l’ansatz cellulaire (15) · 2017. 10. 6. · l’ansatz cellulaire (15) xgv labri, bordeaux 22...
Post on 01-Feb-2021
3 Views
Preview:
TRANSCRIPT
-
L’Ansatz cellulaire (15)
XGVLaBRI, Bordeaux
22 Juin 2012LaBRI
Petite école de combinatoire
Chapitre 5 Compléments, perspectives
-
Physique
UD = DU + IdWeyl-Heisenberg
commutationsréécritures
objets combinatoires
planarisés
représentationpar opérateurs
placements de tourspermutations
tableaux alternatifs
bijections
Q-tableauxex: ASM, FPL
pavages, 8-vertex
automates planaires
paires Tableaux Youngpermutations
histoires de Laguerre
histoiresde fichiers
algèbre quadratique Q
polynômesorthogonaux
?
RSK
DE = qED + E + DPASEP
"normal ordering"
planarisation
"L’Ansatz cellulaire"
-
Heisenbergoperators
U, D
UD = DU + I
(recalling)
-
+
UD YD
U XU
Y
U
Y
X
XD D
Y YX
X
UD = qDU + I
-
UD
-
UD
-
UD
-
UD
-
UD
-
UD
-
UDY
-
UDY
-
UDY
-
UDY
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDY
X
-
UDI
Ipermutation as a complete Q- tableau
-
permutation as a Q- tableau
-
another Q-tableau:Rothe diagram of a permutation
-
representation with operators U and D
UD●
●
●
●
●
●
Young lattice
addingor deleting
a cell ina Ferrersdiagram
-
The Robinson-Schensted correspondence between permutations and pair of (standard) Young tableaux with the same shape
-
how to guess such representation ?
-
data structure
integrated cost
Françon, Flajolet, Vuillemin 1980, .....
data structure history«histoires de fichiers», Françon 1976
-
Primitive operations for “dictionnaries” data structure:
add or delete any elements, asking questions (with positive or negative answer)
-
number of choices for eachprimitive operations
-
this corresponds to the n!“restricted Laguerre histories”
-
this valuation corresponds to the (n+1)!“enlarged Laguerre histories”
-
dictionaryand the PASEP algebra
DE = ED + E +D
-
E
D
-
this corresponds to the n!“restricted Laguerre histories”
this valuation corresponds to the (n+1)!“enlarged Laguerre histories”
-
K
S
S = +
-
A
K
A
S
SJ A
S
416978352
1 1 2
1 3 241 3 241 352416 352
416 7 352416 78352
1
2
34
416978352
5
6 78
D = A + J
D= A+ J
E = S + K
-
prority queuesand the Weyl-Heisenberg algebra
UD = DU + I
-
priority queue
Polya urn
-
A S - S A = I
-
Hermite
-
demulplication of the equationsin a quadratic algebra
-
U
D
X
Y
U D= D U+ Y X
-
UD
U
D
-
Operators U and D
UD●
●
●
●
●
●
Young lattice
addingor deleting
a cell ina Ferrersdiagram
-
1 2 3
1
2
3
1
2
1 2
4 2 1 5 3
-
demultiplication
positions
-
another demultiplicationof the algebra UD=DU+Id
-
(recalling bijection Hermite histories --involutions without fixed points )
-
Rooks placement
Involution with fixed points in 2 colors
-
many demultiplicationsof the algebra UD=DU+Id
-
demultiplication of the PASEP algebra
-
alternating sign matrices (ASM)
-
Alternatingsign
matrices
ASM
-
alternating sign matrices (ex-) conjectureMills, Robbins, Rumsey (1982)
-
BA
-
B
A
AʼBʼB
A
Bʼ
Aʼ AʼBʼ
A
B
A
BAʼ
BʼA Aʼ
B
+
+
Bʼ
“planarisation” of the “rewriting rules”
-
BA
-
BA
-
BA
-
BA
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
The 8-vertex algebra(or XYZ - algebra)
(or Z - algebra)
-
(XY)Z-tableauxand
B.A.BA configurations(or XYZ- configurations)
(see PEC5, 21 Oct 2011)
-
BA
Aʼ
Bʼ
-
demultiplication forthe algebra related to
XYZ configurations and ASM
-
BA
Aʼ
Bʼ
-
BA
Aʼ
Bʼ
-
ASM
-
a new beginning ....
-
The (happy) end of
the petite école
first day summer 2012
top related