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