Stanley-hyperplane.pdf

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1、AnIntroductiontoHyperplaneArrangementsRichardP.StanleyContentsAnIntroductiontoHyperplaneArrangements1Lecture1.Basicdenitions,theintersectionposetandthecharacteristicpolynomial2Exercises12Lecture2.Propertiesoftheintersectionposetandgraphicalarrangements13

2、Exercises30Lecture3.Matroidsandgeometriclattices31Exercises39Lecture4.Brokencircuits,modularelements,andsupersolvability41Exercises58Lecture5.Finiteelds61Exercises81Bibliography893IAS/ParkCityMathematicsSeriesVolume00,0000AnIntroductiontoHyperplaneArrang

3、ementsRichardP.Stanley12R.STANLEY,HYPERPLANEARRANGEMENTSLECTURE1Basicdenitions,theintersectionposetandthecharacteristicpolynomial1.1.BasicdenitionsThefollowingnotationisusedthroughoutforcertainsetsofnumbers:NnonnegativeintegersPpositiveintegersZintegers

4、QrationalnumbersRrealnumbersR+positiverealnumbersCcomplexnumbers[m]thesetf1;2;:::;mgwhenm2NWealsowrite[tk](t)forthecoecientoftkinthepolynomialorpowerseries(t).Forinstance,[t2](1+t)4=6.AnitehyperplanearrangementAisanitesetofanehyperplanesinsomevector

5、spaceV=Kn,whereKisaeld.Wewillnotconsiderinnitehyperplanearrangementsorarrangementsofgeneralsubspacesorotherobjects(thoughtheyhavemanyinterestingproperties),sowewillsimplyusethetermarrangementforanitehyperplanearrangement.MostoftenwewilltakeK=R,butaswe

6、willseeevenifwe'reonlyinterestedinthiscaseitisusefultoconsiderothereldsaswell.Tomakesurethatthedenitionofahyperplanearrangementisclear,wedenealinearhyperplanetobean(n1)-dimensionalsubspaceHofV,i.e.,H=fv2V:
v=0g;whereisaxednonzerovectorinVand
vistheus

7、ualdotproduct:X(1;:::;n)
(v1;:::;vn)=ivi:AnanehyperplaneisatranslateJofalinearhyperplane,i.e.,J=fv2V:
v=ag;whereisaxednonzerovectorinVanda2K.IftheequationsofthehyperplanesofAaregivenbyL1(x)=a1;:::;Lm(x)=am,wherex=(x1;:::;xn)andeachLi(x)isahomogeneouslin

8、earform,thenwecallthepolynomialQA(x)=(L1(x)a1)


(Lm(x)am)thedeningpolynomialofA.Itisoftenconvenienttospecifyanarrangementbyitsdeningpolynomial.Forinstance,thearrangementAconsistingofthencoordinatehyperplaneshas