Maps and Permutations in Toni Machí’s lifedept-info.labri.u-bordeaux.fr/~cori/Publications/toniFest.pdf18 Graphical representation A hypermap is represented by a bicouloured map,

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Maps and Permutations in Toni Machí’s life

Robert CoriLabri, Université Bordeaux 1

Groups and languages, Roma, september 9 2010

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One day of 1979 ...


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Or perhaps one day of 1979 ...


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Part I: Factorization of permutations into two cycles


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Large cycles

A permutation π is a large cycle if one of two conditions below holds:

it is cyclic (i.e. has a unique cycle).Example: (1, 5, 4, 3, 2)

it has two cycles, one of which being afixed point. Example: (1, 3, 2, 5)(4)


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Two main problems

1. For a given permutation σ find the number of factorizations σ = αβ such that α

is cyclic and β is a large cycle.

Related to calculations of the characters in the symmetric group

2. For a permutation σ find the number of factorizations σ = θβ such that θ has k

cycles and β is cyclic

Considered recently by Stanley (2009) giving polynomials with nice properties


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The number of factorizations σ = αβ

such that α is cyclic and β is a large cycle

• This number denoted aλ here depends on the cyclic type λ of σ.

• Two cases to consider λ even or odd


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Main fact: the odd case

Theorem (G. Boccara, 1980) Let σ be a an odd permutation with cycle type λ then

aλ = 2(n− 2)!

Combinatorial proof by Antonio Machí (European Journal of Combinatorics 1992)13, 273-277.


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The even case

Theorem (Combinatorial proof by RC, M. Marcus, G. Schaeffer) Let σ be a aneven permutation with cycle type λ then

aλ =1

n(n + 1)φ(λ)


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The value of φ(λ)

1. Let λ be a partition of n consisting of k parts and let I = {1, 2, . . . , k}

2. For any X ⊆ I denote SX =∑

i∈X λi

Then:

φ(λ) =∑X⊆I

(−1)SX(λ)+|X|(SX(λ))!(n− SX(λ))!


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An example

1. Take λ = [4, 3, 2] so that I = {1, 2, 3}

2. Hence the possible X are ∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.

3. Giving for the SX : 0, 4, 3, 2, 7, 6, 5, 9

4. And φ(λ) = 0!9!− 4!5! + 3!6!− 2!7!− 7!2! + 6!3!− 5!4! + 9!0! = 708480

Hence:

aλ =708480

90= 7872


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Don Zagier main results (1995).

1. The probability that the product of two cyclic permutations of Sn has exactly m

cycles is given by:

2sn+1,m

(n + 1)!

where sn,k is the number of permutations of Sn with k cycles

2. For any partition λ with f fix points:

2(n− 1)!

n + 2− f≤ aλ ≤

2(n− 1)!

n + 1929− f


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Don Zagier central formula (1995).

For any partition λ:

n∑k=1

Φk(z)bk(λ) =Πi=1,p(1− zλi)

(1− z)n+2

Where:

Φn(z) =Descn(z)

(1− z)n+1

and bk(λ) is the number of factorizations of a permutation of cycle type λ as aproduct of a cyclic permutation and a permutation with k cycles.


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Part II: Hypermaps, Riemann-Hurwitz theorem


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Ici a vécu le professeur Antonio Machí entre 1981 et 1982


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What is a hypermap?

A Hypermap (S, σ, α):

σ, α are permutations acting on the set S, such that the group they generate actstransitively on S.


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Graphical representation

A hypermap is represented by a bicouloured map, vertices are coloured blue andred and edges have endpoints of different colors.

The edges are numbered by the integers {1, 2, . . . , n} so that they represent the twopermutations.

• To each cycle of σ corresponds a blue vertex such that when going around thisvertex in the trigonometric positive direction, the edges are met in the same orderas they appear in the corresponding cycle of σ.

• To each cycle of α corresponds a red vertex such that the edges met around itsatisfy the same property as above with respect to α.


