 DecomposeDessin - Maple Help

Home : Support : Online Help : Mathematics : Group Theory : DecomposeDessin

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 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 $\left({g}_{0},{g}_{1},{g}_{\mathrm{\infty }}\right)$ of elements of ${S}_{n}$that generate a transitive subgroup of ${S}_{n}$ and satisfy ${g}_{0}\cdot {g}_{1}\cdot {g}_{\mathrm{\infty }}=1$.
 • Two $3$-constellations $\left({g}_{0},{g}_{1},{g}_{\mathrm{\infty }}\right)$, $\left({h}_{0},{h}_{1},{h}_{\mathrm{\infty }}\right)$ are conjugated if there exists $\mathrm{\tau }$ in ${S}_{n}$ with $\mathrm{\tau }\cdot {g}_{i}\cdot {\mathrm{\tau }}^{-1}={h}_{i}$ for each $i$ in $\left\{0,1,\mathrm{\infty }\right\}$ (or, equivalently, each $i$ in $\left\{0,1\right\}$).
 • The branch pattern of $\left({g}_{0},{g}_{1},{g}_{\mathrm{\infty }}\right)$ is a triplet $\left({B}_{0},{B}_{1},{B}_{\mathrm{\infty }}\right)$ where ${B}_{i}$ is a partition of $n$ giving the cycle-structure of ${g}_{i}$. We include $1$-cycles so FindDessins can find $n$ by taking the sum of the entries of each ${B}_{i}$.
 • 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 $\left({g}_{0},{g}_{1},{g}_{\mathrm{\infty }}\right)$ will be represented by the list $\left[{g}_{0},{g}_{1}\right]$, since ${g}_{\mathrm{\infty }}$ can be computed as ${\left({g}_{0}\cdot {g}_{1}\right)}^{-1}$.
 • 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 $0$, $1$, $\mathrm{\infty }$ is [1$39], [2$14], [7$4]. This means that $f$ should have $1$ root of order $1$ and $9$ roots of order $3$, $f-1$ 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.  > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$  > $B≔\left[1,3$9\right],\left[2$14\right],\left[7$4\right]$
 ${B}{≔}\left[{1}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}{,}{3}\right]{,}\left[{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}\right]{,}\left[{7}{,}{7}{,}{7}{,}{7}\right]$ (1)
 > $S≔\mathrm{FindDessins}\left(\left[1,3$9\right],\left[2$14\right],\left[7$4\right]\right):$$\mathrm{NumberOfDessins}≔\mathrm{nops}\left(S\right)$  ${\mathrm{NumberOfDessins}}{≔}{1}$ (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.  > $d≔{S}_{1}$  ${d}{≔}\left[\left({2}{,}{3}{,}{4}\right)\left({5}{,}{6}{,}{7}\right)\left({8}{,}{9}{,}{10}\right)\left({11}{,}{12}{,}{13}\right)\left({14}{,}{15}{,}{16}\right)\left({17}{,}{18}{,}{19}\right)\left({20}{,}{21}{,}{22}\right)\left({23}{,}{24}{,}{25}\right)\left({26}{,}{27}{,}{28}\right){,}\left({1}{,}{2}\right)\left({3}{,}{5}\right)\left({4}{,}{8}\right)\left({6}{,}{11}\right)\left({7}{,}{14}\right)\left({9}{,}{17}\right)\left({10}{,}{20}\right)\left({12}{,}{22}\right)\left({13}{,}{16}\right)\left({15}{,}{23}\right)\left({18}{,}{21}\right)\left({19}{,}{26}\right)\left({24}{,}{27}\right)\left({25}{,}{28}\right)\right]$ (3) Now let's check that d = [ g0, g1 ] has branch pattern B.  > $\mathrm{g0}≔{d}_{1}$  ${\mathrm{g0}}{≔}\left({2}{,}{3}{,}{4}\right)\left({5}{,}{6}{,}{7}\right)\left({8}{,}{9}{,}{10}\right)\left({11}{,}{12}{,}{13}\right)\left({14}{,}{15}{,}{16}\right)\left({17}{,}{18}{,}{19}\right)\left({20}{,}{21}{,}{22}\right)\left({23}{,}{24}{,}{25}\right)\left({26}{,}{27}{,}{28}\right)$ (4) g0 indeed has cycle-structure [1,3$9] (a $1$-cycle and $9$ $3$-cycles)

 > $\mathrm{g1}≔{d}_{2}$
 ${\mathrm{g1}}{≔}\left({1}{,}{2}\right)\left({3}{,}{5}\right)\left({4}{,}{8}\right)\left({6}{,}{11}\right)\left({7}{,}{14}\right)\left({9}{,}{17}\right)\left({10}{,}{20}\right)\left({12}{,}{22}\right)\left({13}{,}{16}\right)\left({15}{,}{23}\right)\left({18}{,}{21}\right)\left({19}{,}{26}\right)\left({24}{,}{27}\right)\left({25}{,}{28}\right)$ (5)

g1 has cycle-structure [2$14] ($14$ $2$-cycles)  > ${g}_{\mathrm{∞}}≔{\left(\mathrm{.}\left(\mathrm{g0},\mathrm{g1}\right)\right)}^{-1}$  ${{g}}_{{\mathrm{\infty }}}{≔}\left({1}{,}{4}{,}{10}{,}{22}{,}{11}{,}{5}{,}{2}\right)\left({3}{,}{7}{,}{16}{,}{12}{,}{21}{,}{17}{,}{8}\right)\left({6}{,}{13}{,}{15}{,}{25}{,}{27}{,}{23}{,}{14}\right)\left({9}{,}{19}{,}{28}{,}{24}{,}{26}{,}{18}{,}{20}\right)$ (6) Has cycle structure [7$4] ($4$ $7$-cycles).

