algèbres d’opérateurs et physique combinatoire · xavier viennot cnrs, labri, bordeaux...
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Xavier ViennotCNRS, LaBRI, Bordeaux
Algèbres d’opérateurset
Physique combinatoire
12 Avril 2012colloquium de l’IMJ
Institut mathématiques de Jussieu
(part 2)
guided construction of a bijectionfrom a combinatorial representation
of the algebra Q
The cellular Ansatzsecond part:
The RSK correspondence
G. de B. Robinson, 1938C. Schensted, 1961
Robinson-Schensted-Knuth
The algebra UD=DU+Id
The Robinson-Schensted correspondence between permutations and pair of (standard) Young tableaux with the same shape
RSK with Schensted’s insertions
representation of the operators U, D
Sergey Fomin(with C. K.)
Operators U and D
UD●
●
●
●
●
●
Young lattice
addingor deleting
a cell ina Ferrersdiagram
U = + +
D += . .
UD = DU + I
●
●
●
I
●
●
●
●
●
combinatorial “representation” of the commutation relation UD = DU + I
U = + +
D +=
UD = + ++ + +
U = + +
D +=
UD
DU
=
=
+ ++
++ +
+ +
+
U = + +
D +=
UD
DU
=
=
+ ++
++ +
+ +
+
U = + +
D +=
UD
DU
=
=
+ ++
++ +
+ +
+
(UD-DU) =
U
D
I
I
1 4235
1 423
123
12
1
1 4235
123
45
U
D
I
I
number of alternative tableaux
n = 12
alternative tableau
combinatorialrepresentation
of the operatorsE and D
PASEP algebraDE = qED + E + D
K
S
S = +
S S
S S S
Bijection Laguerre histories
permutations
Françon-X.V., 1978
bijectionpermutations
alternativetableaux
416978352
416978352
1 1 2
1 3 241 3 241 352416 352
416 7 352416 78352
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
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
A
K
A
S
SJ A
S
K
A
A
K
A
S
SJ A
S
K
A
AS
A
K
A
S
SJ A
S
K
A
AS
JI
A
K
A
S
SJ A
S
K
A
AS
JI
KI
A
K
A
S
SJ A
S
K
A
AS
JI
KI
KJ
A
K
A
S
SJ A
S
K
A
AS
JI
KI
KJ
K
I
A
K
A
S
SJ A
S
K
A
AS
JI
KI
KJ
K
I
K
J
A
K
A
S
SJ A
S
K
A
AS
JI
KI
KJ
K
I
K
J
K
I
A
K
A
S
SJ A
S
K
A
AS
JI
KI
KJ
K
I
K
J
K
I
K
J
416978352
inverse bijection
The “exchange-fusion” algorithm
equivalent bijection:
4 2 67 8 9 5 13
4
2
67
8
9 5 13
4 2
67
8
9 5 1
3
(43)2
67
8
9 5 1
2
6
7
8
9 5 1
(43)
2
6
7
8
9
51(43)
2
6
7
8
9
5
1(43)
2
6
7
8 91
5
(43)
2
6
7 (89)1
5
(43)
2
6
7
1
5
(43)(89)
6
7
1
5
(432)(89)
6
7
1
5
(432)(89)
6
7
1
5
(432)(89)
6
7
5
(89)
(4321)
6
7
5
(89)
(4321)
6
5(789) (4321)
“exchange-fusion”
algorithm
(789) (4321) 5 6
equivalent to a bijection
S. Corteel, P. Nadeau (2009)
with permutations tableaux
«canonical bijections»