iii. if linear representation : : { symmetric { symmetric group - - PowerPoint PPT Presentation

SLIDE 1

Spin

characters

and

enumeration

• f

maps

÷:

"::*::

"

" exact

formulas

for

asymptotic

problems

"

SLIDE 2

linear spin

V

• linear

space

PCV )

• projective

space

linear representation

• iii. if

:÷÷÷÷÷÷÷÷÷:

{ symmetric

group

{

symmetric

group

projective

representations

• f

S

. =

linear

representations

• f

I

SLIDE 3

symmetric

group

Sn

symmetries

• f

Cnn )

• simplex

) " "

#

transpositions

t

. . . . .

.tn

• e

with

ti

• Ci

,

ite )

[ animated

contents ]

• )

f

.

transposition

ta

• ( 2,3 )

ti

• 1

I

SLIDE 4 ~

spin group

Sn

151=2

n !

rotations

• f

(

n

• t )
• simplex

ER

"

double

cover

• f

Sn

with

a

ribbon

attache

,

\

generate ,¥yangp

,

,↳

, ,

[ animated

#

tents ]

tr

. . . .

.tn

, l

€91800

T

I "

transposition

" ta

• ( 2,3 )

"

E.

imam

SLIDE 5 ~

spin group

Sn

151=2

n !

rotations

• f

(

n

• t )
• simplex

ER

"

double

cover

• f

Sn

with

a

ribbon

attache

,

\

generate ,#gang

,

,↳

, , ,

[ animated

#

tents ]

tr

, . . . , tn

• y

l l 360°

# go

!

ti

=

z

↳§

z

=

360°

twist

• f

the ribbon at

¥..

SLIDE 6 ~

spin group

Sn

151=2

n !

rotations

• f

(

n

• t )
• simplex

ER

"

double

cover

• f

Sn

with

a

ribbon

attache

,

\

generate ,.gg

, ,p ,

,↳

, , ,

[ clever

trick

to

untangle

the

ten

, . . . , tn

• y

↳

bbon ]

.

€97200

#

I

4

2 i

t

, =

z

=

1

a

µ

2=360

twist

• f

the ribbon

imma

SLIDE 7

• symmetric

spin

conjugacy

classes

interesting

conjugacy

classes

• f

the symmetric

group

Sn

• f

the

spin

group

I

are

indexed

by

are

indexed

by

in :÷÷¥

:*

, "

"

" ,

IT

, t

• t

Te

= n

It

. . . t

Te

= n

SLIDE 8

symmetric

spin irreducible representations

irreducible

representations

• f

Sn

• f

I

are

almost

"

indexed

by

a

:÷"÷÷÷!! .es/sn*e.....w;..ngn...ag;;e

,

+

An >

• He

¥¥

,

5ns

. "

> se

¥

SLIDE 9

symmetric

"fix

conjugacy

class

,

spin

normalized

irreducible

function

• n

ate

~

character

• f

Sn

Young

diagrams

"

Sn

O

if

nsk

chit

" " !

it

.

"

Chi

:

" =

gym

. .

.

.

.

.

.

. .

|g↳÷÷÷±⇒⇒

.

• tht

k

"

conjugacy

class

"

• tht

k

jvgacy

class

"

• dd

partition

SLIDE 10

for

any

• dd

partition

it

t

k

alternative

definition

Chin

is

the

unique

symmetric

function

FIX

. . . . . )

sit

. :

• F

E

E[

Pa

, Ps ,

ps

, . . . ] " F is

supersymmetric

"

• Ffg )
• O

for

5 .

>

• >

ese

1512k

• F

is

• f

degree

K

and

[ homogeneous

top

• degree

part ]

F

=

pit

SLIDE 11

alternative

definition

→

Hall

• Littlewood

symmetric

functions

for

E-

• 1

→

Schur

P

• functions ]

Pit

= ?

X' fit)

Pg

\

MAGIC

as:

" =

xena

. . . "

• 1-

I > f-

n

Young

diagram

[

I # k

k

"

conjugacy

class

"

SLIDE 12

problem

:

• ?

Chi:

"

lest EE

,

Nts

les )

s

%

SLIDE 13

problem

:

• ?

Chi:

"

lest EE

,

Nts

les )

es

graph

• n

a

surface

SLIDE 14

6

, =

( 1,5 , 4,2 )

( 3)

pair

• f

permutations

( 2,82 )

• V

W

IT

=

8

.

82=11,2

,

34,5 )

62=12,3 , 5) ( 1,4)

n

YE

,

t

Q

@l.q.z

g

bicolored

graph

G )

I*7

.

÷:* .

