The Art of Bijective Combinatorics
Résumé
This series of lectures is at the crossroad of algebra, combinatorics and theoretical physics. I shall expose a new theory I call "cellular Ansatz", which offers a framework for a fruitful relation between quadratic algebras and combinatorics, extending some very classical theories such as the Robinson-Schensted-Knuth (RSK) correspondence between permutations and pairs of standard Young tableaux. In a first step, the cellular Ansatz is a methodology which allows, with a cellular approach on a square lattice, to associate some combinatorial objects (called Q-tableaux) to certain quadratic algebra Q defined by relations and generators. This notion is equivalent to the new notions (with a background from theoretical computer science) of "planar automata" or "planar rewriting rules". A Q-tableau is a Young (Ferrers) diagram, with some kind of labels in the cells of the diagram. The first basic example is the algebra defined by the relation UD = qDU+Id (or Weyl-Heisenberg algebra, well known in quantum physics). The associated Q-tableaux are the permutations, or more generally placements of towers on a Young diagram (Ferrers diagram). The second basic example is the algebra defined by the relation DE = qED+E+D which is fundamental for the resolution of the PASEP model ("partially asymmetric exclusion process"), a toy model in the physics of dynamical systems far from equilibrium, with the calculus of the stationary probabilities. Many recent researches have been made for the associated Q-tableaux, under the name or "alternative tableaux", "tree-like tableaux" or "permutations tableaux". Note that these last tableaux were introduced ("Le-diagrams") by Postnikov in relation with some positivity problems in algebraic geometry. Another algebra is the so-called XYZ algebra, related to the 8-vertex model in statistical physics, and is related to the famous alternating sign matrices, plane partitions, non-crossing configurations of paths and tilings on a planar lattice, such as the well-known Aztec tilings. In a second step, the cellular Ansatz allows a "guided" construction of bijections between Q-tableaux and other combinatorial objects, as soon one has a "representation" by combinatorial operators of the quadratic algebra Q. The basic example is the Weyl-Heisenberg algebra and we can recover the RSK correspondence with the Fomin formulation of "local rules" and "growth diagrams", from a representation of Q by operators acting on Ferrers diagrams (i.e. the theory of differential posets for the Young lattice). For the PASEP algebra, we get a bijection between alternative tableaux (or permutations tableaux) and permutations. The combinatorial theory of orthogonal polynomials (Flajolet, Viennot) plays an important role, in particular with the moments of q-Hermite, q-Laguerre and Askey-Wilson polynomials. Particular case is the TASEP (q=0) and is related to Catalan numbers and the Loday-Ronco Hopf algebra of binary trees.
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