Combinatorial operators and quadratic algebras · quadratic algebras IMSc, Chennai 1 March 2012...
Transcript of Combinatorial operators and quadratic algebras · quadratic algebras IMSc, Chennai 1 March 2012...
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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
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combinatorialrepresentation
of the operatorsE and D
PASEP algebraDE = qED + E + D
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K
S
S = +
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S S
S S S
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Bijection Laguerre histories
permutations
Françon-X.V., 1978
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bijectionpermutations
alternativetableaux
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inverse bijection
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quadratic algebraoperators
data structuresand orthogonal polynomials
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Primitive operations for “dictionnaries” data structure:
add or delete any elements, asking questions (with positive or negative answer)
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number of choices for eachprimitive operations
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priority queue
Polya urn
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A S - S A = I
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The cellular Ansatz
From quadratic algebra Qto combinatorial objects (Q-tableaux)
and bijections
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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
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Thank you!