hpeybme for Epzo far you lady Let E = A (E) naturally graded : Definition We 2 ( et reap ) = Basics on 2 : E E Eo=R Et = Let A = {

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1 Ytolomm linen fr Bsics on Orbk lgebr Let { He Her } be centrl rrngement in vector you 1km Let R be commuttive ring We denote by Et +0 Re 2hm free R He module where elements lµ me in binge otion with hyeylnes Let E (E) be exterior lgebr tht heybme nturlly grded : We notice tht Et EoR E Ezo < lµ E 9 F iekn > R s 9+j > N / l ldy > R Definition We define R my 2 : E E by e +1 F Het for 22 2 ( et re ) jtmeµ e with Hn H

2 tyle Given or hyybms S ( H H ) we Kent 151 lee +j e + e E n S H n H Nottion Tf o n S ( ) es 1 n S Item gmen mby µ in Ikn we denote by r ( U ) COMMENSION U Definition We cll 5 molyenhnt if r( S )/S/ mhbembnt if rts Let denote set ll tybs let S hkos

3 We Definition Let t be rrngement s bove We denote by I I () idel E generted by elements des for ll deendent S E S centrl hgeybme We notice tht I hence it grded I # generted by homogeneous elements idel : In E Orhk Solomon introduced in 1980 following lgebr Definition Let he centrl hygeylne rrngement s bove define () Et We denote by y : E rotection we write y ( e* ) # if ( es ) y ( E ) g F H E F S E S

4 Exerce If S E S H E S we home lg l + d es We observe tht grded nti commuttive lgebr R +0 Rn Het only deendent elements des EE o or In E me s ( H 10 } H ) for HE From exerce we home tht 0 F > m once if > m every dimension mbient nectr Let L () he hyyey lnes in Eon xe Lot) let let oet tyle in S deendent n sce ll intersections ordered by reverse inclusion Sj E E R es Sts being { se S In sx

5 Lohmn We observe tht E TO Ex XEL Herein For X t L ( ) we ut g ( Ex ) Zhen TO w ( ) Renouf Zhe Orbit lgebr cn be defined lso for ffine hyeylne rrngements difference tht one tkes idel I des ( s olyenhnt) by es ( with One min generted by ns ) Hemet ( Orbk Ylmon 1980 ) Let be on vector n s V Let # ( met ) in hyeylome rrngement comte MC ) V H ) ± ft ) coefficients in R

6 Remmk_ th orem holds lso with integer coefficients : H* (MC ) 2) If t coefficients in RZ Remould Let be Eov every H e rrngement control let 2 such tht H { re V 1 n not > Mtt µ he functionl in V* [ ) + ( 40 } ifengie#fieieiif gives omorhm in Rhm cohomology Lee book Orlik Gmo

7 t He rnold s lgebr Let us consider brid rrngement Pm Tt rrngement m V hyeybms Hog { xu ( y me will Let n us defined in V ) x Y µ ) gwen by o } identify V with { ki ixm ) c / Exiif comlement Configurtions rrngement m C m ( ) { ( ixn ) c 4m / Exeo x f4 } In 1969 rnold nel tht cohomology lgebr H* ( cm ( ) ) reltions nmoyhic to with lgebr genertors sexi { y ne i < sg m } Ny Run c sk + n o F 1 E lese ken Remde it ctully ws ored for integer coefficients From now on we focus on comlex coefficients

8 rnold Igeh omorhic to ( m) 1 show Zhen From Orlvk Yslomn result it follows tht Zhe following exerces ro T comlex coefficients more on from bunny Let us consider brid rrngement Pm Let S ( H H ) be deendent if only if form ( Huy tyle it contins Hhs Ho s ) with 5 subsequence The lte F h1 s Ly ( i e sequence cycle ) bunny rnold reltions 0 cn be written s des o for ll S e deendent mininnyly delete n th mens tht if we hyeylme from trile wht remins indeendent Yxnuse3_ For brid rrngement Pm iobl I in E coincides with noted generted by des for every S e deendent

9 bs for ( Pm) us consider following my : it 13 zzi34 l l : 1! 1 Mm %m 3M lmtm 1 Unsung bns for } h Rich H mens tht born for ll monomils obtined ech row my ( Pm) lys / 2 ± Ty < < J { m bove ( m ) tking Jen tht one gmen by element rnold s reltions # one Using Exerce monomils bove re set genertors for ( Prn) from tht }

10 We notice tht on lgebr Sm ction tht ermutes induces in y ( Bn) re nturl ( NOTTION uy Rgi ) One yuties gols th th course to illustrte some interesting reresenttion Sm configurtion sys tht comes from geometry