XAVIER VIENNOT
  • PART I
    • abstract
    • contents
    • Ch0 Introduction to the course
    • Ch1 Ordinary generating functions
    • Ch2 The Catalan garden
    • Ch3 Exponential structures and genarating functions
    • Ch4 The n! garden
    • Ch5 Tilings, determinants and non-intersecting paths
    • list of bijections
    • index
  • PART II
    • abstract
    • contents
    • Ch1 Commutations and heaps of pieces: basic definitions
    • Ch2 Generating functions of heaps of pieces
    • Ch3 Heaps and paths, flows and rearrangements monoids
    • Ch4 Linear algebra revisited with heaps of pieces
    • Ch5 Heaps and algebraic graph theory
    • Ch6 Heaps and Coxeter groups
    • Ch7 Heaps in statistical mechanics
    • Some lectures related to the course
  • PART III
    • abstract
    • contents
    • Ch0 overview of the course
    • Ch1 RSK The Robinson-Schensted-Knuth correspondence
    • Ch2 Quadratic algebra, Q-tableaux and planar automata
    • Ch3 Tableaux for the PASEP quadratic algebra
    • Ch4 Trees and tableaux
    • Ch5 Tableaux and orthogonal polynomials
    • Ch6 Extensions: tableaux for the 2-PASEP quadratic algebra
    • Lectures related to the course
    • references, comments and historical notes
  • PART IV
    • introduction
    • contents
    • Ch0 Overview of the course
    • Ch1 Paths and moments
    • Ch2 Moments and histories
    • Ch3 Continued fractions
    • Ch4 Computation of the coefficients b(k) lambda(k)
    • Ch5 Orthogonality and exponential structures
    • Ch6 q-analogues
    • Lectures related to the course
    • Complements
    • references
  • Epilogue

The Art of Bijective Combinatorics    Part II

Commutations and heaps of pieces with interactions in physics, mathematics and computer science

 

The Institute of Mathematical Sciences, Chennai, India  (January-March 2017)

Ch 6   Heaps  and  Coxeter  groups

Ch 6 a

23 February 201

slides       (pdf  26  Mo)          video ch6a

 

the heap monoid of a Coxeter group   3

        definition of a Coxeter group  4    

        the associated Coxeter graph  6

        definition: the heap of a Coxeter group   7

        equivalent definition with the fibers over a vertex  s  and over an edge {s,t}  10

reduced decomposition  13

heaps of dimers and the symmetric group  17

        a non-reduced decomposition   23-25

        permutation  associated to a heap of dimers  29

fully commutative elements  (FC)  in Coxeter groups   31

        definition of a FC element and a FC heap  32

        strict heaps  34

        convex chain  37

        Stembridge’s characterization of FC heaps  43

        the list of FC-finite Coxeter groups  47

fully commutative elements for the symmetric group  49

the stair decomposition of a heap of dimers   50

        definition of a stair  52

        the stair decomposition  53

        the bijection heaps of dimers -- heaps of segments  55

exercise  58

        Dyck paths, Lukasiewcz paths 

        pyramids of dimers, of segments, of oriented loops  (for Dyck paths)

total order of the stairs in a heap of dimers   66

the stair lemma   75

fully commutative heap of dimers  79

        characterization   82

bijection  FC  heaps -- Dyck paths  83

exercise  86

        heaps enumerated by n! 

bijection  FC heaps and parallelogram polyominoes  (=staircase polygons)  89

        reminding of chapter 2a, course IMSc 2016   97,98

        q-enumeration of FC elements in Symmetric group  99

exercise  102

        another characterization of FC elements for the symmetric group  102

the end  106

Ch 6 b          

27 February  2007

slides    (pdf  13  Mo)              video Ch6b

complements:      slides      (pdf )

                  

from the previous lecture   3

bijection fully commutative (FC) heaps --  (321)-avoiding permutation   12

The Temperley-Lieb algebra  TL_n(beta)  20

        definition with relation and generators  21

        reduced words  25

        reduced heaps  26

        planar diagram  D(H) associated to a heap  H  of dimers  32

        proposition:  bijection  reduced heap -- planar diagram  35

        product of planar diagrams  44

        Kauffman generators  45

        Basis of Temperley-Lieb algebra  48-49

        planar diagram associated to a skew-Ferrers diagram  56-57

exercise: RSK and FC heaps  58

nil-Temperley-Lieb algebra  66

        definition  67

        representation with operators acting on Ferrers diagrams  71

the end  (of the first part of the lecture)  73

 

complements: relation with symmetric functions

definition of the symmetric function F_sigma  associated to a permutation  9

symmetric functions and (321)- avoiding permutations   13

        in this case, F_sigma is a skew Schur function

        bijection  skew (semi-standard) Young tableau and preheap

Jacobi-Trudi identities  26

        for homogeneous and elementary symmetric functions

        superposition of two dual configurations of non-intersecting paths  33

        duality in paths  34-37

        relation Jacobi-Trudi dual configurations of paths and Fomin-Kirillov construction

                      for F_sigma with sigma (321)-avoiding permutation  39-41

the end  43