GroupTheory - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : GroupTheory/Dessins

GroupTheory

  

FindDessins

  

find all dessins d'enfants with a specified branch pattern

  

DecomposeDessin

  

find all decompositions of a Belyi map represented by a dessin

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

FindDessins( B0, B1, Binf )

DecomposeDessin( d, L, Gr )

Parameters

B0, B1, Binf

-

three lists of positive integers, each with the same sum n

d

-

list [ g0, g1 ] representing a conjugacy class of 3-constellations or, equivalently, a dessin

L

-

(optional) name

Gr

-

(optional) name

Description

• 

Let n be a positive integer. A 3-constellation of degree n is a triplet g0,g1,g of elements of Snthat generate a transitive subgroup of Sn and satisfy g0g1g=1.

• 

Two 3-constellations g0,g1,g, h0,h1,h are conjugated if there exists τ in Sn with τgiτ−1=hi for each i in 0,1, (or, equivalently, each i in 0,1).

• 

The branch pattern of g0,g1,g is a triplet B0,B1,B where Bi is a partition of n giving the cycle-structure of gi. We include 1-cycles so FindDessins can find n by taking the sum of the entries of each Bi.

• 

Given B0, B1, Binf as input, FindDessins computes one representative from every conjugacy class of 3-constellations with branch pattern (B0, B1, Binf). Each 3-constellation g0,g1,g will be represented by the list g0,g1, since g can be computed as g0g1−1.

• 

A conjugacy class of 3-constellations corresponds 1-1 with a dessin d'enfant, as well as with a Belyi map (up to equivalence). A Belyi map is a holomorphic function from a compact Riemann surface to the Riemann sphere that only ramifies above {0,1,infinity}. So we can count how many dessins, or how many Belyi maps, exists for a given branch pattern by counting the output of FindDessins.

• 

FindDessins implements the strategy of Section 4 in arXiv:1604.08158 with a number of additions. Progress is reported during the computation by setting infolevel['FindDessins'] to 1 or 2.

Examples

Suppose we want to know if there exists a Belyi map f whose branch pattern above 0, 1,  is [1$39], [2$14], [7$4]. This means that f should have 1 root of order 1 and 9 roots of order 3, f1 should have 14 roots of order 2, f should have 4 poles of order 7, and f should be unramified outside of {0,1,infinity}. We can determine if such f exist (and if so, how many) as follows.

withGroupTheory:

B1,`$`3,9,`$`2,14,`$`7,4

B1,3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7

(1)

SFindDessins1,`$`3,9,`$`2,14,`$`7,4:NumberOfDessinsnopsS

NumberOfDessins1

(2)

Found 1 conjugacy class of 3-constellations (i.e. 1 dessin), so there exists a Belyi map (unique up to equivalence) with branch pattern B.

dS1

d2,3,45,6,78,9,1011,12,1314,15,1617,18,1920,21,2223,24,2526,27,28,1,23,54,86,117,149,1710,2012,2213,1615,2318,2119,2624,2725,28

(3)

Now let's check that d = [ g0, g1 ] has branch pattern B.

g0d1

g02,3,45,6,78,9,1011,12,1314,15,1617,18,1920,21,2223,24,2526,27,28

(4)

g0 indeed has cycle-structure [1,3$9] (a 1-cycle and 9 3-cycles)

g1d2

g11,23,54,86,117,149,1710,2012,2213,1615,2318,2119,2624,2725,28

(5)

g1 has cycle-structure [2$14] (14 2-cycles)

gg0·g11

g1,4,10,22,11,5,23,7,16,12,21,17,86,13,15,25,27,23,149,19,28,24,26,18,20

(6)

Has cycle structure [7$4] (4 7-cycles).

DecomposeDessind

indecomposable

(7)

The Belyi map for d is indecomposable.

Example with decompositions

SFindDessins`$`3,10,`$`2,15,`$`6,5

S1,2,34,5,67,8,910,11,1213,14,1516,17,1819,20,2122,23,2425,26,2728,29,30,1,42,73,105,136,168,199,2211,2012,1814,2515,2817,2621,2923,2724,30,1,2,34,5,67,8,910,11,1213,14,1516,17,1819,20,2122,23,2425,26,2728,29,30,1,42,73,85,106,119,1312,1614,1915,2017,2218,2321,2524,2826,2927,30

(8)

NumberOfDessinsnopsS

NumberOfDessins2

(9)

dS1

d1,2,34,5,67,8,910,11,1213,14,1516,17,1819,20,2122,23,2425,26,2728,29,30,1,42,73,105,136,168,199,2211,2012,1814,2515,2817,2621,2923,2724,30

(10)

DecomposeDessind

F = F1(deg 2) = F2(deg 2) = F3(deg 2) = F4(deg 2) = F5(deg 2) = F6(deg 5) = F7(deg 10) = F8(deg 15)

(11)

The Belyi map for S[1] has 8 decompositions. With additional arguments, DecomposeDessin returns a list with information on each Fn, and a decomposition graph.

DecomposeDessind,L,Gr

F = F1(deg 2) = F2(deg 2) = F3(deg 2) = F4(deg 2) = F5(deg 2) = F6(deg 5) = F7(deg 10) = F8(deg 15)

(12)

L5

F5: P1-->P1,F5 = F7(deg 5),Degree=15,BranchPattern = ([3$5], [1$3, 2$6], [3, 6$2]),Dessin=1,2,34,5,67,8,910,11,1213,14,15,2,43,76,108,139,1112,15

(13)

L6

F6: EllipticCurve-->P1,F6 = F7(deg 2) = F8(deg 3),Degree=6,BranchPattern = ([3$2], [2$3], [6]),Dessin=1,2,34,5,6,1,42,53,6

(14)

Decomposition graph:

Gr

F1 .. F5 have the same dessin so they represent the same Belyi map (of degree = 15). The reason for listing all five is because their degree = 2 decomposition factors differ.

Compatibility

• 

The GroupTheory[FindDessins] and GroupTheory[DecomposeDessin] commands were introduced in Maple 2019.

• 

For more information on Maple 2019 changes, see Updates in Maple 2019.

See Also

GroupTheory

http://oeis.org/A112948