Counting Gelfand - Gettin type string polytopes

Yunhyung Cho

( Sungkyunkwan Univ. )

2020. 6 . 6

Osaka Combinatorics seminar

Contents : I.

Gelfand - Gettin polytopes

I. String Polytopes

IN.

Indices of reduced words

1- main theorem

References : arXiv i 1904.00130

1912.00658

I Gelfand - Gettin polytope .

• Given D= Cd , ? - . . I Xn ) i Sequence of real numbers)

ncntc )considerI

IR with coordinates ( Xy ) : LEI , j

it ] Intl• ~

743 Defn.

O ,:= § C sci ; ) / Kith ;

E Kiis,

. o o Ki, ]

E Kiijti

←y742 222•

•o Kzinti - i

= di }Xi , Kz , Kz , IS called a Gelfand

/- Gettin polytope .

( n =3 )increasing

fromy

tight to left

bottom to top

Example a ( n=z ) D= ( i. o )

742=1 On = [ 0,1 ]for

Vl

Xi , I Xz ,= O

IT"

"*

• ( n =3 ) D= ( 2,0 ,.

0 )742

743=2 2On

VI VI

7422222=0 7422 O

VI VI VI 11

ku ? Xz ,

2kg,

= o K " ? O = OK "

743=2

• ( n =3 ) D= ( 2. o. -2 )

Vl

7422122=0VI VIXIIIXz ,

2kg,

= - 2

743=2

• ( m =3 ) D= ( 2. o. -2 )

Vl

742 I 122=0VI VI

Ku I Xz ,Z Kz ,

= - 2

←when a i regular .

• • # of inequalities = # facets of a ,

•

• \( o.o.

o ) = f saz-

- o } n { Xz ,= o } n { Xu = 242 }

••

n { 24 ,= Xz , }

Da ( 6 facets )

Interaction with geometry

• Consider Hm i set of Mxn Hermitian matrices

- r

= deal Lie alg . of Un )

,,

g ,

flag variety,

conj• U Cn) A Hn -

( Eco - adjoint action )⇒ An = Y Un ) . La

,I

, =/ !'

.

.

. !n)W/ di ? ooo IX n

.

O

I I

Well - known facts no L M , s n z s - . - Cnn,

Kz = ni - ni - i

Suppose d , = . . . = An,

7 An ,+ ,= . . - = Xn

,2 . - - 2 dnr

. , ,= . - i = dnr

" I th = U%ck , × . . . × Ock ,( dinner Ox = of - EKE .

= 2 - dim On )

hi ,I

pwj . embedding Ox ↳ IP" "

cpliicker embedding )

An ,Ki

-

Tn,

K2 # free variables = It ( m2 - EKE ) = dims ,

O

. - a O

G

dnr Kr

m - the

gprincipal minor mat

.

Zeal ,:-(iii ) Guillemin - Sternberg , an ,m I - the largest

Ioa : 0×-7 IR-

e. v. of

A ↳ § , CA) ) Aae; - is

* ZHAI.am , - i

= Xi

Theorem Im 8,

= On .

-

I central role in mirror symmetry / syrup . geom

of flag varieties.

I String polytopes

° It turned out that GC polytope is just one of certain special

polytopes called"

string polytopes"

.

← arise from representationtheory

of highest WE Utm - modules

deg nti

° Consider Sm ,i

symmetricgroup ( = Weyl group of Utne ) )

- S ,= ( 1,2 )

,- - -

,Sn = C ninth ) simple transposition

- word : arrangement of outs of { 1,2 ,- - - in } allowing repetition

( e.g .12321

,2254

,- - - )

-

- Each word assigns an ett of Sn+ ,( For WE Sna .

set

↳Siszszszs ,

Ew ] : set of words pre

- word is called"

reduced"

if it has - seating w )

minimumlength among EHS Tn Cw ].

Reduced words & wiring diagram• There is a natural way of visualizing ( reduced ) words :

I 2 - - - Mtl• Let Wo = ( ) Esme ,

Ntl n - . - I

↳ longest element (e -9 ' ( II? )= C 1.3 )

= ( I .27 C 2

- 3) C I. 2)↳

% = ( 1,2 ,,

, 3,2 , I,

- - -

,n

,n - I

,. . -

,I )

c , to Ssszs ,

C- [ Wo ]( = Sas , S2 )

/ 2 3 4 µcalled

"

wiring diagram

I prop o I is reduced 'off2

,

9=121321 each lie, Lj meet

3 at most once.

2

I

4 3 2 I ← one - line expression . EE Cwo ] iff linlgtoof Wo for Vitj .

T T T

I 2 3

Example 1=213213

Tits'

Thin Any two reduced words

I 2 3 4

representing the same ett are

related by

{2- move C commutation )

Sis = Ssi for ti - j I > I

3- move ( braid more )

4 3 2 I

Si Siti Si = Site Si Siti

121321 Using wiring diagram-

→ 212321rn

→ 21 323 I~

→ 213213 t

• String polytope is determined by % and a- Cd , ,- . .

,An ) E IRI

.

-

I dominant ett• Gelfand - Cette polytope

( in the pos . Weyl

&Cart . - - tan

,dat . . tan

,

. . .

,any

I Dq Cd )chamber I

w/ &= ( 1.2 , I,

- - - M ,- . -

, I ).

° All notions can be generalized to any Lie types .

