G- Tutte polynomial and Lie Abelian arrangements group ( - - PowerPoint PPT Presentation

SLIDE 1

FJV

2018

Nha

Trang

.

17/09/2018

G-

Tutte

polynomial

and

Abelian

Lie

group

arrangements

Masahiko

Yoshinaga

( Hokkaido

V

. )

Based

• n

joint

work

(

arXiv

:

1707

.

04551

)

with

Ye

Liu

and Tan what

Tran

.

SLIDE 2

Yet

another

relationship

Counting points

↳

topology

SLIDE 3

Yet

another

relationship

Counting points

↳

topology

Exampte

ft

I

>

• 4.

skinny }¥%¥?¥

:}

X

(

74gal 'll

's

± of 4*5=010/3×-15--01

SLIDE 4

Yet

another

relationship

Counting points

↳

topology

Exampte

ft

I

>

• 4.

skinny }¥%¥?¥

:}

X

(

74gal 'll

's

± of 4*5=010/3×-15--01

SLIDE 5

Yet

another

relationship

Counting points

↳

topology

Exampte

ft

I

>

• Hi

2)

E 17%212/2=10

,

xtyto

E- 38+2

if

8=1,5

mod

6

{

3×+2

#

• modo

}=

q2

• 38+3

8=2

Y

nod 6

( "

ga ) 'll

's -=ol4xey=ofu/3×+y=o

I ¥38

• 14

8=-3

mods

( 82-38+5 8=-6

mod 6

SLIDE 6

Iml

Kamiya

• Takemura
• Terao

)

For

integral

linear

forms

di

lui

, . . . , ke

)

=

Aiplit

• .
• t

Gie

Ke

(

i

• 4.2

; . i. n ) ,

3-

Pso

,

tf

, It ) , . . . , felt ) E

2ft ]

sit

.

forts

>

• ,

# f

1748241

lxldilxtonodq

})

=

fi

CG )

,

where

it

Ecmodp !

"

characteristic

quasi

• polynomial

" .

SLIDE 7

Background

:

Tutte

&

characteristic

polynomials

.

SLIDE 8

Background

:

Tutted

characteristic

polynomials

.

Data

:

d

. , . . . ,

Ln

E

El

. in

Il

.

A :={

Li

,

• , ten }

. The

rank

• f

the

subs!YP For

SE

[

n ]

fir

. . . . in } ,

Rs

:=

rank I

di

lies

)

.

We

consider

Li

as

a homomorphism

Il

• 12

,

by

di

=

I a

ii.

air , . . . ,

Gie

)

turd

Li

:

2h

→

2

, xi→j€9ijKj .

SLIDE 9

Background

:

Tutte

&

characteristic

polynomials

. d. , . . . , Ln E

El

.

A

:= f Li ,

• .

,

Ln }

.

For

S E

[

n ] :=

fir

. . . . in } , Rs : =

rank I

di I it s ) .

di

= I a ii. air , . . . , hi e ) turd

Li

:

2h

→ 2 , at

,€hij Kj

.

Det

Till

Tutte

polynomial

• f

A )

Ta

C

• x. y )

: = §q

I x

• c)

%

• rs

. ( y

• 1)

n

• ts

.

( 2 )

C

characteristic

polynomial

• f

A )

Xa I t )

: =

f IYA .tl

• rt

. Tall

• t

, O) =

⇐

I

• t )

# S

. te

• rs

.

SLIDE 10

Background

:

Tutted

characteristic

polynomials

. d. , . . . , Ln E

El

.

A

:=fd

,

• .

. ,

Ln }

. Rs :=

rank I

di

lies

)

.

di

= I a ii. air , . . . , Gie

)

turd

Li

:

2h

→ 2 , at

,¥hijKj

.

Talky

)i=§g

( x

• c)

hrs

. ( y . 1) n

• rs

Xslt

)

filtrate

• ra

. Taft

• t

,o)=

⇐

f-

I ) # S . te

• rs

.

Ext

Finite

graphs

.

( Vi

E

)

IV

=

④

I

. v . VEV

z.to

. ,

V

= { ' '

2.3

}

in

For

e

• C vi.

met

,

C-

=/ I 1.23 , 12.331

E 2

"

de

V ,

• Vz

E

I

"

(

• r

K

• Y )

.

SLIDE 11

Background

:

Tutte

&

characteristic

polynomials

.