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A hypermap on a set of 10 points

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σ = (1, 7, 3, 4)(2, 10)(5, 6)(8, 9)


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σ = (1, 7, 3, 4)(2, 10)(5, 6)(8, 9) α = (1, 6)(2, 5, 4)(3, 9, 10, 8)(7)


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Faces

The number of faces of the hypermap is the number of cycles of the permutation ασ


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σ = (1, 7, 3, 4)(2, 10)(5, 6)(8, 9) α = (1, 6)(2, 5, 4)(3, 9, 10, 8)(7)

ασ = (4, 6)(1, 7, 9, 3, 2, 8, 10, 5)


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Genus

The value of the genus is given by the following relation

z(σ) + z(α) + z(ασ) = n + 2− 2g(σ, α)

where z(σ), z(α), z(ασ) are the numbers of cycles of the permutations σ, α, ασ

On the example:

4 + 4 + 2 = 10 + 2 + g


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Automorphisms

Two hypermaps are isomorphic if one can be obtained from the other by relabellingthe set of points.

Or if (S, σ, α) and (S ′σ′, α′), acting S ′ are such that there exists a bijection φ of S

onto S ′

σ′ = φσφ−1, α′ = φαφ−1

Anautomorphism of the hypemap (S, σ, α) is a permutation φ satisfying

σ = φσφ−1, α = φαφ−1


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Quotient hypermap

If G is a subgroup of the automorphism group of {α, σ} then the permutations α

and σ act on the orbits of G giving a quotient hypermap.


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Riemann Hurwitz Formula

Let {σ, α} be a hypermap and {σ, α} its quotient by a subgroup G of itsautomorphism group.

The genus g of the hypermap {σ, α} and that γ of its quotient are related by theformula;

2g − 2 = |G|(2γ − 2) +∑

φ6=1∈G

χ(φ)

where χ(φ) is the total number of cycles of σ, α, ασ fixed by φ.


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Part III: Riemann-Roch Theorem

A combinatorial version, from Mathew Baker and Serguei Norin paper (Advancesin Mathematics 2007)


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Divisors or the vertex subspace

• Let G = (X, E) be a connected (undirected) graph with n vertices and no loops

• A divisor, or an element of the vertex subspace is a formal sum

u =n∑

i=1

uixi

where the ui are integers in Z and the xi formal variables

• The degree of a divisor u is the sum:

deg(u) =n∑

i=1

ui

This is the free abelian group with n generators.


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Laplacian divisors

For a vertex xi we denote di (degree of the vertex) the number of edges having xi asan end point and ei,j the number of edges with end points xi, xj . The divisor

∆i = dixi −∑j 6=i

ei,jxj

will be called a Laplacian divisor, there are n different Laplacian divisors. Theysatisfy the relation:

n∑i=1

∆i = 0

The subset spanned by the Laplacian divisors will be called the Laplacian subspace,all its elements have degree equal to 0. We denote it: ∆(G)


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The equivalence relation ∼

• Two divisors u, v are equivalent by ∼ if u− v ∈ ∆(G). This is denoted u ∼ v.

• This could be translated into a game with chips on a graph.

1. To each vertex of the graph G = (X, E) is associated an integer positive ornegative

2. A firing of vertex xi consists in one of the two following operations:

• Add 1 to each of the neighbours of xi and substract di to xi

• Substract 1 one to each of the neighbours of xi and add di to xi

3. A divisor belongs to the Laplacian subspace if and only if starting from theconfigurations it defines on G, the configuration 0 can be reached by a sequenceof firings.


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The dimension of a divisor

• The dimension of an element of the Laplacian subspace is 0

• The dimension of a divisor with negative degree or with degree 0 and which is notin the Lplacan subspace is 0

• A divisor u is effective if all the ui are non negative.

• The dimension of a divisor u is k if for all effective divisor v of degree k thedivisor u− v is equivalent to an effective divisor, and there exists an effectivedivisor w of degree k + 1 such that u−w is not equivalent to an effective divisor.

the dimension of a divisor u is denoted r(u).


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A useful Lemma

Lemma For two divisors u and v one has:

r(u + v) ≥ r(u) + r(v)

The canonical divisor K is given by:

K =n∑

i=1

[di − 2]xi


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Riemann Roch theorem for graphs

Theorem For any divisor u in a graph

G = (X, E)

with n vertices and m edges one has:

r(u)− r(K − u) = deg(u) + m− n


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Applications

Corollary

• Let u define an initial configuration of the game such that deg(u) > m− n, thenthere exists a sequence of firings giving an effective divisor

• For any k ≤ m− n there exists a divisor of degree k wich is not equivalent to aneffective configuration.


See you for the nextcompleano

2019?

Documents

Maps and Permutations in Toni Machí’s lifedept-info.labri.u-bordeaux.fr/~cori/Publications/toniFest.pdf18 Graphical representation A hypermap is represented by a bicouloured map,