⊙. ThealcoveAcontainingthispointiscalled
thefundamentalalcove, andisboundedbythehyperplanesH0 (for1 6 i < s)andH1 .
i(i+1) 1s
We define an s-point to be a point p ∈ Ps whose coordinates are integers which are pairwise
incongruent modulo s. Obviously each s-point is contained in some alcove, and as we shall see
below,eachalcovecontainsauniques-point.
Example.In the case s = 3, we can draw a picture of part of the plane Ps with 3-points and
hyperplanesmarkedasfollows.
• • •
3,−1,1 2,1,0 1,3,−1
• • • •
3,−2,2 2,0,1 1,2,0 0,4,−1
• • • •
2,−2,3 1,0,2 0,2,1 −1,4,0
• ⊙ •
1,−1,3 0,1,2 −1,3,1
• • •
0,−1,4 −1,1,3 −2,3,2
• • • •
0,−2,5 −1,0,4 −2,2,3 −3,4,2
Foranyi,j,kasabove,letrk denotetheorthogonal(withrespecttotheusualinnerproducton
ij
Rs)reflectioninthehyperplaneHk;thisisgivenby
ij
rk : p7−→p−(p −p −ks)(e −e),
ij j i j i
wheree ,...,e arethestandardbasisvectors. rk preservesthesetofhyperplanesH;indeed,one
1 s ij
cancheckthat
Hn ({i,j}∩{l,m}= ∅)
lm
rk(Hn )= Hn−k (i = l, j , m)
ij lm H2jmk−n (i = l, j = m).
Hence the group generated by all the rkprelmserves the set of alcoves. It also preserves the set of
ij
s-points,andwecanregarditasactingonthesetofalcovesorthesetofs-points,asappropriate.
6 MatthewFayers
ffi
3.2 The a nesymmetricgroup
Nowweconsidertheaffinesymmetricgroup. ThisisthegroupSˆ withgeneratorsσ ,...,σ
s 0 s−1
andrelations
σ2 = 1 fori = 0,...,s−1,
i
σσ = σ σ fori . j±1(mod s),
i j j i
σσ σ = σ σσ fori ≡ j+1 . j−1(mod s).
i j i j i j
AleveltactionofSˆ
s
There is a well-known action of Sˆ on Ps given by mapping the generators σ ,...,σ to the
s 0 s−1
reflections in the walls ofthe fundamentalalcove. In fact, we give a more generalversion ofthis
action. Foranypositiveintegert,definetheleveltactionψ ofSˆ by
t s
σ 7−→ r0 fori = 1,...,s−1,
i i(i+1)
σ 7−→ rt .
0 1s
Using the above formula for rk(Hn ), one obtains conjugacy relations between the reflections rk,
ij lm ij
andfromtheseitiseasytocheckthatthisreallydoesgiveanactionofSˆ . Givenσ ∈ Sˆ ,weshall
s s
write σˇ for theimage ofσ underψ , if tis understood. Wemay view ψ as an action on thesetof
t t
alcoves,oronthesetofs-points,asappropriate.
Itisworthwhiletowritedownexplicitlytheactionofthegeneratorsσ ,...,σ :
0 s−1
σˇi : (p1,...,ps)7−→(p1,...,pi−1,pi+1,pi,pi+2,...,ps) fori = 1,...,s−1;
σˇ : (p ,...,p )7−→(p −st,p ,...,p ,p +st).
0 1 s s 2 s−1 1
Thenextlemma,whichiswell-known,concernsthecaset = 1.
Lemma3.1.Theimageoftheactionψ includesallthereflectionsrk,andistransitiveonthesetofalcoves.
1 ij
Proof. LetGdenotetheimageofψ . Thenwehaver1 ∈ G,andwealsohave
1 1s
r0 = r0 r0 ...r0 r0 r0 ...r0 r0 ∈ G.
