combinatorial operators and quadratic algebras · quadratic algebras imsc, chennai 1 march 2012...

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!