countingsv2-mat.ist.osaka-u.ac.jp/~higashitani/string_polytopes...2020/06/06 · tits ' thin...
TRANSCRIPT
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Counting Gelfand - Gettin type string polytopes
Yunhyung Cho
( Sungkyunkwan Univ. )
2020. 6 . 6
Osaka Combinatorics seminar
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Contents : I.
Gelfand - Gettin polytopes
I. String Polytopes
IN.
Indices of reduced words
1- main theorem
References : arXiv i 1904.00130
1912.00658
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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
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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
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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 )
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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 )
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An ,Ki
-
Tn,
K2 # free variables = It ( m2 - EKE ) = dims ,
O
. - a O
G
dnr Kr
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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.
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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 ].
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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
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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
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• 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 .
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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÷ .
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/ 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,
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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.
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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.
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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.
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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
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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
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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)
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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
34 I 2 3 I - A - Index :L
=231£30
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C
2112137D
2=12• en EE [ win " '
] → Cockle Cui'sD
:
.
→I
✓ D- index
y231231 I A
A- indexI
A.b
.
↳I
~- J O o
z Cpl 'd) o
CACHE Civil] → z④±= CEI . Iota )
as a, QQ QQ
D A LOf ①O o AID
.
:I
①c
O O O ofD - index of Cocoa
Get tree HID If- OOf D - index of %
A&p
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% 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.
- 4. kin - Lee - Park ) for any DE CIRZOT,
- -
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.
7hm ( Co - J.
kin - Lee ) # of GC type I 's are I-
!-
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A
%= (
€232) → I
/I
#*
①¥ 0%44 # acey
X '31¥)
Dio DioDio
I¥71
121. - 121=95%3- I -
12312, LA :30gal = GC
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