Geometric Techniques in combinatorial Coxeter Catalan - Theo Douvropoulos de Theories Seminar 're des Groupies @ LAMFA
enumeration formulas Some : Thm [ Hurwitz 1892 ] • ⇐ Ltd , # { shortest festoniizatwignstiofro ;¥y :p ; ; ( 23 ) ( l 3) ( 123J , = t - n ) ( 123 }=nnyYIYi¥IYY . . . - generated Now coxeter element irreducible given , well in a an c , complex reflection W =h of with ord (c) ranrn group : , Thm [ Bess is 2006-2016 ] .tn } # { shortest reflection factorization c=t = " , , IWI
enumeration Some formulas : Thm[ 2016 ] intersection Given D. flat X . an # { , start .at#eiYIrIaetiyinoEtjotewifoyyx3=hdEd*H [ Nx :W× ] # L ]=>[ W=An ¥4 - I [ Basis Hurwitz : NY :
Combinator ics ( oxeter Catalan - ° ) A complex reflection W is subgroup of finite GLCVKGLN (e) group a " pseudo reflections generated by " that ti l :} the form of are % ) ¢[V]W:={ , .gg for root of some an . - Shephard . Todd ] o ) Invariant Theory Khevalley A finite subgroup WIGLCV ) a complex reflection is group , and only its invariant sub algebra if if EW } fcwtxkfcxj f ECKV ] fx fw EV : , a polynomial algebra . In fact is " WW , then ¢ a
- example An example and : non a G={ id G={ id via via > C- X. y ) > C- × ( X. y ) y ) ( x ,y ) - ,a4]~N - - , ,a4]7lR2 1 t.ie#i3i - " t.FI ( 3.1 ) . 1 fixed → - 1) C- 3 Invariant polynomials : , Invariant polynomials : ' fz=y ,=X f ftX2.f@F.X , 's degcfitdegcfiklcd * |f,.fz=f3=
Discriminant Hyper surfaces . . . : How does the act GIT Q the I ,= ( f , fn ) map on ... , reflecting hyper planes ? It A a hyper them together " glues in " : surface , discriminant called the . if lit =Ilµe cuts It and linear that In particular form is a , order the associated reflection the then of CH , f. " D: it G- invt a polynomial in the fits ) is is e. .
: The Swallow 's Tail . Discriminant Hyper surfaces . . ( 34 ) ( 14 ) ( 13 ) sina.EE?IIiiiiii#H " (B)
" The " most beautiful aesthetic World theory the in Rene Thom 's Catastrophe Theory Salvador Dali ' For , .
elements eh Coxeter Springer via 7mW C oxeterelts characterized by having are which lies eigenvector ' . hyper plane T refl an no on ' zni , with eigenvodue J MRI = h= , " " Proof : E 6 E E8 7
towards topological a ... construction the of coxeter element . . .
a topological Towards element construction Coxeter of a Steinberg 's Theorem Vee " Wr ' pIxµ : €€f Eff acts W Vre9 ✓ freely :=V\UH on e :YeIeInYY :p l##%#x " ⇒ a . IC B. ✓ # I :=C×i :×n ) > PCW ) 1- 'I>>W→1 BCWJ " - |P , t jiregj wiivreg ( a ( ( en HH a # ( ficx 't . ifncx 'D (W\v)Ien . finsenigtioimpaoiwuiirretatoiieforaii " a covering of p , which is map explicitly the fins given via =p(oµ , .