1s (s−1)s (s−2)(s−1) 23 12 23 (s−2)(s−1) (s−1)s
Byrepeatedlycomposingr0 andr1 ,wefindthatrk ∈ Gforallk. Andnowwecanseethatrk ∈ G
1s 1s 1s ij
foralli,j,kwithi< jbydownwardsinductionon j−i: if j−i< s−1,thenwehaveeitheri> 1or
j < s. Ifi > 1,thenthenbyinductionrk ∈ G,andhencerk = r0 rk r0 ∈ G. On theother
(i−1)j ij (i−1)i (i−1)j (i−1)i
hand,if j< s,thenrk ∈ Gbyinduction,andhencerk = r0 rk r0 ∈ G.
i(j+1) ij j(j+1) i(j+1) j(j+1)
Toseethattheactionistransitiveonalcoves,wenotethatwecangetfromanyalcoveBtoany
otheralcove C by crossingsomefinitesequenceofhyperplanesin H. Applyingthereflectionsin
eachofthesehyperplanesinturntakesBtoC. (cid:3)
Since the fundamental alcove A clearly contains a unique s-point (namely the origin ⊙), we
see that each alcove contains exactly one s-point. This gives a useful one-to-one correspondence
betweens-pointsandalcoves.
Thet-coreofans-core 7
AsecondleveltactionofSˆ
s
Now we assume that s,t are coprime, and consider another level t action of Sˆ on the set of
s
s-points. Given an s-point p and given i ∈ {0,...,s−1}, let j,k be theunique elementsof {1,...,s}
suchthat
p ≡ (i−1)t, p ≡ it (mod s).
j k
Defineσ˜ (p)tobethepointobtainedbyreplacingp withp +t,andp withp −t. Clearly σ˜ (p)is
i j j k k i
ans-point,anditisroutinetoverifythatthemap
χ : σ 7−→ σ˜
t i i
definesanactionofSˆ onthesetofs-points. Givenanyσ ∈Sˆ ,wewriteσ˜ fortheimageofσunder
s s
χ ,iftisunderstood.
t
Note that the maps σ˜ are not isometries, and there is no natural way to extend them to give
an action on the whole of Ps. However, using the natural correspondence between alcoves and
s-points, we may abuse notation and regard χ as an action ofSˆ on the setof alcoves. Recalling
t s
thatAdenotesthealcovecontainingtheorigin(0,1,...,s−1),wethenhavethefollowinglemma,
whichiseasytocheck.
Lemma3.2.
1. Iftisapositive integercoprimetos,thentheactionsψ ,χ onthesetofalcovescommute.
t t
2. Ift= 1andi∈ {0,...,s−1},thenσˇ (A)= σ˜ (A).
i i
NowsaythattwoalcovesareadjacentifthereisonlyonehyperplaneinH separatingthem.
Corollary3.3.SupposeBisanalcove,andi ∈ {0,...,s−1},anddefineσ˜ usingthelevel1actionχ . Then
i 1
σ˜ (B)isadjacenttoB.
i
Proof. Writeσˇ fortheimageofσ ∈Sˆ underthelevel1actionψ . Sincethisactionistransitiveon
s 1
thesetofalcoves,wecanwriteB = σˇ(A)forsomeσ. Hence
σ˜ (B)= σ˜ (σˇ(A))
i i
= σˇ(σ˜ (A)) byLemma3.2(1)
i
= σˇ(σˇ (A)). byLemma3.2(2)
i
ClearlyAandσˇ (A)areadjacent,andsinceσˇ isanaffinetransformationofPs,itpreservesadjacency
i
ofalcoves. (cid:3)
Usingaverysimilarargument,onecanshowthatifpisans-pointandBthealcovecontaining
it,theneachalcoveadjacenttoBcontainsthepointσ˜ (p)forsomei.
i
3.3 s-sets
Defineans-settobeasetofsintegerswhicharepairwiseincongruentmodulosandwhosesum
s
is . Thereisans!-to-1mapfroms-pointstos-sets,givenbyforgettingtheorderofcoordinates.
2
W(cid:16)he(cid:17)nrestrictedtothesetofs-pointsinthedominantregion,thismapbecomesabijection. Given
thecorrespondencebetweens-pointsand alcoves, wehave an s!-to-1map fromthesetofalcoves
tothesetofalcovesinthedominantregion;thisisgivenby‘folding’Ps alongthehyperplanesH0
ij
foralli,j. Thisfoldingwillbeusefulinunderstandingsymmetrylater.
Note that our second action χ of Sˆ on the set of s-points descends to an action on s-sets
t s
(although the action ψ does not). We use the same notation χ (and σ˜) for this action on s-sets
t t
withoutfearofconfusion.
8 MatthewFayers
4 The t-core of an s-core
Now we come to themain part ofthe paper. We supposes,t are coprime integerswith s > 2,
and we compare the t-cores of different s-cores. By representing s-cores as s-points, we use the
geometricsymmetryofthelastsectiontoseethesymmetryoft-coresofs-cores.
