FJV 17/09/2018 2018 Nha Trang . G- Tutte polynomial and Lie Abelian arrangements group ( Hokkaido . ) Masahiko Yoshinaga V ) ( Based joint work 04551 1707 arXiv : on . Tran Ye Liu and Tan what with .
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relationship Yet another points topology ↳ Counting Exampte ft I > o 4. skinny }¥%¥?¥ :} X ( 74gal 'll ± of 4*5=010/3×-15--01 's
relationship Yet another points topology ↳ Counting Exampte ft I > o 4. skinny }¥%¥?¥ :} X ( 74gal 'll ± of 4*5=010/3×-15--01 's
relationship Yet another points topology ↳ Counting Exampte ft I > o I ¥38 E- 38+2 if 8=1,5 }= { E 17%212/2=10 xtyto Hi 2) , mod 6 3×+2 modo # o q2 8=-3 8=2 -38+3 Y nod 6 ( 82-38+5 8=-6 's -=ol4xey=ofu/3×+y=o ga ) 'll ( " -14 mods mod 6
Iml ) Kamiya Terao Takemura - - forms linear integral For ( ) di lui n ) Aiplit i Gie Ke -4.2 ke t = . - . . . - ; . i. , , , tf 3- , felt ) Pso , It ) 2ft sit 1748241 ] forts E . , . . > o . , , # f } ) lxldilxtonodq fi CG ) where it Ecmodp ! = , " " - polynomial characteristic quasi .
& Background characteristic polynomials Tutte : .
Tutted Background characteristic polynomials : . Data El d Ln : E . . . . , , . A :={ Il , ten } in Li - - - . , . subs !YP of the rank The in } fir For SE [ n ] rank I lies ) di Rs := . . . . . , a homomorphism Il We Li -12 by consider as , , xi→j€9ijKj 2h ) di Li I a turd 2 Gie : = air → ii. . . . . , ,
& Background characteristic polynomials Tutte : . := f El A Ln } d. Li Ln E . . . - - . , , . , , . in } fir S For [ n ] rank I s ) E it di I Rs := : = . . . . , . ,€hij Kj 2h e ) Li di I a turd 2 at air hi : = → ii. . . . , . , , Det Till A ) polynomial of Tutte = §q % rs n ts - c) - 1) - I x - . ( y Ta x. y ) C : . A ) polynomial of characteristic ( 2 ) C f IYA .tl - rt . Tall Xa I t ) , O ) t : = - t ) # S ⇐ . te I rs - = - .
Tutted Background characteristic polynomials : . rank I lies ) :=fd El A Ln } di d. Rs := Ln E . . . - . . , , . . , , . ,¥hijKj 2h di ) Li I a turd at 2 Gie = air : ii. → . . . , . , , hrs )i=§g n rs - c) . 1) - ( x . ( y Talky - ra . Taft I ) # S filtrate ⇐ . te ) f- rs Xslt ,o)= - t - . Ext graphs ( Vi Finite ) E IV ④ I v . = . = { ' . } 2.3 VEV V z.to ' in For C vi. met e =/ I 1.23 - " , 12.331 - E 2 . C- , , de " V Vz E I - , ( Y ) K or - .
& Background characteristic polynomials Tutte : . Ext graphs C V. Finite IV ⑦ ) I E v = . . ' Vfb = { ' } 2.3 , .£• V ' For C vi. e WEE us - - =/ 11.23 , " , 12.331 E 2 C- , ( Y ) " I de K C- v u or - - , ( V. E ) chromatic of polynomial ) the Then Xslt is , of IV. E) polynomial Tutte the Tak . y ) is and . muumuu ↳ specializations many e. g .
& Background characteristic polynomials Tutte : . Ext graphs C V. Finite IV ④ ) I E v = . . ' V fu V =L } 2.3 ' , .£• ' For C vi. e WEE us - - = { 11.23 , " , 12.331 E 2 C- , ( Y ) " I de K C- v vz or - - , ( V. E ) chromatic of polynomial ) the Then Xslt is , of IV. E) polynomial Tutte the Take . y ) is and . muumuu ↳ specializations many e. g . . poly expectation of ch of random • . subgraphs . function of Partition Ising model • . of • alternating poly Bracket knots . . .
& Background characteristic polynomials Tutte : . rank I s ) lie :=fd El A Ln } di d. Rs := Ln E . . . - . . , , . . , , . , xi→j€9ij Kj 2h e ) Li di I a Emt 2 air hi : = → ii. . . . . , , x. y )i=§q . yn Hrs rs - c) I x - . ( y Ta l film . te - ra . Taft I ) # S ⇐ . te f- rs Xslt ) , o ) - t = - . Ge ) ( 2e④G± Let G be abelian an group . Ge di G G ④ : → , Get tf Ker ( di MIA G) , G ) : = .
