spannedlet美人2

Tightrelative2-designsontwoshellsinJohnsonassociationschemeYanZhuJointwithEiichiBannaiandEtsukoBannaiShanghaiJiaoTongUniversityMay24,2014AssociationschemeX=aniteset,{R0,R1,Rd}=thesetofrelationsonX(i.
e.
,RiX*X).
R0={(x,x)|x∈X}.
R0R1.
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Rd=X*X,andRiRj=ifi=j.
tRi=Rjforsomej∈{0,1,d},wheretRi={(y,x)|(x,y)∈Ri}.
Given(x,y)∈Rk,then|{z∈X|(x,z)∈Ri,(z,y)∈Rj}|=pki,j.
ThenX=(X,{Ri}0≤i≤d)isanassociationscheme.
Moreover,XissymmetriciftRi=Ri.
2/21AdjacencymatrixThei-thadjacencymatrixAiofXisdenedby(Ai)xy=1,if(x,y)∈Ri0,otherwiseA0=I.
A0+A1Ad=J.
tAi=Ajforsomej∈{0,1,d}.
AiAj=di=0pki,jAk=AjAi.
{A0,A1,Ad}formanalgebrawhichiscalledtheBose-Mesneralgebraoftheassociationscheme.
3/21MatrixversionSymmetricassociationscheme:X=(X,{Ri}i=0,.
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,d).
Adjacencymatrices:A0,Ad.
Primitiveidempotents:E0,Ed.
Bose-Mesneralgebra:C[A0,Ad]=C[E0,Ed]AiAj=di=0pki,jAkandEiEj=1|X|di=0qki,jEk.
Eigenmatrices:(A0,Ad)=(E0,Ed)P(E0,Ed)=1|X|(A0,Ad)Q4/21Denition1.
1LetXbeacollectionofd-elementsubsetsof[v]withd≤v2.
(x,y)∈Riif|x∩y|=di.
ThenX=(X,{Ri}0≤i≤d)isasymmetricassociationschemeofclassdandiscalledJohnsonassociationschemeJ(v,d).
u0∈X:axedpointarbitrarily.
Xi={x∈X|(u0,x)∈Ri},thenX0,X1,XdarecalledshellsofX.
F(X):thevectorspaceconsistsofalltherealvaluedfunctionsonX.
Lj(X):thesubspaceofF(X)spannedbyallthecolumnsofEj.
F(X)=L0(X)⊥L1(XLd(X).
5/21Denition1.
2[1]Let(Y,w)beaweightedsubsetofXwithpositivefunctionwonY.
(Y,w)iscalledarelativet-designwithrespecttou0ifthefollowingconditionholds.
pi=1x∈XriWri|Xri|f(x)=y∈Yw(y)f(y)foranyfunctionf∈L0(X)⊥L1(XLt(X),whereWri=y∈Yriw(y),i=1,2,p.
Let{r1,r2,rp}={r|XrY=}andS=Xr1Xr2.
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Xrp.
DenoteYri=YXri,i=1,2,p.
XiscalledaQ-polynomialschemewithrespecttoE0,E1,Ed,ifthereexistsomepolynomialsvi(x)ofdegreeisuchthatEi=vi(E1).
6/21Theorem1.
3[1]Let(Y,w)bearelative2e-designofaQ-polynomialscheme.
Thenthefollowinginequalityholds.
|Y|≥dim(L0(S)+L1(S)Le(S))()whereLj(S)={f|S,f∈Lj(X)},j=0,1,e.
Denition1.
4Ifequalityholdsin(),then(Y,w)iscalledatightrelative2e-designwithrespecttou0.
Theorem1.
5[2]XisaQ-polynomialschemeandG=Aut(X).
Let(Y,w)beatightrelative2e-designwithrespecttou0.
AssumethatthestabilizerGu0actstransitivelyoneveryshellXr,1≤r≤d.
ThenweightfunctionwisconstantoneachYri(1≤i≤p).
7/21Theorem2.
1TakeasequenceelementsfromXasu0={1,2,d},ui={1,2,d1,d+i+1},(1≤i≤vd1)ui={1,2,d,d+1}\{i(vd)+1},(vd≤i≤v1)i.
e.
,u1={1,2,d1,d+2}u2={1,2,d1,d+3}.
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uvd1={1,2,d1,v}uvd={2,3,d1,d,d+1}uvd+1={1,3,d1,d,d+1}.
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uv1={1,2,d1,d+1}Then{φ0|S,φ1|Sφv1|S}isabasisofL0(S)+L1(S),whereS=Xr1Xr2.
8/21Somenotationsφ0(x)=φ(0)u0(x)=|X|E0(x,u0)≡1,φi(x)=φ(1)ui(x)=|X|E1(x,ui).
