Combinatorial operators and
quadratic algebras
IMSc, Chennai 1 March 2012
Xavier G. ViennotCNRS, Bordeaux, France
part III: bijection alternative tableaux -- permutationsfrom a combinatorial representation of
the PASEP algebra DE=qED+E+D
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
quadratic algebraoperators
data structuresand orthogonal polynomials
Primitive operations for “dictionnaries” data structure:
add or delete any elements, asking questions (with positive or negative answer)
number of choices for eachprimitive operations
priority queue
Polya urn
A S - S A = I
The cellular Ansatz
From quadratic algebra Qto combinatorial objects (Q-tableaux)
and bijections
Physics
UD = DU + IdWeyl-Heisenberg
commutationsrewriting rules
combinatorialobjects
on a 2d lattice
representationby operators
towers placementspermutations
tableaux alternatifs
bijections
Q-tableauxex: ASM,
(alternating sign matrices)FPL(fully packed loops)
tilings, 8-vertex
planarautomata
pairs of Tableaux YoungpermutationsLaguerre histories
data structures"histories"
quadratic algebra Q
orthogonal polynomials
?
RSK
DE = qED + E + DPASEP
"normal ordering"
planarisation
"The cellular Ansatz"
Koszul algebrasduality
Thank you!
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