Monstrous Moonshine Æ v Zyokoemon.web.fc2.com/misc/20110716CB.pdfMonstrous Moonshine と計算機数論 Shun’ichi Yokoyama (Kyushu University) Kyushu Univ. Combinatorics Seminar

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Sage OverviewApplication

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Monstrous Moonshine と計算機数論

Shun’ichi Yokoyama(Kyushu University)

Kyushu Univ. Combinatorics Seminar / July 16th, 2011

Shun’ichi Yokoyama (Kyushu University) Monstrous Moonshine と計算機数論

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概要

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1 計算機システム Sage の紹介

Sage の概要デモンストレーション

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2 代数的組合せ論の問題から

Monstrous Moonshine と保型関数数論とのつながり


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What’s Sage / Why Sage?Demonstration

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What’s Sage

System for Algebra and Geometry Experimentation の略称.

W. Stein (Washington) らが中心となって開発.

　　

開発動機： Magma, Maple, Mathematica (何れも計算機代数システム), MATLAB の代替となるフリーかつオープンソースなソフトウェアを提供すること


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“Building the Car”


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Sage

Notebook interface


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Sage

3D Plot (like Mathematica)


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Sage

http://www.sagenb.org/


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Sage - Linux Live CD “Knoppix/Math”

CREST 日比チームを中心として開発, 最新版は 2011.

100を超える数式処理ソフトウェアと TEX 環境無料で入手可能 http://www.knoppix-math.org/

Axiom, GiNaC, Maxima, Risa/Asir(OpenXM), CoCoA, Yacas, GAP,Magnus, KANT/KASH, NZMATH, PARI/GP, Singular, Macaulay2,

KNOT, Knotscape, Orb, SnapPea, CHomP, Sage, etc.


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デモンストレーション（1/7）

“Fibonacci iterator”

> def fibonacci iterator(a=0, b=1):

> while True:

> yield b

> a, b = b, a+b

> f = fibonacci iterator()

> f.next()

1

> f.next()

1

> f.next()

2

...3,5,8,...


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> while True:

> yield b

> a, b = b, a+b


> f.next()

1

> f.next()

1

> f.next()

2

...3,5,8,...


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> while True:

> yield b

> a, b = b, a+b


> f.next()

1

> f.next()

1

> f.next()

2

...3,5,8,...


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> while True:

> yield b

> a, b = b, a+b


> f.next()

1

> f.next()

1

> f.next()

2

...3,5,8,...


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> while True:

> yield b

> a, b = b, a+b


> f.next()

1

> f.next()

1

> f.next()

2

...3,5,8,...


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Permutations

> P = Permutations(3)

> P

Standard permutations of 3

> P.cardinality()

6

> P.list()

[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3,

1, 2], [3, 2, 1]]

> Permutations(1000)


> P = Permutations(7, avoiding=[2,1,4,3])

Standard permutations of 7 avoiding [[2, 1, 4, 3]]

> P.cardinality()

2761


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Permutations


> P


> P.cardinality()

6

> P.list()

[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3,

1, 2], [3, 2, 1]]





> P.cardinality()

2761


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Permutations


> P


> P.cardinality()

6

> P.list()

[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3,

1, 2], [3, 2, 1]]





> P.cardinality()

2761


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Permutations


> P


> P.cardinality()

6

> P.list()

[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3,

1, 2], [3, 2, 1]]





> P.cardinality()

2761


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Permutations


> P


> P.cardinality()

6

> P.list()

[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3,

1, 2], [3, 2, 1]]





> P.cardinality()

2761


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Permutations


> P


> P.cardinality()

6

> P.list()

[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3,

1, 2], [3, 2, 1]]





> P.cardinality()

2761


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Permutations


> P


> P.cardinality()

6

> P.list()

[[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3,

1, 2], [3, 2, 1]]





> P.cardinality()

2761


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Partitions

> P = Partitions(4)

> P

Partitions of the integer 4

> for p in Partitions(4):

> print p

[4]

[3, 1]

[2, 2]

[2, 1, 1]

[1, 1, 1, 1]


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Partitions

> P = Partitions(4)

> P



> print p

[4]

[3, 1]

[2, 2]

[2, 1, 1]

[1, 1, 1, 1]


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Partitions

> P = Partitions(4)

> P



> print p

[4]

[3, 1]

[2, 2]

[2, 1, 1]

[1, 1, 1, 1]


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Compositions

> for c in Compositions(4):

> print c

[1, 1, 1, 1]

[1, 1, 2]

[1, 2, 1]

[1, 3]

[2, 1, 1]

[2, 2]

[3, 1]

[4]


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Dyck Word

> D = DyckWords(4)

Dyck words with 4 opening parentheses and 4

closing parentheses

> D.cardinality()

14

> for dw in DyckWords(4):

> print dw

()()()()

()()(())

()(())()

()(()())

()((()))

(())()()

(())(())

(()())()

...Shun’ichi Yokoyama (Kyushu University) Monstrous Moonshine と計算機数論

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Dyck Word

> D = DyckWords(4)


closing parentheses

> D.cardinality()

14


> print dw

()()()()

()()(())

()(())()

()(()())

()((()))

(())()()

(())(())

(()())()


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Dyck Word

> D = DyckWords(4)


closing parentheses

> D.cardinality()

14


> print dw

()()()()

()()(())

()(())()

()(()())

()((()))

(())()()

(())(())

(()())()


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Example: Vexillary involution

avoids the pattern 2143,

is an involution (i.e., p = p−1).