Ext

Finite

graphs

. C V. E )

IV

= ⑦

I

. v Vfb ' , .£• , V = { ' ' 2.3

}

us

For

e

• C vi.

WEE

, C- =/ 11.23 , 12.331 E 2 "

de

v ,

• u

(

• r

K

• Y )

C-

I

"

Then

Xslt

)

is

the

chromatic

polynomial

• f

( V. E )

,

and

Tak

. y )

is

the

Tutte

polynomial

• f
• IV. E)

. muumuu

↳

many specializations

• e. g

.

SLIDE 12

Background

:

Tutte

&

characteristic

polynomials

.

Ext

Finite

graphs

. C V. E )

IV

= ④

I

. v V fu ' , .£• , V =L ' ' 2.3

}

us

For

e

• C vi.

WEE

, C- = { 11.23 , 12.331 E 2 "

de

v ,

• vz

(

• r

K

• Y )

C-

I

"

Then

Xslt

)

is

the

chromatic

polynomial

• f

( V. E )

,

and

Take

. y )

is

the

Tutte

polynomial

• f
• IV. E)

. muumuu

↳

many specializations

• e. g

.

• expectation
• f

ch

. poly .

• f

random

subgraphs

.

• Partition

function

• f

Ising

model

• Bracket

poly

. of

alternating

knots

. . .

SLIDE 13

Background

:

Tutte

&

characteristic

polynomials

. d. , . . . , Ln E

El

.

A

:=fd

,

• .

. ,

Ln }

. Rs :=

rank I

di

lie

s ) .

di

= I a ii. air , . . . , hi e ) Emt

Li

:

2h

→ 2 , xi→j€9ij Kj .

Ta l

• x. y )i=§q

I x

• c)

Hrs

. ( y . yn

• rs

Xslt

)

film

. te

• ra

. Taft

• t

, o ) =

⇐

f-

I ) # S . te

• rs

.

Let G

be

an

abelian

group

.

( 2e④G±

Ge )

di

④

G

:

Ge

→

G

,

MIA

, G ) : =

Get tf

Ker ( di

G)

.

SLIDE 14

Background

:

Tutte

&

characteristic

polynomials

. d. , . . . , Ln E

El

.

A

:=fd

,

• .

. ,

Ln }

. Rs : =

rank I

di

lie

s ) .

di

= I a ii. air , . . . , hi e ) Emt

Li

:

2h

→ 2 , xi→j€9ij Kj .

Ta

C

• x. y ) i=§q

( x

• c)

hrs

. ( y . 1) n

• rs

Xslt

)

filtrate

• ra

. Taft

• t

, o ) =

⇐

f-

I ) # S . te

• rs

.

MIA

, G ) : =

Get II

Ker ( di

Gl

.

¥

.

(

Gore sky

• MacPherson

,

Bjorn

er )

GER

'

Inca

.

colt

)

=

I

• t

'

• ' le

. Xa (

• ¥

)

,

T

Poincare

'

poly

.

which

general

ises

Zaslausky 's

Chamber

counting

formula

(

et )

and

Orlik

• Solomon 's

result

(

c=2

,

G=E

)

.

SLIDE 15

Background

:

Tutte

&

characteristic

polynomials

. d. , . . . , Ln EE e .

A

:=fd

,

• .

. ,

Ln }

. Rs : =

rank I

di I it s ) .

di

= I a ii. air , . . . , hi e ) Emt

Li

:

2h

→ 2 , xi→,€9ij Kj .

Ta

C

• x. y )

: = §g

( x

• c)

he

• rs

. ( y . 1) n

• rs

Xslt

)

: =

f IYA

. te

• me

. Taft

• t

, o ) =

⇐

I

• l )

# S . te

• rs

.

MIA

, G ) : =

Get tf

Ker ( di

Gl

.

Def (

Arithmetic

Tutte

I

char

.

poly

.

by

L

.

Mou )

Tsa

x.

y )

i =

£ ,

Mls )

. ( x

• 1)

hits

. I y

• Dh
• b

,

xajithit ,

: =

⇒

C

• IFS

. miss .tl '

b

,

where

Mls )

# 1% , Ifor

.

SLIDE 16

Background

:

Tutte

&

characteristic

polynomials

. d. , . . . , Ln E

El

.

A

:=fd

,

• .

. ,

Ln }

. Rs : =

rank I

di I it s ) .

di

= I a ii. air , . . . , hi e ) turd

Li

:

2h

→ 2 , at

,€hij Kj

.