 > $\mathrm{DecomposeDessin}\left(d\right)$
 ${"indecomposable"}$ (7)

The Belyi map for d is indecomposable. Example with decompositions

 > $S≔\mathrm{FindDessins}\left(\left[3$10\right],\left[2$15\right],\left[6\$5\right]\right)$
 ${S}{≔}\left\{\left[\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}{,}{6}\right)\left({7}{,}{8}{,}{9}\right)\left({10}{,}{11}{,}{12}\right)\left({13}{,}{14}{,}{15}\right)\left({16}{,}{17}{,}{18}\right)\left({19}{,}{20}{,}{21}\right)\left({22}{,}{23}{,}{24}\right)\left({25}{,}{26}{,}{27}\right)\left({28}{,}{29}{,}{30}\right){,}\left({1}{,}{4}\right)\left({2}{,}{7}\right)\left({3}{,}{8}\right)\left({5}{,}{10}\right)\left({6}{,}{11}\right)\left({9}{,}{13}\right)\left({12}{,}{16}\right)\left({14}{,}{19}\right)\left({15}{,}{20}\right)\left({17}{,}{22}\right)\left({18}{,}{23}\right)\left({21}{,}{25}\right)\left({24}{,}{28}\right)\left({26}{,}{29}\right)\left({27}{,}{30}\right)\right]{,}\left[\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}{,}{6}\right)\left({7}{,}{8}{,}{9}\right)\left({10}{,}{11}{,}{12}\right)\left({13}{,}{14}{,}{15}\right)\left({16}{,}{17}{,}{18}\right)\left({19}{,}{20}{,}{21}\right)\left({22}{,}{23}{,}{24}\right)\left({25}{,}{26}{,}{27}\right)\left({28}{,}{29}{,}{30}\right){,}\left({1}{,}{4}\right)\left({2}{,}{7}\right)\left({3}{,}{10}\right)\left({5}{,}{13}\right)\left({6}{,}{16}\right)\left({8}{,}{19}\right)\left({9}{,}{22}\right)\left({11}{,}{20}\right)\left({12}{,}{18}\right)\left({14}{,}{25}\right)\left({15}{,}{28}\right)\left({17}{,}{26}\right)\left({21}{,}{29}\right)\left({23}{,}{27}\right)\left({24}{,}{30}\right)\right]\right\}$ (8)
 > $\mathrm{NumberOfDessins}≔\mathrm{nops}\left(S\right)$
 ${\mathrm{NumberOfDessins}}{≔}{2}$ (9)
 > $d≔{S}_{1}$
 ${d}{≔}\left[\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}{,}{6}\right)\left({7}{,}{8}{,}{9}\right)\left({10}{,}{11}{,}{12}\right)\left({13}{,}{14}{,}{15}\right)\left({16}{,}{17}{,}{18}\right)\left({19}{,}{20}{,}{21}\right)\left({22}{,}{23}{,}{24}\right)\left({25}{,}{26}{,}{27}\right)\left({28}{,}{29}{,}{30}\right){,}\left({1}{,}{4}\right)\left({2}{,}{7}\right)\left({3}{,}{8}\right)\left({5}{,}{10}\right)\left({6}{,}{11}\right)\left({9}{,}{13}\right)\left({12}{,}{16}\right)\left({14}{,}{19}\right)\left({15}{,}{20}\right)\left({17}{,}{22}\right)\left({18}{,}{23}\right)\left({21}{,}{25}\right)\left({24}{,}{28}\right)\left({26}{,}{29}\right)\left({27}{,}{30}\right)\right]$ (10)
 > $\mathrm{DecomposeDessin}\left(d\right)$
 ${"F = F1\left(deg 2\right) = F2\left(deg 2\right) = F3\left(deg 2\right) = F4\left(deg 2\right) = F5\left(deg 2\right) = F6\left(deg 5\right) = F7\left(deg 10\right) = F8\left(deg 15\right)"}$ (11)

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

 > $\mathrm{DecomposeDessin}\left(d,'L','\mathrm{Gr}'\right)$
 ${"F = F1\left(deg 2\right) = F2\left(deg 2\right) = F3\left(deg 2\right) = F4\left(deg 2\right) = F5\left(deg 2\right) = F6\left(deg 5\right) = F7\left(deg 10\right) = F8\left(deg 15\right)"}$ (12)
 > ${L}_{5}$
 $\left[{"F5: P1-->P1"}{,}{"F5 = F7\left(deg 5\right)"}{,}{"Degree"}{=}{15}{,}{"BranchPattern = \left(\left[35\right], \left[13, 26\right], \left[3, 62\right]\right)"}{,}{"Dessin"}{=}\left[\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}{,}{6}\right)\left({7}{,}{8}{,}{9}\right)\left({10}{,}{11}{,}{12}\right)\left({13}{,}{14}{,}{15}\right){,}\left({3}{,}{4}\right)\left({5}{,}{7}\right)\left({6}{,}{8}\right)\left({9}{,}{10}\right)\left({11}{,}{13}\right)\left({12}{,}{14}\right)\right]\right]$ (13)
 > ${L}_{6}$
 $\left[{"F6: EllipticCurve-->P1"}{,}{"F6 = F7\left(deg 2\right) = F8\left(deg 3\right)"}{,}{"Degree"}{=}{6}{,}{"BranchPattern = \left(\left[32\right], \left[23\right], \left[6\right]\right)"}{,}{"Dessin"}{=}\left[\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}{,}{6}\right){,}\left({1}{,}{4}\right)\left({2}{,}{5}\right)\left({3}{,}{6}\right)\right]\right]$ (14)

Decomposition graph:

 > $\mathrm{Gr}$ 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.