Tag: groups

more iguanodons via kfarey.sage

Published December 11, 2007 by lieven

For what it is worth, Ive computed some more terms in the iguanodon series. Here they are

$L_2(7),M_{12},A_{16},M_{24},A_{28},A_{40},A_{48},A_{60},A_{68},A_{88},A_{96},A_{120},A_{132},A_{148},A_{164},A_{196},\ldots $

By construction, the n-th iguanodon group $Ig_n $ (corresponding to the n-th Farey sequence) is a subgroup of the alternating group on its (half)legs. Hence to prove that all remaining iguanodons are alternating groups boils down to proving that they are sufficiently transitive, for example, by showing that there are permutations of certain cycle-types in the group. Im sure any grouptheorist can crack this problem over lunch, so if you did please drop a comment.

Clearly, I didnt do the calculations in the archaic way of the previous post (as depicted on the left) which consisted in adding a pair of new legs at the proper place in the spine for every new Farey number, write down the two generating permutations, giving them to GAP and check simplicity and the isomorphism type.

Instead I used a nice SAGE-package to compute with Farey-symbols written by Chris Kurth and available from his website. As this package is a good tool to experiment hunting for other dinosaur-series of simple groups coming from series of Farey-symbols, Ill include the details for $Ig_3 $ (the example used to outline the construction of the Iguanodon-series ).

First we need to have the n-th Farey-sequence $F(n) $. There are several short Python programs around to do this, for example this one from the Python-Cookbook. Save it to your sage-directory and name it fareyseq.py and load it into sage via load fareyseq.py. Then typing farey(3) to the sage-prompt spits back

sage: farey(3)
[(1, 3), (1, 2), (2, 3)]

That is, 0 and 1 are not included and Farey-numbers are represented by numerator-denominator couples. The iguanodon-series uses the Fareys upto 1/2, identifies the edges connecting 0 and 1 to $\infty $ and makes all other intervals odd. That is, the corresponding Farey symbol for F(3) is

[tex]\xymatrix{\infty \ar@{-}[r]_{1} & 0 \ar@{-}[r]_{\bullet} & \frac{1}{3} \ar@{-}[r]_{\bullet} & \frac{1}{2} \ar@{-}[r]_{\bullet} & 1 \ar@{-}[r]_{1} & \infty}[/tex]

(to add to the confusion, I denote odd intervals by a black-bullet whereas in Kulkarni’s paper they are white…) Anyway, get Kurth’s kfarey-package and save the folder as kfarey in your sage-folder. Kurth uses the following notation for Farey-symbols

The Farey Symbol is a list [a,b,p] where
a is a list of numerators, b a list of denominators, and p the pairing
information. If x[i]=a[i]/b[i]:
inf x0 x1 x2 ... xn inf
  p0 p1 p2 ... pn pn+1
So p[i] is the pairing of the side between x[i-1] and x[i]. The p[i]’s can be
positive integers, indicating pairing between sides, or -2 or -3, meaning
an even or odd pairing respectively.

The above Farey-symbol is therefore represented as
[[0,1,1,1],[1,3,2,1],[1,-3,-3,-3,1]]. The kfarey-function LRCosetRep(F) returns two permutations L and R giving the permutation action of the two generators of the modular group $PSL_2(\mathbb{Z}) $

$~~~L = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}~\qquad \text{and} \qquad~R = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} $

on the half-legs of the inguanodon (the dessin corresponding to the Farey-symbol). Here’s the sage transcript

sage: load kfarey/farey.sage
sage: load kfarey/conggroups.sage
sage: load kfarey/LR.sage
sage: ig3=[[0,1,1,1],[1,3,2,1],[1,-3,-3,-3,1]]
sage: LRCosetRep(ig3)
[(1,2,3,9,10,11,6,7,8,4,5)(12), (1,8,4,2,11,6,3,12,10,7,5)(9)]

Giving these two generators to GAP one verifies that they indeed generate $M_{12} $

gap> ig3:=Group((1,2,3,9,10,11,6,7,8,4,5)(12), (1,8,4,2,11,6,3,12,10,7,5)(9));
Group([ (1,2,3,9,10,11,6,7,8,4,5), (1,8,4,2,11,6,3,12,10,7,5) ])
gap> IsSimpleGroup(ig3);
true
gap> IsomorphismTypeInfoFiniteSimpleGroup(ig3);
rec( series := “Spor”, name := “M(12)” )

kfarey has plenty of other useful functions. One can even create an .eps file of the fundamental domain specified by the subgroup of the modular group encoded by the Farey symbol using MakeEpsFile(F). For the above example it returns the picture on the right. Not quite as nice as the one on the left, but surely a lot easier to create.

the iguanodon dissected

Published November 9, 2007 by lieven

Here the details of the iguanodon series. Start with the Farey sequence $F(n) $of order n which is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. Here are the first eight Fareys