> def is involution(p):

> return p == p.inverse()

> P = Permutations(4,

avoiding=[2,1,4,3]).filter(is involution); P

Filtered sublass of Standard permutations of 4

avoiding [[2, 1, 4, 3]]

> P.cardinality()

9

> P.list()

[[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 4,

3, 2], [3, 4, 1, 2], [2, 1, 3, 4], [4, 2, 3, 1],

[3, 2, 1, 4], [4, 3, 2, 1]]


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avoiding [[2, 1, 4, 3]]

> P.cardinality()

9

> P.list()

[[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 4,

3, 2], [3, 4, 1, 2], [2, 1, 3, 4], [4, 2, 3, 1],

[3, 2, 1, 4], [4, 3, 2, 1]]


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avoiding [[2, 1, 4, 3]]

> P.cardinality()

9

> P.list()

[[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 4,

3, 2], [3, 4, 1, 2], [2, 1, 3, 4], [4, 2, 3, 1],

[3, 2, 1, 4], [4, 3, 2, 1]]


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avoiding [[2, 1, 4, 3]]

> P.cardinality()

9

> P.list()

[[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 4,

3, 2], [3, 4, 1, 2], [2, 1, 3, 4], [4, 2, 3, 1],

[3, 2, 1, 4], [4, 3, 2, 1]]


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avoiding [[2, 1, 4, 3]]

> P.cardinality()

9

> P.list()

[[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 4,

3, 2], [3, 4, 1, 2], [2, 1, 3, 4], [4, 2, 3, 1],

[3, 2, 1, 4], [4, 3, 2, 1]]


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Graph Theory: Cayley Graphs

> G = DihedralGroup(5)

> C = G.cayley graph(); C

Digraph on 10 vertices

> C.diameter()

3

> C.show()


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SetupOur ResultExample

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概要

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1 計算機システム Sage の紹介

Sage の概要デモンストレーション

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2 代数的組合せ論の問題から

Monstrous Moonshine と保型関数数論とのつながり


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Hecke-type Faber Polynomial (HFP) Pn,g

Conway-Norton 予想（Borcherds により解決, 1992）

Conway-Norton のリスト（171種類）


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Hecke-type Faber Polynomial (HFP) Pn,g

Conway-Norton 予想（Borcherds により解決, 1992）

– Monster 群 M（の各共役類）に対して定義される「保型関数」

McKay-Thompson 級数　 g ∈ M

Tg (z) :=1

q+

∞∑k=1

χk(g)qi (q = e2πiz)

：twisted Hecke operator T ′n が作用

Hecke-type Faber 多項式

T ′n (Tg (z)) =

1

nPn(Tg (z)) and Pn(Tg (z)) ≡ q−n (mod qZ[[q]])


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Classification

Pn,g (z) の零点分布に関する結果（Bannai-Kojima-Miezaki）

全て実軸上：30個　※ g = 1A, 2Aのときは証明済全て虚軸上：3個2直線上：8個3直線上：5個1点から 3方向へ拡散：8個ほぼ実軸上：11個　　　　　（以上 65個）それ以外：106個

　 g = 2A


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Classification



　 g = 6F


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Classification



g = 33A


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Conjecture

Pn,g (z) の（殆ど）全ての零点　 on　 Tg (BFg )

BFg : type g ∈ M の “良い”基本領域の境界

　type 1A → Γ0(1) = SL(2,Z) → T1A(z) = j(z)

T1A(BF1A) ⊂ R


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Results

Pn,g (z) の（殆ど）全ての零点　 on　 Tg (BFg )

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Theorem

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Conway-Norton の 171 種類のリストのうち, 幾つかの type について上の予想は成り立つ.（e.g. Γ0(N), Fricke 群 Γ∗0(N) に対応するケース）

全ての type g ∈ M に対して, 予想が成立する？？

　※「BFg を与える全半円周の像とした場合」

　⇒ BFg の部分だけを抜き出す・・・N に依存して困難


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Fundamental Domain of H/Γ

> G := Gamma0(39);

> H<i,rho> := UpperHalfPlaneWithCusps();

> tri := [H|Infinity(),i,rho]; tri1 := [H|0,i,rho];

> C11 := CosetRepresentatives(G);

> triangles := [g*tri : g in C11]

> cat [g*tri1 : g in C11];

> DisplayPolygons(triangles,"(Directory)");


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Fricke Group Γ∗0(29) as CSG


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Example

Missing part (real axis, positive area)

Exceptional zeros


Documents

Monstrous Moonshine Æ v Zyokoemon.web.fc2.com/misc/20110716CB.pdfMonstrous Moonshine と計算機数論 Shun’ichi Yokoyama (Kyushu University) Kyushu Univ. Combinatorics Seminar