Course  X.V.  IMSc 2017


Ch6   Heaps  and  Coxeter  groups

    For each item, the first number refers to the page of the slides, clicking on «video» takes you directly to the corresponding slide in the video.  (to be completed)


                     23 February 2017

                                    slides        (pdf  26  Mo)          video

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


Ch6b           27 February  2007

                               slides    (pdf  13  Mo)              video

                              complements:      slides      (pdf  8 Mo)


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

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