4.1 s-cores ands-sets
Recallthatapartitionisans-coreifandonlyifinitss-runnerabacusdisplayeverybeadhasa
beadimmediatelyaboveit. Thismeansthatifwetakeans-coreλandthenforeachi = 0,...,s−1
definea to be thenumber ofthehighestunoccupied positionon runneri, then a ≡ i (mod s) for
i i
eachi,and
a +···+a = 0+1+2+···+s−1 = s .
0 s−1
2
(cid:16) (cid:17)
Hencetheset{a ,...,a }isans-set. Wedenotethiss-setQ(λ),andweletp bethecorresponding
0 s−1 λ
s-pointinthedominantregion,i.e.thepointwhosecoordinatesaretheelementsofQ(λ)arranged
inascendingorder.
For example, taking s = 5 and returning to the 5-core (5,2,2,1) from the example in §2.2, we
have
Q((5,2,2,1)) = {5,−4,2,−2,9}, p = (−4,−2,2,5,9).
(5,2,2,1)
Itiseasytocheckthatanys-setisobtainedfromans-coreinthisway: givenans-setQ,construct
an abacus display in which there is a bead at position b if and only if there is an element of Q
below bon thesame runner; thisis thentheabacus displayofan s-core λ, with Q(λ) = Q. Hence
wehaveanaturalbijectionbetweens-coresands-sets,andthereforebetweens-coresandalcoves
ffi
in thedominantregion. Usingthisbijection, wemay regardtheaction χ ofthea nesymmetric
t
groupons-setsasanactiononthesetofs-cores.
Example.InFigure1weillustratethebijectionbetween3-coresandalcovesinthedominantregion
ofP3,bydrawingtheYoungdiagramofa3-coreinsidethecorrespondingalcove. (Recallthatwe
usetheemptydiagramforthepartition(0,0,...).)
The aim of this paper is to compare the t-cores of different s-cores, when s and t are coprime
integers. Ifwetaket = 4,andexpandandredrawFigure1witheach3-corereplacedbyits4-core,
wegetthediagramonthefirstpageofthispaper.
Remark.In the case t = 1, there is a more familiar description of the action χ on s-cores; this is
1
described in [L, §4] and elsewhere. Given a partition λ, say that a box in [λ] is removable if it can
beremovedtoleavetheYoungdiagramofapartition(i.e.itconstitutesarim1-hook),whileabox
notin[λ]isanaddableboxofλifitcanbeaddedto[λ]togiveaYoungdiagram. Theresidueofthe
box(i,j)isdefinedtobetheresidueof j−imodulos.
Now when t = 1, the action of σ˜ on an s-core λ is given by adding all the addable boxes of
k
residue k to λ, or removing all the removable boxes of residue k (an s-core cannot have addable
and removable boxesofthesame residue). Intermsoftheabacus display forλ, thiscorresponds
tointerchangingthe(k−1)thandkthrunnersoftheabacus(withaslightmodificationinthecase
k = 0).
This interpretation can be generalised to t > 1, if one considers addable and removable rim
t-hooks,withasuitablenotionofresidue. Weleavetheinterestedreadertoworkoutthedetails.
Thet-coreofans-core 9
Figure1: thecorrespondencebetween3-coresanddominantalcovesinP3
4.2 s-cores havingthe samet-core
Incomparingthet-coresofdifferents-cores,thefollowingpropositionwillbecrucial.
Proposition 4.1.Suppose s,t are coprime positive integers. Suppose λ,µ are s-cores, and that there is a
bijection φ : Q(λ) → Q(µ) such that φ(b) ≡ b (mod t) for every b ∈ Q(λ). Then λ and µ have the same
t-core.
Proof. We use induction on the size of λ. If λ is not itself a t-core, then there is some b in the
beta-setforλsuchthatb−tisnotinthebeta-setforλ. IfweletabetheelementofQ(λ)whichis
congruent to b modulo s, then the element of Q(λ) congruent to b−t is a−t−ds for some d > 0.
Letλ− bethes-coredefinedby
Q(λ−)= Q(λ)∪{a−t,a−ds}\{a,a−t−ds}.
Thebeta-setforλ− isobtainedfromthebeta-setforλbyreplacing
a−s, a−2s, ..., a−ds
with
a−t−s, a−t−2s, ..., a−t−ds.
