417.se#esofknuth/PacticequivaeceGLl3) crystal 1130×1130113 : GL crystal 1130×1130×113

Yamanauchi words : 1,1 , I - 112-122--222Yamanashi words : -

-. 1,1

,I

• 1,1,I

" I→ 112- 122- 222 113- I 23 223

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t''

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I' ! ! ÷:;D.iii.

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t ."

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"ti "

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121- 221 § µit" tie. ." m, y

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¥ 12 441

µ442

331=332321y,

34332,44 ,421

321 It431¥32

↳ LH) crystal 1130×1130×1130×1133,2, 1,11-3212

Questions: Bla,I,1)

=

y r X.

T.int?i.9npiiiiicwoqdsu::fYttehse9iin?p.ne.ts ? 1 4211¥42 ' '

! 3213¢ !Yad :

O

÷:÷i÷÷s±¥*÷:i÷¥: .

"Ie Theorem 8. 6. 8.7 . gas -_ rw (PHD \t

324• 2,2

,1,1 Iko 2. 1,2, I - ° 4314

type type & Qcx)=Qcy) f- YEE .

• 3 3. 1,2 . '

\, Exempts :• " " ' t C = i. 3. 2,11-2321 D X -- 1432 . 47=4312

"ma , ¥724,1 Pas -- Eff QIN -- ¥07

3321

t.tt/intenstingged9wsti ¥, Oi ! , 2432 2) x ' --3432,0147=4323 ) '

a : let +⇒ y be Knoth equivalent Yamauchi Ix .

pay.PE?acxt--#words

, w/ igom of : component Component (Y)4421 34320

3431#\If ×

, y are k Knuth nieces apart , is the scene \ I -54432true for all pairs z

,lolz) ? [special case : K =D 4431

In example , is GCD always 2 Knuth moves

from x ? Can we deduce this from RSK ?

Maite

ThmI : PCH determines P lactic class,

rw ( PM) E X

• Thin 8.4• inductive insertion style argument

TIME: Qu) determines Connected Comp.• Raising flowering ops do it affect G Cx)• consider sig . rule us. Sch en stead insertion•

"

localizes " to d. stl argument-

Exercise : -Verify both t'

for small example : Compute ( Pix) , QCxDfor all elts ( perhaps in Parallel ? ]

Next time :- Ex 3.2 (Schur functions)- General facts abt

.

Schur functions- Section 8.4- More examples

8.4 : skew shapes true reading words,• and these give crystals too !

Skew shapes :

T E±µ=O.

A -

-

, Hal.ITJust like partitions

.ae/ueue:u=D,1=ffyfu=qfssYT3Hln) : I -92121 I -93231

wt -- (2,20) wt 'll, 1,2)

• Skew Tableau E> 3121 -92132

¥ wt :(2,1, D wt -- ( 1. 2,1 )

• Standard, semi standard E> 21313132

wt -- 11,12)

wt ( 2. 1,1 )

• Weight ofa tableau are> 31313232

wt ) II. s:3 'm> 41,1)'

wt -- 140,2) wt -192,2)

wt ) -- i. a:3 '

→ (2. o.O I ⇒ 3221

wt:( 1. 2,1)

Littlewood - Richardson Coefficients ExercisedFor n -- 3

,M

-

- O.A -

- ¥ , should be 213micame # ftw: IT!!In) with compote Bun teconfirudin books

rwlt) a Yaman ouch : word

2121 is Yuan .

3121 3221 3231T i

I'→ HEI -4ft

theorem : For fixed n,the

'

21312132 ✓

set B×µ=ssYItHn) is a %

crystal by identification w/ '

zig , ↳ 313W 3232O Qiistiiiisoberestal ? " →

RR ( Byu) EIB""m

CA : seems so,

not 3

explicitly addressed ) T

Ig : RRC ) = 2×010×2 I Exercise : should be a to 132,2

ME 4in Find Yamauchi words(confirmed in books

n xpin 4 letters from SSYT (Hn)

Prop#l0 For large enough n '. ".si/.3.i.2.l/Byn=OtBfitVwt : 2.2 2,1 . '

V

Hseconduseoflhel-Rcooffia.at# Example n=4,1=-0

" Recall " that ⑥ is the direct product FET→ HEcorresponding to dir

. prod . of root systems . X f¥¥we can Levi Branca a Guns crystalto a GLCK) x Glen -K) crystal :

-

z

-

Branch at r=2 Et't'

'

!!¥"kiHs"ktnroy Creme ⇒ arrows ) X

Ete. HEx. x.

Theorem 8.14 : for It,Kaen : EEF-5"E' "Y" I continued

Branch, n- r (By) =④ Bu Byu next- page

MEX X-

= H "Idea :n . .".....

"

w/ entries 1. . .

.

,r

IMIHVI -- K and you is remainderIlmen

,laser -r

u

F- (41,1 ) ,n=4 ,

GULD branch to Gus)xGLCD- -

Az A. Ht

Fft-1. Et

3By BE, =

ETw

HE also

* *

:÷: ' EE'

¥⇒By Bato =

,

Foot 're IET→ Etty .

Bo Bogo -- FEI -

Eko EFE*on 2 letters