* We will define two strongly rational convex cones

Cq : string cone

⇒ Oecd ) := CenterTo : a - come me

kidstring polytope .

String cone Cq Collect inequalities as follows i Denote by WE

step I : Let Week ) :"

oriented"

wiring diagram-

- wing diagram

St Li,

. - .

,Lk T C upward )

I 2 3 4 Lea,

- . - ,dn+ , I ( downward )

✓ In each Wyck ),

startsat K

Step 2 : Find a path T which La ends at Kel

✓

§"

S.t. µ,

orare avoided

4 3 2 I

""

"

:::÷:*:÷÷i÷ .

/ 2 3 4steps label each mode by ti C from top to bottom)

I

z v ( # nodes = # ti 's = I ninth )3

✓" I express V by Li

,-2 - . . → lis

"D ' ' Da

,Zo

T T

g3 node node

.{¥~e.¥life4¥13✓ 6

° If l , → I . →

+t

4 3 2 I to ta, to

~

t ,

J [" J '

k

27J ,→ - tk

Wgn ⇒ - tatty - tf ZoThen lol := I Signed tie

.

tri node"

step 3'

"

: label each chamber by Diof ,

( ti : top of the chamber Di )& lb ) Zo

runi String mega .

Set led : = E Di

Dz C region enclosed by r /<

related by uni nodular

Set LG ) Zo transformation,

I 2 3 4

I

2 v Dz-1134 = - titty -

tsttzttc-

te ,

= tz - tf

3in me

a= Dy Dz

✓4

D a5£}r Di 5-7 E tie - E tis whereDu treat TKEJ

6

4 3 2 I

Wqcz,T : set of nodes in Di lying on the

{ same column as ti

-

: otherwise.8

• ti : node variable

Di ..

chamber variable.

Example 9=4.2 .I )

an :-. fecnzo : a appeal

tz - t32o tho t , Io

or Ot or

Dz Zo DctDzIo Dz ? o

• # Gp paths = # facets of Cq.

o Cq contains at least # node - number of facets

Ai " ease . -next :*

'

e.

j'

: Smallest a " Lj Khj ' meet ' a.

A - cone Given A = Cdi,

- - - .hn ) E RIO,

Collect the following inefce :

• For each node tj , Lj = E Dk. i. ← t ,

is on the

Lj = IKLy - the column

KI -

I 2 3 4J

I •

2 t ,•

I• Le = Dit Dst Df

3 o

2o Lz = Dzt Ds

I o

4 . 3 2 I

T T T d - inequalities : lyEdi ; .

I 2 3

^ ' d 2 " 3Exactly # node number of a - Inequalities

⇐ :-. cone defined by a - Trefualities-

Smooth come.

String polytope Og CX) : = Ce n Cio.

* Origin : Lattice points in Oecd ) t? dual canonical basis of

U,

i ar.

Ulm - rep w/ high .

wt a.

-

T

Oef .vectors of

Example ( GC polytope ) 9=(1.2-1) dominant cuts.

a = ( 2.2 )

x+120

/

"

¥4Ck = f Dit D2 Zoo 13220 . Ds Zo } H tilts Zo.

X CE = f bit Ds S2,

17312,

Dz E 2. }

I Ite t ,

- tztztz S2

• ts E 2 )• °

tz - t 312

Ck = f DIED 220 ,Dzzo , pzzo } ← ,

tito⇐

× Ft "

+2=4tilt 320

. En'

= f Dit Ds S2,

DzE2,

Dae z. }

⇒+ "

it ti - tztztzsz 2-3 ← Hz- Xi tfex ,-6/-2*+8EX

III. ezKian× " "o

£320

cnn.mn -- c : is

m . ill '¥ .tl:7( Me , Miz ) = ( -1 o - I )

X , = - to - tze 4÷tE:III . ..

tz = - Xzt 4

Il Indices of %.

- a

Given EE Cw ! ']

,

⇒natural ways of producing new

-

T

longest ett in

SIIe [ win" '

],

or Ee Cw ! "I as follows

,-

-

I 2 3

45# I

2 I .

I •1=(2/3213)• 4 ^

21 3213• 3 A4FI 3

# . 3

2

break lines at •'

s 2

'4

35 4 2g 2 I5 4 3 2 I

/f→ ¥ You = (

zi44324)

prop . ( c.

- Kimi Lee - Park ) For any EE Curl " "

],

up to z - move,

.

we may rearrange numbersso that= → CEI

.EI - II

.

is again% n ( II n.nu ,- - .

,1. Lot ) -

=

==arednce#dDn i descending subword

-

→ ceiaaiiia 'and also

an ( Ea ,, .

Eta I-

An- a

Ex.

(3.l.tt?-43)We can :Dexof %

( 3,1 . 4.2,

I, 3,2 . 4

.

3.

1)

just &i :

H-index:

( 3,1 . 4.2,

I, 3,2 - 4

.

3.

I ( 213213 ):

:#apt =3

- D - index : I

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A&p

% Ind : [ won"

] → ( Ezo F is cared"

total index"

= I i→ binaryI rooted trees C labelled )I

.

Thu ( C.

- Kin - Lee ) Ind is Injective .

-

Thin C C.

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- -

DECA) I ACCT ) I = C dit . - tan,

- - -

,an )

- cviuvodular -

equiv .in Ind LE )-

TffF-

height - increasing path from root to some leaf-

where labels of all vertices are zero.

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