Henceλ−isobtainedfromλbyremovingdrimt-hooks. Soλ−issmallerthanλ,andhasthesame
t-core. λ− andµsatisfythehypothesesoftheproposition,sobyinductionλ− hasthesamet-core
asµ,andwearedone.
Sowemayassumethatλisat-core. Symmetrically,wemayassumeµisat-core,andwemust
showthatλ= µ. Inotherwords,weneedtoshowthatapartitionλwhichisbothans-coreanda
t-coreisdetermineduniquelybytheintegers
n = a ∈ Q(λ) a ≡ i(mod t)
i
(cid:12)(cid:12)(cid:12)(cid:12)n (cid:12)(cid:12)(cid:12) o(cid:12)(cid:12)(cid:12)(cid:12)
10 MatthewFayers
fori= 0,...,t−1.
Since s and t are coprime, we can write the elementsof Q(λ) as b ,...,b in such a way that
0 s−1
b ≡ −tj (mod s) for each j. The fact that λ is a t-core means that for every a in the beta-set for λ
j
we have a−t also in the beta-set for λ, and this gives b > b −t for each j (reading subscripts
j j−1
modulos). Soifwewriteb = b −t+m sforeach j,thenm ,...,m arenon-negativeintegers
j j−1 j 0 s−1
whichsumtot.
Nowconsiderthefollowingcyclicsequenceoflengths+t:
S= (b −t, b −t+s, b −t+2s, ..., b ,
0 0 0 1
b −t, b −t+s, b −t+2s, ..., b ,
1 1 1 2
.
.
.
b −t, b −t+s, b −t+2s, ..., b ).
s−1 s−1 s−1 0
ThestepsbetweenconsecutivetermsofSare either−t(s times)or +s(ttimes). Hencemodulot,
the steps are 0 or +s. Since s generates the cyclic group of integers modulo t, this means that S
containsatleastonetermofeachresidueclassmodulot. Moreover,thetermsinSchangemodulo
tonlyttimes,sothetermsinagivenresidueclassmodulotmustbeconsecutiveinS.
Claim. Foreachi,Scontainsatleastn +1termscongruenttoimodulot.
i
Proof. We have just seen that this is true if n = 0, so suppose n > 0. S contains all the n
i i i
elementsofQ(λ)congruenttoimodulot,andifweletadenotethesmallestoftheseintegers,
thenSalsocontainsa−t< Q(λ).
Since (n +1) = s+t, theremustbeexactlyn +1termsin Scongruenttoimodulotfor eachi.
i i i
AndnPowSisdetermineduptotranslationandcyclicpermutationbytheintegersni: startingfrom
the largest term divisible by t, S consists of n +1 terms divisible by t (with steps of −t between
0
them),andthenastepof+s,thenn +1termscongruenttosmodulot(withinterveningstepsof
s
−t),andthenastepof+s,andsoon.
Sotheintegersn determineSuptotranslationand cyclic permutation,andhencedetermine
i
s
Q(λ) up to translation. But the sumof the elementsof Q(λ) must be , so Q(λ), and hence λ, is
2
determined. (cid:16) (cid:17) (cid:3)
Usingthisproposition,weseethattheactionχ ofSˆ onthesetofs-corespreservesthet-core
t s
ofapartition.
Proposition4.2.Suppose sandt are coprime positive integers withs > 1, andfor i ∈ {0,...,s−1} write
σ˜ fortheimageofσ undertheactionχ . Ifλisans-core, thenλandσ˜ (λ)havethesamet-core.
i i t i
Proof. By definition, σ˜ does not change the multiset of residues modulo t of the elements of an
i
s-set. HencebyProposition4.1,itdoesnotchangethet-coreofthecorrespondings-core. (cid:3)
We shall refer to an orbit in the set of s-cores under the action χ as a level t orbit. From
t
Proposition4.2,weseethattwos-coreshavethesamet-coreiftheylieinthesameleveltorbit. In
fact, we shall see that they have the same t-core only if theylie in the same level t orbit; the way
wedo thisis toshowthateach leveltorbit contains at-core. Beforewe dothis, it will behelpful
tointroducesomemorenotation: fors,tcoprime,defineRs tobetheleveltrhomboid
t
Rts = p∈ Ps 1 6 pi+1 −pi 6 tfori= 1,...,s−1 .
n (cid:12) o
(cid:12)
(cid:12)