Tami 'T

x. y ) : =

£ ,

Mls )

. ( x

• 1)

hits . ( y

• Dh
• b

,

xajithitli-sc.EC

• IT'S

. miss .tl ' b ,

where

Mls )

:=

# I%

g)

for

MIA

, G ) : =

Get II

Ker ( di

Gl

.

Thin (

De

Concini

• Procesi

,

Moci

)

Inca

.

ex , It )

=

I

• tie

. Xaarithf

• tt )

.

SLIDE 17

Background

:

Tutted

characteristic

polynomials

. d. , . . . , Ln E

El

.

A

:=fd

,

• .

. ,

Ln }

. Rs :=

rank I

di

lies

)

.

di

= I a ii. air , . . . , Gie

)

turd

Li

:

2h

→ 2 , xi→j€9ijKj .

Summary

and remarks

.

A

SLIDE 18

Background

:

Tutted

characteristic

polynomials

. d. , . . . , Ln E

El

.

A

:=fd

,

• .

. ,

Ln }

. Rs :=

rank I

di

lies

)

.

di

= I a ii. air , . . . , Gie

)

turd

Li

:

2h

→ 2 , at

,€hijKj

.

Summary

and remarks

.

A

{

V

C

arith

. )

char

. poly

Xaaithftl-sfyhfts.mcss.tt

"

SLIDE 19

Background

:

Tutted

characteristic

polynomials

. d. , . . . , Ln E

El

.

A

:=fd

,

• .

. ,

Ln }

. Rs :=

rank I

di

lies

)

.

di

= I a ii. air , . . . , Gie

)

Emt

Li

:

2h

→ 2 , xi→,€9ijKj .

Summary

and remarks

.

A

my

"

G

• plexification

"

§

MIA

.G)=GelUker4i④G )

C

arith

. )

char

. Poly

Xaatithftl-sfghfts.mcss.tt

"

SLIDE 20

Background

:

Tutted

characteristic

polynomials

. d. , . . . , Ln E

El

.

A

:=fd

,

• .

. ,

Ln }

. Rs :=

rank ( di

lies

)

.

di

= I a ii. air , . . . , Gie

)

turd

Li

:2l→2

, xi→j€9ijKj .

Summary

and remarks

.

A

my

"

G

• plexification

"

§

MIA

.G)=G4Uker4i④G

)

C

arith

. )

char

. poly

e

Xaatithft )=€afy# s.mg, .tl

• rs

this

, 4=1

• t "lXaf
• fa , )

Inca

. 1=1

• t )?xj"

th fifty .

f.

It )

• Xslt )

( Athanasia dis)

SLIDE 21

Background

:

Tutted

characteristic

polynomials

. d. , . . . , Ln E

El

.

A

:=fd

,

• .

. ,

Ln }

. Rs :=

rank ( di

lies

)

.

di

= I a ii. air , . . . , Gie

)

turd

Li

:

2h

• 32

, at

,€,9ijKj

.

Summary

and remarks

.

A

my

"

G

• plexification

"

§

MIA

.G)=GelUker4i④G )

C

arith

. )

char

. poly

e

Xaatithft )=€afy# s.mg, .tl

• rs

this

, 4=1

• t "lXsf
• fa , )

Aime

:

unify

them ④

" ' A .

" 't

THX;

" "f¥y .

f.

It )

• Xslt ) I Athanasia dis)

SLIDE 22

G-

Tutte

I

characteristic

polynomial

SLIDE 23

G-

Tutte

I

characteristic

polynomial

T

:

finitely

generated

Abelian

group

( e. g.

F- 2e )

A={

d

, , . . . , dnt a

list

• f

elements

in

?

u

S

,

ts

: =

rank

( s >

A

subgroup of

T

generated by S

.

G

:

an

Abelian

Lie

group with finite

components

.

• e. g

.

f-

=p

' ,

S

' ,

Ex

,

7482

, .

• .

(

genera

"

"

GE ! ? !

III.

'

se

:aY

:O

. )

Ree

We

• nly

need

GED ]i=fgtG/gd= It

is

finite

for

"

de

• Bo

.

SLIDE 24

G-

Tutte

I

characteristic

polynomial

T

:

finitely

generated

Abelian

group

( e. g.

F- 2e ) A =L di , . . . , dnt a list

• f

elements

in

?

rs

. . = rank

( s >

for

S CA

G-

=

I s

' )PxRExF ,

where

F is a

finite

Abelian

group

Det

For

SEA

,

Mls

, G) ' =

# Hom I I Tks ) for

,

It )

.

