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 4  Linear algebra revisited with heaps of pieces

Ch 4a

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’’

        

Ch 4b

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’’

Ch 4c

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: