Applications of geometric techniques Coxeter combinator Catalan in ics . Theo Douvropoulos Thesis Defense Minnesota University of
Some enumeration formulas : lay • |YsYdYIYhY Thm I Hurwitz , . , ( 23 ) ( l 3) ( 123J = - # { of | Redd shortest transpositions } factorization - cycle c) I " " of := s an n " = . .tn ti wl ti . . , . n ) ( 123 . . . - generated Now coxeter element irreducible given , well in a an c , complex reflection W with ord =h of (c) group ranrn : , Thm [ Bess is 2006-2016 ] n ¥ T . .tn } # { shortest I Redwccsl reflection factorization c=ti := = ,
c=w win Some # a enumeration formulas : Thm[ 2016 ] intersection Given D flat # Z . an . , NWCH # { shortest I Factwttt factorization Z } trite l= dim Z . := , , wit reflections & the . orbit of W ; heh [ :Wz ] L ]=>[ W=An ¥4 - I [ Basis nm ) Hurwitz : : ( h=n )
" mnultiplicity The simplest way to of prove [ Hurwitz and ] Cagley perhaps to count the is these the quasi . homogeneous of . Looijengoc map Lyashro - V. 1. Arnot 'd . X-ray a polynomial map of :E→ E p : theorems al - @ of @ Ei¥IEnt÷F*i⇐E7 @ " around " monodromy ( 134 ) ( 4 D : . ( 134 ) ( z } ) ( 1234 ) " monodromy = " around ( 11 ( 23 ) (4) : D
Is there construction ? inverse an Riemann 's Theorem Existence : # { ( lengthy polynomial { { ... ,w . } Ee Ese . } p maps : w .my#aEoHtnWitItIeID } , , p¥oY¥I :* critical ( values p - before ) , monodromy as of P # So Polynomials degree of with |Redq( of I c) = n , , fixed - I distinct critical values n , .
: The Deft . Looijenga sends Lyashro CLL ) morphism a polynomial E Blyn Zhtazznt to its p = not an + multi set values { www.wr ] of critical . = # ( where critical multcwil ) pts ( counted multiplicity of with zj ) . th pczjkwi s . I Redqcdl Riemann 's Theorem Existence of fiber size generic ⇒ = of LL
Presentation Coordinate for LL Target Domain Multiset critical values of p= Etaizmtooiancpolyn { wbuoswn . , } p r ✓ . ( 012,013 an ) " " EE polynomial + WD ( t ( ttwn . , ) n ; . , . . tmtb tmtu but = . o , q . ;bn ( b , , ) em E . , . . ;Wn bi=C . D i. ; ( where w Ci e. , , . polynomial ) ( ith elementary the Is symm
Geometry the LL of map LL > ( b , ( ED as , CED ) an ) ( , bn : Ohio := ; - on . , AL LL ' en . : , Properties ÷ L ° polynomial is a map - homogeneous b) LL weights with is quasi : " ;h) . ;bn & - Dn ) ( Qzu an ) ( 2,3 , ( b . , )< ( n ,2n , , Cn ← → , ; - u on , c) finite LL morphism is a
Geometry the LL of map b) Domain : Consider the scalar action Pdyn : on ftp.T.p ) HEE coordinates : y*P=Yu( in ton ) zntazzn 't . - = 42-5+047242-52 + and " Too . set zifz ' jn2+ : + aid ' " =( zgn @ and + ... " ) . ;an)=( i. * ( ago aid ?o Y ; and C. : .
Geometry the LL of map Target b) : . . . The critical values of y*p⇒?p : { w , ; ;wr]→{ ; Now . } How , ;o . In coordinates : . ;D "wr ) biG*p=CiC PW , ; . C :C www.wrj-yni.bi Mi =L . " bn , )=4" boo ;D h*Cb , ;o;bn ' . , ) So " . , #
Geometry the LL of map a) Polynomial ity : critical value is has W if pets a w - double root That w ) Discz ( pots =O if is a - . , root Disczcpczs - t ) of is ⇒ W a " has ' (8) ( only c) ' critical Finiteness =5 all L[ z < : > = values equal to 0 ) Bezout 's thm Consequence via : descutwwttkw.tk#aInnnjEnneFfI=nni
Basics reflection of complex groups VIE " Wr ' . generated , complex pTKµ Wis well a µ##y reflection acting Vic " group on if W=< two GLCVI ;tn>< with Gkvtstiafl ,g ] ¥ E ( x , . ) µ ,× := ... ... t " pseudo . reflection ) y=| Ctiisa . ifncx 'D (W\YIen ( ficx 't . µ > V Wr Wr E[ v ] > ⇒ via w*f:=fCw .v ) ' ' _\ art :={ :¥tf¥w } fears :=e(oµ ,
Basics reflection of complex groups " V±E Wr . Todd ' µ . Chevalley pIxµ Shephard : E[V]W=E[f #x## . ;fn] , ; fundamentatnvariants ' ' " di :=degCfis We write I :=C×s :×n ) and order them µ d. Edztoootdn " t For Wweltgen 'd ,h:=dn is ( ficx 't (W\v)Ien 'D ;fat . . number the Gxeter . Significance : Implies that the fibers the =\ of map / > CFCXJ . ;fnCx→j ) ;Xnk p :C × ; - . , It :=e( UHI - orbits > precisely the W are .
