Course X.V. IMSc 2017
Ch4 Linear algebra revisited with heaps of pieces
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.
Ch4a
6 February 2017
slides_Ch4a (pdf 12 Mo ) video Ch4a
inversion of a matrix 4 video 2’ 08’’
examples 16
bounded Dyck paths 18 video 26’ 23’’
exercise: directed paths on the square lattice 25 video 47’ 10’’
MacMahon Master theorem 27 video 50’ 54’’
inversion lemma: heaps of cycles 31 video 52’ 13’‘
heaps of cycles and rearrangements 38 video 53’ 00’’
MacMahon formulation 40-41 video 54’ 19’’
relation with quivers and gauge theory in physics 42 video 1h 1’ 02’’
complements: an identity of Bauer for loop-erased random walks 43 video 1h 3’ 13’’
research problem: substitution in heaps 53 video 1h 14’ 52’’
the end 54 1h 17’ 16’’
Ch4b
9 February 2017
slides_Ch4b (pdf 19 Mo ) video Ch4b
correction to exercise 3, p65, Ch3b 3 video 7’
from the previous lecture 4 video 2’ 16’’
from Ch2d: the logarithmic lemma 10 video 4’ 37’’
a paradox ? 16 video 8’ 24’’
proof of Jacobi identity 17 video 11’ 03’’
Jacobi identity with exponential generating function 25 video 19’ 25’’
discussion on species, labeled pyramids and exponential generating functions video 38’ 25’’
end discussion 42’ 11’’
beta extension of MacMahon Master theorem 35 video 44’ 40’’
Cayley-Hamilton theorem 42 video 49’ 03’’
another weight preserving involution 53 video 1h 5’ 02’’
complement and exercise: a general transfer theorem 57 video 1h 7’ 39’’
the exercise 62 video 1h 14’ 43’’
next lecture: Jacobi duality video 1h 16’ 25’’
the end 65 1h 17’ 42’’
Ch4c
13 February
slides_Ch4c (pdf 23 Mo ) video Ch4c
Jacobi duality 4 video 44’
the main theorem 6 video 2’ 14’’
special case 1: I and J have only one element 9 video 10’ 51’’
deducing Jacobi identity from the main theorem 13-17 video 12’ 52’’
a Lemma expressing minors 14 video 13’ 29’’
an example 15 video 17’ 32’’
special case 2: no cycles 23 video 20’ 26’’
the LGV Lemma (from the course IMSc 2016, Ch5a) video 21’ 7’’
a simple example 34 video 25’ 22’’
another example: binomial determinants video 26’ 50’’
proof of the LGV Lemma 48 video 29’ 56’’
proof of the main theorem: introduction 55 video 34’ 31’’
how to handle this mixture of cycles, an idea coming from physics:
discussion for defining a simultaneous loop-erased process video 38’ 34’’
the problem for defining the involution 64 video 40’ 50’’
proof of the main theorem: first step with Fomin theorem 65 video 41’ 47’’
proof of the main theorem: second step 75 video 51’ 15’’
end of the proof 78 video 58’ 32’’
another way to prove the Jacobi duality identity 80 video 59’ 07’’
main theorem with crossing condition 87 video 1h 2’ 31’’
the end (of the video) 92 1h 10’ 44’’
about the terminology «LGV Lemma» 92 (not in the video)
the end 98
corrections:
go to:
the IMSc 2016 bijective course website