Det

C G

• Tutte

I

G

• char

.

poly

.)

Tat

'

His

)

Eames

. G)

I x

• i )

"

• b.

( y

• if
• b

Xatlt )

: =

Esq

C

• IFS

.

MCs

, G) . te

• ts

.

SLIDE 25

G-

Tutte

I

characteristic

polynomial

T

:

finitely

generated

Abelian

group

( e. g.

F- 2e ) A = { di , . . . , dnt a list

• f

elements

in

?

rs

. . = rank

( s >

for

S CA

G-

=

I s

' )PxRExF ,

where

F is a

finite

Abelian

group Mls

, G) ' =

# Hom I I Tks ) for

,

It )

.

Tak

His

)

Eames

. G)

I x

• I )

"

• b.

ly

• IT

? Xilt

) : =

⇐

C

• IFS

.

mcs.GI.tl

• ts

.

Thm1_

Cl )

T 's

"

(

x.

2)

=

Talk

. 's )

(2)

Tas

'

ix.

y )

=

Ta

"

ix.

g)

= Taavi "

• ix. y )

.

proof

:

( I )

Mls

,

{

• } )

=/

.

(

z )

Use

the

fact

:

#

Houff

,

s

' ) =

#

F

for

a

finite

abelian

group

F

.

H

SLIDE 26

G-

Tutte

I

characteristic

polynomial

T

:

finitely

generated

Abelian

group

( e. g.

F- 2e ) A =L d , , . . . , dnt a list

• f

elements

in

?

rs

. . = rank

( s >

for

S CA

G-

=

I s

' )PxRExF ,

where

F is a

finite

Abelian

group Mls

, G) ' =

# Hom I I Tks ) for

,

It )

.

Tak

His

)

Eames

. G)

I x

• I )

"

• "

.

ly

• IT

?Xi

It ) : =

⇐

C

• ITS

. mcs.CH . te

• ts

.

Thm

Let

felt )

be

the

h

• th

constituent

• f

characteristic

quasi

• poly

.

Then

felt

)

=

XIA

"

It )

.

Furthermore

,

felt )

=

Xss

'

It )

= Xaatithft) .

SLIDE 27

G-

Tutte

I

characteristic

polynomial

T

:

finitely

generated

Abelian

group

( e. g.

F- 2e ) A =L di , . . . , dnt a list

• f

elements

in

?

ts

: = rank

( s >

for

S CA

G-

=

I s

' )PxRExF ,

where

F is a

finite

Abelian

group Mls

, G) ' =

# Hom I I Tks ) Hor

,

It )

.

Tak

IX. s )

Eames

. G)

I x

• I )

"

• b

.

ly

• IT

?Xi

It ) : =

⇐

C

• IFS

. mcs.CH . te

• ts

.

Det

( CT

• plexifi

cation )

Hai

. ↳ '

• = { 4

t Hom IT

, G) 14141=0 }

C Honk

,

G )

Mla

, G) ' . =

Hom I T

, G)

I

Hai

. a .

Rein

.

This

is

consistent

with

previous

Mld

• 9. Mla

. Et

SLIDE 28

G-

Tutte

I

characteristic

polynomial

T

:

finitely

generated

Abelian

group

. A = { d , , . . . , dnt a list in

?

ts

. . = rank

( s >

for

S CA .

G-

=

I s

' )P×RExF ,

where

F :

finite

Abelian

Mls

. G) ' =

# Hom ( I Tkssltor

, G ) .

Xilt

) : =

Esq

C

• IFS

. MCs . Gt . te

• ts

.

Hai

. ↳ '

• = {

4tHomH

, 4-1/4141=0 } C Honk , G)

MIA

, G) ' . =

Hom I T

, G)

I Hai

. a .

Iet

(

Deletion

• Contraction

)

man

in

, . . .

.am

. } . ,

mhm

,

Alan

: =

{ I

, .

• .

,

In

. , }

in

T

" :=

Than ,

.

Th

Xie

Itt

• Xa

It )

• XII

.

It )

.

M I

Alan

, G) =

MIA

. G ) w

Ml

Alda

.

G )

.

SLIDE 29

G-

Tutte

I

characteristic

polynomial

A={

d , , . . . , dnt a list in

? G

=

I s

' )PxRExF ,

where

F :

finite

Abelian

Mls

. G) ' =

# Hom ( I Tkssltor

, G ) .

