Heaps  of  pieces

(with interactions in mathematics and physics)

     

In relation with combinatorial and probabilistic properties of rearrangements of sequences,  Cartier and Foata introduced some monoids defined by generators and some partial commutations relations. These Cartier-Foata monoids are also called trace monoids. They were also introduced as model in computer science for concurrency access to data structures and parallelism. Heaps of pieces have been introduced by the speaker in 1985 as a geometric interpretation of such monoids. The spatial visualization of elements of the monoids in term of heaps makes it very versatile for applications. Since the introduction of heaps, many authors have made various contributions in applying the heaps point of view to combinatorics, algebra and theoretical physics. The course introduces the basic definitions and lemma, and show these various possible applications, especially in physics.

some references

the two historical papers:

1. -X.G.Viennot, Heaps of pieces, I: Basic definitions and combinatorial lemma, in « Combinatoire énumérative », eds. G. Labelle et P. Leroux, , Lecture Notes in Maths. n° 1234, Springer-Verlag, Berlin, 1986, p. 321-325.       pdf
   

2. - X.G.Viennot, Problèmes combinatoires posés par la physique statistique, in Séminaire Nicolas BOURBAKI, exposé n°626, Astérisque n°121-122, Soc. Math. France, 1985, p. 225-246.      pdf
   

for an introduction see:

100. C.Krattenthaler, “The theory of heaps and the Cartier-Foata monoid",  appendice to the reedition of the monograph "Commutation and Rearrangements" by P.Cartier et D.Foata, SLC, 2006,  and the references within, 
     

  see the web page of   SLC  (Séminaire Lotharingien de Combinatoire)

another pictorial introduction with slides and video:

1. -X.G.Viennot, Introduction to the theory of pieces with applications to statistical mechanics and quantum gravity in workshop “Combinatorial Identities & their Applications in Statistical Mechanics”,  Isaac Newton Institute for Mathematical Science, Cambridge, 7 April 2008          slides  and  video  (1 hour)
   

A 24 hours classes in University of Talca, Chile, December 2013-January 2014  see the     slides here

Curso 1   Commutation  monoids and heaps of pieces: 

                                basic definitions

            29 January 2014        slides      (pdf   6,2 Mo  )

Curso 2   Generating function for heaps of pieces, heaps and paths

                                  (The  3  basics  lemma)

             30 January 2014         slides      (pdf   6,7 Mo  )

Curso 3   Heaps of pieces and  physics

            31 January 2014        slides      (pdf   14,3 Mo  )

Curso 4   Heaps, Temperley-Lieb algebra, Coxeter groups and

          representation of Lie algebra with operators on heaps

            31 January 2014        slides      (pdf   7,8 Mo  )

Epilogue:  from Euler triangulations to Lorentzian triangulations

references:  to be completed