Abel群上Cayley圖的譜的綜述報告

Introduction

Abeliangroupsaremathematicalstructuresthatarecommonlyencounteredinvariousbranchesofmathematicssuchasnumbertheory,combinatorics,andalgebraicgeometry.Theyarealsoimportantinphysicsandengineering.ACayleygraphisavisualrepresentationofagroup,whereeachelementofthegroupisrepresentedbyavertexandtheedgesrepresentthegroupoperation.ThestudyofthespectrumofaCayleygraphofanAbeliangroupisimportantinseveralareasofmathematicsandphysics.Inthisreport,wewillprovideanoverviewoftheliteratureonthespectrumofCayleygraphsforAbeliangroups.

PropertiesofAbeliangroups

AnAbeliangroupisagroupthatsatisfiesthecommutativeproperty.Thatis,foranytwoelementsaandbofthegroup,a·b=b·a.Abeliangroupshaveseveralimportantpropertiesthatmaketheminterestingobjectsofstudy.Forexample,ifGisafiniteAbeliangroup,thenGcanbewrittenasadirectproductofcyclicgroups,thatis,G=C1×C2×...×CnwhereeachCiisacyclicgroupofprimepowerorder.ThisisknownasthefundamentaltheoremoffiniteAbeliangroups.AnotherimportantpropertyofAbeliangroupsisthattheyhaveaFouriertransform.

TheFouriertransformforAbeliangroups

TheFouriertransformforAbeliangroupsisageneralizationoftheFouriertransformforfunctionsontherealline.Itisalineartransformationthatmapselementsofthegrouptocomplexnumbers.LetGbeafiniteAbeliangroupandletxibeanelementofG.ThentheFouriertransformofxiisdefinedas

φ(xi)=∑x∈Gχ(xi,x)f(x),

wheref(x)isacomplex-valuedfunctiononG,andχ(xi,x)isacharacterofGdefinedas

χ(xi,x)=e^(2πi(xi,x)/|G|),

where(xi,x)istheinnerproductofxiandx,thatis,(xi,x)=Σi=1nxi_ix_i,and|G|istheorderofthegroup.

SpectrumofCayleygraphsforAbeliangroups

ACayleygraphisagraphthatisconstructedfromagroupGbychoosingasetofgeneratorsSforGandthendefiningedgesbetweenverticesgandgsforeachgeneratorsinSandeachelementginG.ThespectrumofaCayleygraphisthesetofeigenvaluesofitsadjacencymatrix.TheadjacencymatrixAofaCayleygraphforanAbeliangroupisadiagonalmatrixwhoseentriesaretheFouriertransformvaluesofthegroupelements.Thatis,thei-thdiagonalentryofAisgivenby

Aii=φ(xi).

TheremainingentriesofAaregivenby

Aij=φ(xi-xj)

fori≠j.

ThespectrumofCayleygraphsforAbeliangroupshasbeenstudiedextensivelyintheliterature.Oneofthemostimportantresultsisthefollowing:

Theorem.LetGbeafiniteAbeliangroupandletSbeageneratingsetforG.ThenthespectrumoftheCayleygraphΓ(G,S)isgivenby

Spec(Γ(G,S))={λ(ξ,S):ξ∈G},

whereλ(ξ,S)isthesumoftheFouriertransformvaluesoftheelementsinξS,thatis,

λ(ξ,S)=∑g∈ξSφ(g).

ThisresultshowsthatthespectrumofaCayleygraphforanAbeliangroupiscompletelydeterminedbytheFouriertransformofthegroupandthegeneratingset.Inparticular,thesizeofthespectrumisequaltotheorderofthegroup.

Applicationtoquantuminformationtheory

ThestudyofthespectrumofCayleygraphsforAbeliangroupshasseveralapplicationsinquantuminformationtheory.Onesuchapplicationisinthedetectionoferrorcorrectingcodes.Errorcorrectingcodesareusedtoprotectquantuminformationagainsterrorscausedbynoiseinquantumchannels.Theperformanceofanerrorcorrectingcodecanbecharacterizedbyitsdistance,whichistheminimumnumberoferrorsthatcanbedetectedandcorrected.Thedistanceofanerrorcorrectingcodeisrelatedtothespectrumofitsstabilizergroup,whichisanAbeliangroupthatencodestheinformationandtheerrorsthatcanoccur.ThestabilizercodecanbeencodedasaCayleygraph,andthespectrumoftheCayleygraphcanbeusedtocalculatethedistanceofthecode.

Conclusion

Insummary,thespectrumofCayleygraphsforAbeliangroupsisanimportanttopicofstudyinseveralareasofmathematicsandphysics.ThepropertiesofAbeliangroupsandtheFouriertransformprovideausefu