Xilt

) :=

Esq

C

• IFS

.

mcs.GI.tl

• b

.

Hai

,e ,

{ 4EHomlt.tl/4ldit=o3cHom/T.G)MlAiG)i=HomlT

, G)

I Hai .ci

.

Attn

: =

{

di , . . . , dm

• I }

Attn : =

{ I

, " ' ,

In

. , } in T " it

Than ,

.

Th

Xiiltl

• Xa

It )

• Xian

Its .

egg,

GO

, P > 0

MIAKn.fi/=MlA.G)UMlA12n.G

?

⇐

I

Euler

characteristic

• f

MIA

,

G )

. )

IFI

,P=o

.

Let

g

:=

dime

,

f-

peg

)

.

Note

:

elmla.GL/--f-iTk.Xafc-ns.ecctF

.

proof

:

Induction

A

SLIDE 30

G-

Tutte

I

characteristic

polynomial

A={

d , , . . . , dnt a list in

? G

=

I s

' )PxRExF ,

where

F :

finite

Abelian

Mls

, G) ' =

# Hom ( I Tkssltor

,

t )

.

Xilt

) : =

Esq

C

• IFS

. MCs , GI . te

• b

.

Hai

. ↳ '

• = {

4tHomH

, G) 14141=0 } C Honk , G)

MIA

, G) ' . =

Hom I T

, G)

I Hai

. a .

Coe

.

( Euler

characteristic

• f

MIA

,

GI

. )

Let

g

: = dim

G

f-

pig

)

. e I MIA . G ) I

• f- IT

" . Xj ' I e

• ii.

eats )

.

Thiel

( Poincare

poly

. )

If

CT is

non

compact k⇒

8

so ) ,

Ema

. #

HI

=

I

• th

I "

. xftf-REI.tt ) .

Proof

:

Induction

t

Meyer

• Vietoris

arguments

11

SLIDE 31

G-

Tutte

I

characteristic

polynomial

Thiel

( Poincare

poly

. )

It

CT is

non

compact k⇒

8

so ) ,

Ema

. #

HI

=

l

• th

I "

. Xf ' f

• RELIC )

.

SLIDE 32

G-

Tutte

I

characteristic

polynomial

Thiel

( Poincare

poly

. )

It G

is

non

compact k⇒

8

so ) ,

Ema

. #

HI

=

l

• th

I "

. X act f

• RELIC )

.

Ree

Why

the

non

• compactness

is

necessary

?

→

The

fundamental

cycle

• bstructs

induction

.

SLIDE 33

G-

Tutte

I

characteristic

polynomial

Thiel

( Poincare

poly

. )

It G

is

non

compact k⇒

8

s

• )

,

Ema

. #

HI

=

I

• th

) "

. xactf-PEI.tt ) .

Ree

Why

the

non

• compactness

is

necessary

?

→

The

fundamental

cycle

• bstructs

induction

.

Prof

Let

M

be

a

connected

I

• riented

)

n

• dim

mfd

( 2M=0 ! Let

Pi

, ' . ' ,

Pes

C-

M

.

Then

*

hyp

. , . . .pe , It ) = {

& Mlt

)

+

hit

" '

it

M

is

non

apt

IMA

)

• tht

Ih

• 1) t

" '

if

M

is

Cpt

.

SLIDE 34

G-

Tutte

I

characteristic

polynomial

A={

d , , . . . , dnt a list in

? G

=

I s

' )PxRExF ,

where

F :

finite

Abelian

Mls

. G) ' =

# Hom ( I Tks ) Hor

, G ) .

Xilt

) : =

Esq

C

• IFS

. MCs , Gt . te

• b

.

Hai

. ↳ '

• = { 4

t Honk , 4-1/4141=0 } C Honk , G)

MIA

, G) ' . =

Hom I T

, G)

I Hai

. a .

Thiel

( Poincare poly

. )

If G

is non

compact k⇒

8 s

• )

,

Ema

. #

HI

=

I

• th

I "

. X act f- RELIC ) .

Cole

Let

felt

) be

the most

degenerate

constituent

• f

char

.

quasi

• poly

.

Then

Inca

,

ex

,

Itt

I

• tf

. tf

• Htt)

.

More

generally

,

Inca

,

sirs

,

Itt

I

• t 'll .fr/-tf# )

.