& Background characteristic polynomials Tutte : . rank I s ) lie :=fd El A Ln } di d. Rs : Ln E = . . . - . . , , . . , , . , xi→j€9ij Kj 2h e ) Li di I a Emt 2 air hi : = → ii. . . . . , , x. y ) i=§q hrs n rs - c) . 1) ( x - . ( y Ta C filtrate - ra . Taft I ) # S ⇐ . te f- rs Xslt ) , o ) - t = - . Get II Ker ( di MIA Gl , G ) : = . ¥ er ) ( Bjorn colt ' GER Gore sky MacPherson - . , - ' le . Xa ( ) ¥ I - t ' ) Inca = - . T , Poincare ' poly which general Zaslausky 's Chamber ises . formula et ) ( counting and Solomon 's Orlik - ( ) result G=E c=2 , .
& Background characteristic polynomials Tutte : . rank I s ) :=fd e A it Ln } di d. Rs I : Ln EE = . . . - . . , , . . , , . , xi→,€9ij Kj 2h e ) Li di I a Emt 2 air hi : = → ii. . . . . , , = §g he rs n rs - c) - . 1) ( x - . ( y Ta x. y ) C : . te . Taft l ) # S - me ⇐ f IYA . te I rs Xslt ) , o ) - t : = = - - . Get tf Ker ( di MIA Gl , G ) : = . Mou ) Def ( I char Arithmetic Tutte poly L by . . . Tsa . ( x hits - Dh b £ , . I y Mls ) - 1) - y ) x. i = , # 1% b . miss .tl - IFS ' where Mls ) xajithit , ⇒ , Ifor C : = , .
& Background characteristic polynomials Tutte : . rank I s ) :=fd El A it Ln } di d. Rs I : Ln E = . . . - . . , , . . , , . ,€hij Kj 2h e ) Li di I a turd 2 at air hi : = → ii. . . . , . , , Tami 'T . ( x hits - Dh b £ , . ( y Mls ) - 1) - y ) x. : = , b g) . miss .tl # I % - IT'S ' where xajithitli-sc.EC Mls ) := for , Get II Ker ( di MIA Gl , G ) : = . De ) ( Procesi Moci Thin Concini - , . Xaarithf tie Inca ex , It ) I tt ) = - - . .
Tutted Background characteristic polynomials : . rank I lies ) :=fd El A Ln } di d. Rs := Ln E . . . - . . , , . . , , . , xi→j€9ijKj 2h ) Li di I a turd 2 air Gie : = → ii. . . . . , , Summary remarks and . A
Tutted Background characteristic polynomials : . rank I lies ) :=fd El A Ln } di d. Rs := Ln E . . . - . . , , . . , , . ,€hijKj 2h ) Li di I a turd 2 at air Gie : = → ii. . . . , . , , Summary remarks and . A { V . poly . ) char C arith " Xaaithftl-sfyhfts.mcss.tt
Tutted Background characteristic polynomials : . rank I lies ) :=fd El A Ln } di d. Rs := Ln E . . . - . . , , . . , , . , xi→,€9ijKj 2h ) Li di I a Emt 2 air Gie : = → ii. . . . . , , Summary remarks and . " " plexification G A my - § .G)=GelUker4i④G ) MIA . Poly . ) char C arith " Xaatithftl-sfghfts.mcss.tt
Tutted Background characteristic polynomials : . rank ( di lies ) :=fd El A Ln } d. Rs := Ln E . . . - . . , , . . , , . , xi→j€9ijKj :2l→2 ) Li di I a turd air Gie = ii. . . . . , , Summary remarks and . " " plexification G A my - § .G)=G4Uker4i④G MIA ) . poly . ) char C arith - t "lXaf e - fa , ) this , 4=1 . 1=1 Xaatithft )=€afy# s.mg , .tl rs - th fifty t ) ?xj" Inca - . ( Athanasia dis ) f. Xslt ) It ) -
Tutted Background characteristic polynomials : . rank ( di lies ) :=fd El A Ln } d. Rs := Ln E . . . - . . , , . . , , . ,€,9ijKj 2h ) Li di I a turd -32 at air Gie : = ii. . . . , . , , Summary remarks and . " " plexification G A my - § .G)=GelUker4i④G ) MIA . poly . ) char C arith them ④ " 't - t "lXsf e - fa , ) this , 4=1 Xaatithft )=€afy# s.mg , .tl rs - " "f¥y THX ; " ' A . . unify Xslt ) I Athanasia dis ) Aime f. It ) : -
I characteristic polynomial G- Tutte
I characteristic polynomial G- Tutte 2e ) finitely T Abelian ( e. g. generated F- : group A={ , dnt of d ? list elements in a . . , . , u A ( s > generated by S S ts rank subgroup of T : = , . G with finite Abelian Lie components : group an . e. g ' 7482 GE III. ? ! ! ' f- =p Ex S . - . ( , . , , , . ) :aY " genera :O " ' se GED ]i=fgtG/gd= It Ree " finite We for - Bo only need is de .
I characteristic polynomial G- Tutte 2e ) finitely T Abelian ( e. g. generated group F- : A =L di , dnt of ? list elements ( s > in for rs rank CA S a . . . . = , . ' )PxRExF where finite G- I s Abelian F is group = a , Det For SEA , # Hom I I Tks ) for It ) Mls , G) ' = , . Det C G . ) I char poly Tutte G - - . - b. " - if b Tat - i ) ' Eames I x ) His ( y . G) - - IFS . te ts Xatlt ) - MCs , G) Esq C . : = .
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