Innerproductisdenedby=2i=1Wri|Xri|x∈Xrif(x)g(x).
d0=,c0=,for1≤i≤v1c1,5=,for1≤i≤v1c1,1=,for1≤i=j≤vd1c1,2=,forvd≤i=j≤v2c1,3=,for1≤i≤vd1c1,4=,forvd≤i≤v2c2=.
for1≤i≤vd1,vd≤j≤v29/21OrthonormalbasisGram-Schmidt'smethod:{φ1,φv1,φ0}→{1,2,v}.
1=φ1c0,i=1√Di1Di.
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φ1φ2.
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φiTheGramdeterminantDiisgivenbyDi=.
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10/21PropertyoforthonormalbasisMatrixHisindexedbyY*[v]whose(y,j)-entryisdenedbyw(y)j(y).
Then(tHH)j,k=δj,kand(HtH)x,y=δx,yimplyy∈Yw(y)j(y)k(y)=δj,kvj=1w(y)j(x)j(y)=δx,yx∈Xri(i=1,2),x={1,2,dri,d+1,d+2,d+ri}1wri=vs=12s(x).
(1)11/21x,y∈Xriand(x,y)∈Rαi,i=1,2.
x={1,2,dri,d+1,d+2,d+ri},y={1,2,ai,dri+1,2d2riai,d+1,2dαiai,d+ri+1,2ri+αi+ai},(d2ri≤ai≤dri).
vs=1s(x)s(y)=f(Wr1,Wr2,v,d,ri,αi,ai)g(Wr1,Wr2,v,d,ri,αi,ai).
(2)x∈Xr1,y∈Xr2,(x,y)∈Rγ,r1x={1,2,dr1,d+1,d+2,d+r1},y={1,2,a3,dr1+1,2dr1r2a3,d+1,2dγa3,d+r1+1,r1+r2+γ+a3},(dr1r2≤a3≤dr2).
vs=1s(x)s(y)=f(Wr1,Wr2,v,d,r1,r2,a3,γ)g(Wr1,Wr2,v,d,r1,r2,a3,γ).
(3)12/21Determineparametersetvs=1s(x)s(y)=0fordistinctx,y∈Y,i.
e.
,numeratorsof(2)and(3)haveacommonfactork1Wr1k2Wr2suchthatk2k1ispositive.
Step1:Givenv,d,r1,r2,solvetheequationsvs=1s(x)s(y)=0.
Ifthenumeratorofthesethreeexpressionshavesuchcommonfactor,thenkeeptheparametersv,d,r1,r2,α1,a1,α2,a2,γ,a3.
13/21Step2:Assumewr1=1,thenWr1=Nr1,Wr2=(vNr1)wr2.
1wr1=vs=12s(x)forx∈Xr1.
Weobtainwr2(Nr1).
Step3:Substitutealltheparametersaboveintovs=1s(x)s(y)=0forx,y∈Xr1.
SolveNr1(=|Yr1|)andkeeptheintegralsolutions.
14/21Listofpossibleparameters4≤v≤50vdr1r2α1a1α2a2γa3Nr1Nr2wr1wr212424304030102851663540240,14010611996852304113681328123956708324423281281284406024412321210129260602752336157109269049051521136157119112294333107361610169240803151231361610161054080315123115/21vdr1r2α1a1α2a2γa3Nr1Nr2wr1wr23912912931008036310940151015907091337711451281190390,190,13312145151015100901003965465451891811090120423674518918116701203873557451812181127010038721384518151890908042327355018161814290904289145020152012280100446325516/21Example2(16,6,2)design={v,d,r1,r2,Nr1,Nr2}={16,6,3,5,10,6}G=Z4*Z4BaseblockDandB={gD|g∈G}.
Du001230***1*2*3*=01230**1**2**3Xr1={gD|g∈G},whereG={(0,0),(0,1),(0,2),(1,1),(1,3),(2,2),(2,3),(1,0),(2,0),(3,0)}.
Xr2=B\Xr1.
17/21vdr1r2α1a1α2a2γa3Nr1wr1wr21663540240,1401013615710926904905151451281190390,190,133164281418160121601016010361642815211601216061607561962015191604160,1160,176110045222725120250182501845181361536196179024076919993621362301318024092182318/21FutureworkDoeseverytightrelative2-designontwoshellsinJ(v,d)havethestructureofcoherentcongurationArethereanysuchdesignswithnon-constantweight19/21ReferenceEi.
Bannai,Et.
Bannai,Remarksontheconceptsoftdesigns,J.
ApplMathComput.
40no.
1-2,(2012),195-207.
Ei.
Bannai,Et.
Bannai,Hi.
Bannai,Ontheexistenceoftightrelative2-designsonbinaryHammingassociationschemes,arXiv:1304.
5760Ei.
Bannai,Et.
Bannai,S.
Suda,H.
Tanaka,Onrelativet-designsinpolynomialassociationschemes,arXiv:1303.
7163Ei.
Bannai,Ta.
Ito,AlgebraiccombinatoricsI:Associationschemes,Benjamin/Cummings,MenloPark,CA,1984.
Th.
Beth,D.
Jungnickel,H.
Lenz,Designtheory,BibliographischesInstistu,1985.
20/21Thankyou!