Lattimer subsets of Art IE ) field function for Fixed & sub - - PowerPoint PPT Presentation

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Lattimer subsets of Art IE ) field function for Fixed & sub - - PowerPoint PPT Presentation

May 20, 2023 295 likes •438 views

of Characters Independence fields Fixed of I : subgroups Lattimer subsets of Art IE ) field function for Fixed & sub extensions of E subsets et Aut ( E ) lie , inclusion reversing ) " contravariant " not too small ) (

slide-1
SLIDE 1

Independence

  • f

Characters

I :

Fixed

fields

  • f

subgroups

slide-2
SLIDE 2

Lattimer

Fixed

field

function

for

subsets of Art IE)

&

subextensions of E

subsets et Aut ( E)

→

" contravariant "

lie , inclusion reversing)

→ [ E

: EG ) 3141

(fixed feild, not too small)

necessary

tool

: if

Er . .

. ., rn) EAT LE) , then

99 .

. -son)

are

" E
  • independent
"
slide-3
SLIDE 3

NYn

Wut happens

when

we stty

fixed

fields

  • f

subgroups

→

can

we

say

more

about

IE

: EG) and

IGI ?

→

is

fixed field function

  • n

subgraph

" injective " ?
slide-4
SLIDE 4

theorem

( Bigness

matches

her subgroups)

Suppose

GE Atle )

. Then

IE : EG)

= IGI .

PI

Wc already

know

CE

: EG)

> IGI

.

For

contradiction

,

assume

CE : EG)

> 1Gt

.

lets

write

G

  • Er , .
. . , rn)

, so

1Gt - n

.

By

assumption

we

have

{ e , .

. ., Ent . ) E E

that

are

EG

  • independent
.
slide-5
SLIDE 5

we'll

create

a

linear system of equations :

L

=

Tile

;

) x , t

  • to

legit

,)xnt , = O

{

race ,)x , t

.
  • t

lentil xnt,

= 0

Since there

are

more

variables

than equations

in

. £ ,

we

know

he

has

a

non

  • trivial

solution

.

Let

r

be

The

minimal

number of

nonzero

elements

in

a

nontrvinl

solution

to

ye

.
slide-6
SLIDE 6

let's

create

a

" nice "

solution

to

L

with

r may

nonzero

components

.

By

rearranging

columns of Lo

, we

can

assume

we have

a

solution

  • f term

( x,,

.. . .,Xr , O,
  • -yo)

with all xito . Scaling by

Xi

'

still

produces

a

solution

( l, ya ,

. . ., yr , q .
  • , o )

with

all

gi to

.

w

Xzxi

'

Focus

  • n the

row

corresponding

to

idq

,

we see

slide-7
SLIDE 7

idle , ) Lt idler) yzt

  • t

id Cena) yay .

If

all

git EG

, This

violates

EG

  • indepef leg .>em!

So :

some

yi

has

yi

Ef EG

After

carrying

columns

,

can

assume

yr # EG

.

Hence , there

exists

some

ri EG

with

rilyr) t yr

.
slide-8
SLIDE 8

let

ri

act

  • n

each

row

  • f

L

to

create

a

new

system

  • il

grid

: {

tri ( T , (4) Xi

t

  • -

i.

to , lentil xnti )

=D

Oi ( Tale,)x, t

  • t Talent ,) Xna )

=D

= {

circle .)oi(x ,) t;

  • - trio , lent .) rilxnti)
=

Orion

(e.) ri (a) t

  • - t rirnlenti)rilxnti) =D

① (x . ,

. - . xut.)

solves L

iff

( rlx . ) .

. . .ir/xutiDsohesriL

② since

GEAVHE)

,

Erin ,

. - i , rion) :{r, , . . ., on)
slide-9
SLIDE 9

Hence

we

have

( x . .

. . , Xu , )

solves L

iff

(rilx .) ,

. . ., rilxnti))

solves

L

In

particular

( l , ya,

  • -syr,0 .
. -ie) - (rill) ,oily .),

rilyr) , rill, rill)

⇐ ( O , ya
  • oily ) ,
. . ., gr
  • oilyr), o ,
.
  • yo )
  • solves

L

.

to since

  • ilyr) # yr

It

has

too few

non

  • zero

terms

. →←

TIX

slide-10
SLIDE 10

Ouranginalypictre

G

HH et Aut ( E)

subextensions of E

subgroup'

properties

. contravariant
  • preserves
" bigness "
slide-11
SLIDE 11

Cos ( Fixed field function is injection

  • n subgroups)

If

it

, Ge Aut CE)

have

E

't = EG

, then H=G .

PI

claim

E

"

  • EG

gives

E " "

  • E

"

= E "

( you

can

prove This

as

"fun

exercise

")

So

:

htt = IE

: E

't)

= ( E : E

"")

> I Hv G)

So

GE H

.

Run the

same argument

but

reverse

the

roles of

H & G ,

we get

HEG

.

µ

slide-12
SLIDE 12

Ouronginalypictre

G

HH et Aut ( E)

subextensions of E

subgroup'

properties

. contravariant
  • preserves
" bigness "
  • injective