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)
Ch6a
23 February 2017
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
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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