'

neater "

" "

with

face

• type

it

17

>

torus

SLIDE 15

problem

:

• "

number

• f

colorings

"

chits

Z

,

Nts

les )

s

%

SLIDE 16

6

, =

( 1,5 , 4,2 )

( 3 )

( frfr )

is

a

X

• coloring
• f

( 8,82 )

if

T

• f ,

maps

cycles

• f

8 ,

to columns

,

↳

=

( 2

, 3,5 ) ( 1,4)

• fz

maps

cycles

• f

82

to rows ,

FF

• tf

C .

• cycle
• f

Q

,

Cz

• cycle
• f

82

2. >

canoe

# 0

⇒

( file

. ) ,

• flat )e1

*¥⇐¥÷

iii :it

• f. IV)

flw )

SLIDE 17

6

, =

( 1,5 , 4,2 )

( 3 )

(f .,fz )

is

a

X

• coloring
• f G

if

T

T

• f ,

maps white vertices

to columns

,

↳

=

( 2

, 3,5 ) ( 1,4)

• fz

maps

black

vertices

to rows ,

§

• tf

w

• white

vertex

,

b

• black

vertex

y

y

w ,

b

connected ⇒

( file

. ) , flat )e1

• ••GZ

¥i¥

.

"

iiiii :* .

g

• >

few)

• f. (w)

SLIDE 18

symmetric

Stanley

formula for

partition

it

ltl=k

and

Young

diagram

J

Chala ) -52

th

"

No

. . "

( t )

Q , she Sk

• 8,8 .
• I

9

number

• f

X

• colorings

✓

identify partition

IT

• f

(4,82 )

" sum

• ver

with some

permutation

• riented

maps ITE

Sk

with

face

• type

it "

SLIDE 19

symmetric

spin

5

• ¥##

s

fit

f- 1-

Young diagram

1=13/5 )

shifted

Young

diagram }

double

SLIDE 20

spin

Wtic

Stanley

formula for

• dd

partition

it

lit

• k

and

shifted

Young

diagram

f

spin

1

Chile ) -52

z

th

"

No

. . . . ( Dls ))

• G

, ,8zE Sk h

Q8z=t

number

• f
• ly

Dlg )

• colorings
• f

14,82 )

" sum

• ver
• riented

maps

number

• f
• rbits

with

face

• type

IT " in

{ n

. . . . .kz under =

#

connected

action

• f

44,8

, >

components

• f

the map

SLIDE 21

two

equivalent

formulas

PREVIOUS

SLIDE

: n

• # white

vertices

citing ) -52

#!,

ta )

Ngl

Dls ))

• riented

2

components maps with

face

• type

IT

NEW

:

A

T n

• # white

vertices

citing >

=

• Z

ta )

Nixes

)

zeal

s

non

• riented

but

• rientable

maps

with

face

• type

it

SLIDE 22

proof part

1

"

linear

character

define

Tht ( s )

: =

f- Chit ( Dls ))

exported

to

spin world

" #

fun fact

• a. b ,

c , . . . C- {1. 3,5 . . . . }

symmetric

spin

Tha

=

Chs!

" "

split

to at most

two

groups

"

CT

. . =

Chs::

+

Chs:

"

.com:

Tha

, s .

Ch

"? .ec

+

Chinna

.

Ch

" a + +

Ch

': "

chat:

+

ch :

"

ca

.

SLIDE 23

proof

part

2

use

Mobius

invention

:

Hint

:

f

• 15

. ( 2K

• t ) ! !

for

k

= # blocks

China

" =

Cha

Chi:

=

Tha

. .

• Tha
• Eh

.

Chiari :c

=

Tha

. . . .

• Tha

Ths

. .

• Eh

.

Tha

. .

• Tha Tha

, s

t

3 Tha Ths

Thr

now

apply

symmetric

Stanley

formula

&

magic

cancellations

, ]

SLIDE 24

Homework

:

find

better

proof

"

" =?↳ .

n

th

"

Neatness )

8^82=15

Hint

:

Chi

" is

the

unique

symmetric

function

Flx

. . . . . )

st

. :

✓

• F

E

E[

pn

, Ps ,

ps

, . . . ] " F is

supersymmetric

"

7

.

X

.

Ffg )

• O

for

5 .

>

• > ese

1512k

✓

• F

is

• f

degree

K

and

[ homogeneous

top

• degree

part ]

F

=

pit

SLIDE 25

Application random

shifted

Young

diagrams

random

shifted

tableaux

SLIDE 26

psniady.impan.pl/ school

Summer School

• n

Algebraic

Combinatorics

Cracow

,

Poland

.

F

'

July

G

• 10

,

2020

• Valentin

Ferry

• Vic

Reiner

• Anne

Schilling