Topological element construction Coxeter of a Theorem Saito Vee " Bessis W - pIxµ - : > FCF ; ;fn ) . gen 'd⇐ W well Sith c. ✓ t÷## ✓ A is . the : www.etnttt?:steoEtioo.n . ;fn Where EECF , ] di MY ' . X ( × , ;×n ) , . . := . . |p , vEVre9 f Now that pick such : , =fmH=O Fat fn as =/ .it?IoeniiEtnt# ( f. ( Eb (W\HIen . " ;fc x-D , . . eknilhtt path BCH too : ,D . ✓ := " EY " " S :=p( BHDEBCWI ( (5) the Coxeter element is > It :=p( c :# UH )
Topological :* elements factorization Coxeter of s a path O :O '→y ' Pick Y in ¥ . xp \ ¥ Lift to path Do WW . in a " that " stays above " It . f¥¥hjlfg¥fyiI¥i¥¥iiiI¥¥i www.t#3YIFIYa . ✓ by ,×⇒= of ibiioo Define :* , ,
Topological elements factorization Coxeter of s We define the " reduced " label # map . ;Cr) rlblcys ( a := . , ¥µ¥Ytp€n§nIEfyYi:YiiIYYw¥¥wY that Notice : . Dj=S bcy.x.jubcy.x.fb.iq ✓ Lµ#µ€\µ\ C , .Cr= > C = ... . 0=3%1*051 rlbl well-defined ! ! is :* , ,
. Looijenga The Lyashko morphism We define the LL map : - { M configurations centered y - LL : f÷Hyht¥lf¥eff - . miiiiiiiintiiniettten .it#Ca.cys.....angDOliECCfi,ooo,fn 4yTfWl@HyLLiYnem-sEnnen_y-Cfi.fn.D Algebraically ' : ~ - { ertatsgstfnit . .+anys=o] ,}iasfsiyY1b?enM¥ Cfioifn fnn , ] .
rlbl the Properties LL & of maps : line The to transverse It Ly ally is for (y= . . The # m orphism finite LL is - map a . :c If " :i÷¥m¥iIi÷¥÷e¥¥i *f¥iig€lfEEfa only ) ) ) ✓ # . ;fn ¥Y×¥/↳# > ( ( f , , , ) , ( y ) a . o ; . § , LL compatible rlbl and . are : LLC g) ={ x , ;o;Xr } Maltais - with ni := It :L rlbl (g) by and is given ( Goo Cr ) eqn = ; , fnntazfn ' " lrccitni ' then an=0 too + . . E [ fbooo E , ] , fn Oli .
The Trivialization Theorem ( Bess is ) The map tPfEIhhFlEefyuxrenaniiametIYaYsIhisabijection.iefH@fx.l the numerological Depends on coincidence deg ( LD=IRedwl c) I : -
Factorization tf maker Y theses µ"*¥÷(f wn.ru#itve l*#ta*⇒±Ej "Ik¥Ik# . ;¥mfd , #
Primitive Factorization s lift to We the LL flat Z can map : any ( fnce 's ,{ ↳ # multi set Aft ZFCZ : E ;;ZA= , , decorated at FNCEI In coordinates : bra 'D } ) roots of LTCEK 's ]nr[E+b that [ t.f.cz # . + . , L linear t or : relation # u ;Zr ) ( b. ( E 's , CE 's ) l Zi ; b. . . , IT . ocrhs =hmr!=hdim?( dimzs ! , degLT= So , deglbd =h 2h . . counted factorization We [ Nwczs :Wz] have by over . , IFactwczjl.hn#dhzI So : Wz ] [ NWCZS
Towards the Trivialization Theorem uniform proof of a a configuration Pick e={ ;xw ] multiplicities with ni x , ; . . Compare : ' (6) t.mu/tyu,CL4 ' (e) lL[ )=[ nrlbl . degcu " ,Ck) G=( C i ; N compatible | with e . and : ✓ II I , lredwccisl I 1RedwkH= . " ,Ck) G=( C i ; compatible with e .
. sieving . phenomena Some ( CSP 's ) cyclic : Consider following action the reduced reflection factorization on : ' tat " o.tn > ( , ) , .tn ; ;tn ) 0 ' ) ( t ' ( :=ctni tn - : , , . How Old factorization fixed Q by ? : A : ¥1 are many , , fifty )|q=g . ]=e2n%h where - ' lol ,q ) Hilb ( LE
. sieving . phenomena Some ( CSP 's ) cyclic : the following compatibility The is reason of rlbl the action with scalar map a : . rlblcy ) rlblc }*y)=0 }dm!fn . , )=(}d' " ;fn . f. @ 2mi/h 't ( f 3*1=3 , ) & }= on , . ,
" ( E) ° ) For particular point configurations ' symmetric ( E the fiber LL , natural action Cd EE* carries cyclic of a group a . ' Cd ) deformation hand other the fiber On the always L[ is a , ' ( 0 part of its special fiber ' and retains I e± structure the of Lt . Hilbert ) ring the encodes D The of ( graded KCLECOD series its structure e* . .
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