Basics reflection of complex groups VIE " Wr > Eff PTKH Steinberg 's Theorem : 's covering map . # HPCWKSBCWK |p f >W→1 .ae?tri.snIYIdio:tohe@@y FX.it#I:EtsvfeeYwoiIFImtT(8).vXs:=Cx......xnj niviires ) n.ciimvreg.it ,( ( f. C Eb ( wwten enytj 'D ;fat . . Huili a covering p , which is map explicitly the fits given via =p(uµ , .
their geometric Coxeter elements and factorization µ÷Y/ ' Bessis Saito theorem " VIE W 's pIxµ : Wiswell . yen 'dc FCF ; ogfn ) fFj¥ c. ✓ sth mv ⇒ .it?IoeniEtnt_# . ;fn Qicccfso , ] . ) eanon.IE?aYfftI..+an.Xs:=CX.,o...xn |p picrvevressth Now f , "=fniH=O fat f. as =/ . ( f. ( I 's Wwtren x-D ;fc , ( . . eknilhtt path , BCH too : ,D . ✓ := "fIY " at S :=p( BHDEBCWI / isthecoxeter (5) element c It :=p( :# UH )
their geometric Coxeter elements and factorization a path O :O '→y ' Pick Y in ¥ . xp \ ¥ Lift to path Do WW . in a " that above " stays " It . MAIL.tn#tEfyI bcy.x.jo?eobiub5o#*l :¥F¥a¥I¥÷¥xi¥ ✓ Define . ×#}" BEFIT "a :* , ,
their geometric Coxeter elements and factorization " reduced ¥ define We the - ×*\ maprlbl " label : - . LyµLo→\µ Pt#sh#lfI§fI¥±I÷÷⇐¥±¥ bcy.x.j.i.by/.fPoSyoooaS Cioo .Cw=( So ) lrlblis t.FI#Eeaseianm well-defined !
- Looijenga ( LL ) morphism The Lyashro We define the LL : map > { LL :* : :[ Y in : - . *fftEs#¥h÷fynieimi cenotfernedpoinonsfipnurngtionsf multi set nft ↳ > y - T.tl#/yHfiifntrftn+aIFEI+a.w=d a ⇐ - ÷ ¥ . ;fn ( y ) ) ( f , , > ( Oh ( y ) , ) - y= an . oo . , , It :L by is given eqn fnntazfn ' " ' an=0 Too + ° E [ fbooo E , ] , fn Oli .
✓ (y= Properties LL and rlbl of the maps : line to The It transverse Ly ally is for . . The in m orphism finite LL is - map a . :c If " :t÷¥ae¥Ii÷¥÷¥a¥ *f#itg€lfEEfa only ) ) ) # ¥Y×¥/t# . ;fn ( f , , > ( , ) , ( y ) a o ; . . § , LL rlbl compatible and . are : - Llc g) ={ . ;Xr } Maltais with ni x , ; := It :L rlbl (g) by and ( Goo Cr ) is given eqn = ; , fnntazfnn ' then lrccitni an=0 too + . . E E [ fbooo , ] , fn Oli .
is ] The [ trivialization theorem Bess . * III @ Estimated :n . Methylene Redwcdl inaction . ¥ ## Lyf//Lojµ the numerological ! ! Depends on : degCL4= coincidence 1
Factorization tire s Egg w " Eitan III. :# ⇒ Y theater l*¥1i¥⇒ =±,aPnmi µ ( f , ;o;fn . i. F) , #
Factorization Primitive s lift We to the LL flat Z can map : any # ↳ multi set Aft ZFCZ : E ;;ZA= , , decorated at FNCEI In coordinates : bra 'D } ) roots of , { [ LTCEK ( fnce 's 's ]nr[E+b that t.f.cz # . + oo , L linear t or : relation # ( b. ( E 's , CE 's ) 1 Zi ;Zr ) ; br . . , ET . ocrhs =hmr!=hdim?( dimzs ! , degLT= So , deglbd =h 2h . . counted factorization [ Nwczs :Wz] We have by over .hn#dimzI . , Factwczjl So [ NWCZS : Wz ]
Towards the Trivialization Theorem proof of uniform a Picr configuration e={ := Maltais multiplicities ;Xr ] wl x ni a . . , , LL ) =L Compare : ' (6) t.mu/tycaCLL ) ← ' Arlbl I L[ (e) ' ; C k ) ( a 6 = . . , compatible with e and : K I Redwccisl I Redwcdl It - 1 " = ←¥,⇐ , , compatible with e
Thank ! you
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