F(1) = {0⁄1, 1⁄1}
F(2) = {0⁄1, 1⁄2, 1⁄1}
F(3) = {0⁄1, 1⁄3, 1⁄2, 2⁄3, 1⁄1}
F(4) = {0⁄1, 1⁄4, 1⁄3, 1⁄2, 2⁄3, 3⁄4, 1⁄1}
F(5) = {0⁄1, 1⁄5, 1⁄4, 1⁄3, 2⁄5, 1⁄2, 3⁄5, 2⁄3, 3⁄4, 4⁄5, 1⁄1}
F(6) = {0⁄1, 1⁄6, 1⁄5, 1⁄4, 1⁄3, 2⁄5, 1⁄2, 3⁄5, 2⁄3, 3⁄4, 4⁄5, 5⁄6, 1⁄1}
F(7) = {0⁄1, 1⁄7, 1⁄6, 1⁄5, 1⁄4, 2⁄7, 1⁄3, 2⁄5, 3⁄7, 1⁄2, 4⁄7, 3⁄5, 2⁄3, 5⁄7, 3⁄4, 4⁄5, 5⁄6, 6⁄7, 1⁄1}
F(8) = {0⁄1, 1⁄8, 1⁄7, 1⁄6, 1⁄5, 1⁄4, 2⁄7, 1⁄3, 3⁄8, 2⁄5, 3⁄7, 1⁄2, 4⁄7, 3⁄5, 5⁄8, 2⁄3, 5⁄7, 3⁄4, 4⁄5, 5⁄6, 6⁄7, 7⁄8, 1⁄1}

Farey sequences have plenty of mysterious properties. For example, in 1924 J. Franel and Edmund Landau proved that an asymptotic density result about Farey sequences is equivalent to the Riemann hypothesis.
More precisely, let a(n) be the number of terms in the Farey sequence F(n) (that is, a(1)=2,a(2)=3,…,a(8)=23 etc. This is sequence A005728 in the online integer sequences catalog).
Let $F(n)_j $ denote the j-th term in F(n), then the following conjecture is equivalent to the Riemann hypothesis

For every $\epsilon > 0 $ there is a constant C depending on $\epsilon $ such that

$\sum_{j=1}^{a(n)} | F(n)_j – \frac{j}{a(n)} | < C n^{\frac{1}{2}+\epsilon} $

when n goes to infinity. Anyway, let us continue our construction. Farey sequences are clearly symmetric around 1/2 so let us just take half of them, so we jump to 1 when we have reached 1/2. Let us extend this halved Farey on both sides with $\infty $ and call it the modified Farey sequence f(n). For example,

$f(3) = {~\infty,0,\frac{1}{3},\frac{1}{2},1,\infty } $

Now consider the Farey code in which we identify the two sides connected to $\infty $ and mark two consecutive Farey numbers as

[tex]\xymatrix{f(n)_i \ar@{-}[r]_{\bullet} & f(n)_{i+1}}[/tex]

That is, the Farey code associated to the modified sequence f(3) is

[tex]\xymatrix{\infty \ar@{-}[r]_{1} & 0 \ar@{-}[r]_{\bullet} & \frac{1}{3} \ar@{-}[r]_{\bullet} & \frac{1}{2} \ar@{-}[r]_{\bullet} & 1 \ar@{-}[r]_{1} & \infty}[/tex]

Recall from earlier that to a Farey-code we can associate a special polygon by first taking the hyperbolic convex hull of all the terms in the sequence (the region bounded by the vertical lines and the bottom red circles in the picture on the left) and adding to it for each odd interval [tex]\xymatrix{f(n)_i \ar@{-}[r]_{\bullet} & f(n)_{i+1}}[/tex] the triangle just outside the convex hull consisting of two odd edges in the Dedekind tessellation (then we obtain the region bounded by the black geodesics for the sequence f(3)).

Next, we can associate to this special polygon a cuboid tree diagram by considering all even and odd vertices on the boundary (which are tinted red, respectively blue) together with all odd vertices in the interior of the special polygon. These are indicated in the left picture below. If we connect these vertices with the geodesics in the polygon we get a cuboid tree diagram. The obtained cuboid tree diagram is depicted on the right below.

Finally, identifying the red points (as they lie on geodesics connected to $\infty $ which are identified in the Farey code), adding even points on the remaining geodesics and numbering the obtained half-lines we obtain the dessin d’enfant given on the left hand side. To such a dessin we can associate its monodromy group which is a permutation group on the half-lines generated by an order two element indicating which half-lines make up a line and an order three element indicating which half-lines one encounters by walking counter-clockwise around a three-valent vertex. For the dessin on the left the group is therefore the subgroup of $S_{12} $ generated by the elements

$\alpha = (1,2)(3,4)(5,6)(7,8)(9,10)(11,12) $

$\beta = (1,2,3)(4,5,7)(8,9,11) $

and a verification with GAP tells us that this group is the sporadic Mathieu group $M_{12} $. This concludes the description of the second member of the Iguanodon series. If you like to check that the first 8 iguanodons are indeed the simple groups

$L_2(7), M_{12}, A_{16}, M_{24}, A_{28}, A_{40}, A_{48}, A_{60}, \ldots $

the following dissection of the Iguanodon may prove useful

The M(13)-groupoid (2)

Published July 30, 2007 by lieven

Conway’s puzzle M(13) involves the 13 points and 13 lines of $\mathbb{P}^2(\mathbb{F}_3) $. On all but one point numbered counters are placed holding the numbers 1,…,12 and a move involves interchanging one counter and the ‘hole’ (the unique point having no counter) and interchanging the counters on the two other points of the line determined by the first two points. In the picture on the left, the lines are respresented by dashes around the circle in between two counters and the points lying on this line are those that connect to the dash either via a direct line or directly via the circle. In the first part we saw that the group of all reachable positions in Conway’s M(13) puzzle having the hole at the top positions contains the sporadic simple Mathieu group $M_{12} $ as a subgroup. To see the reverse inclusion we have to recall the definition of the ternary Golay code named in honour of the Swiss engineer Marcel Golay who discovered in 1949 the binary Golay code that we will encounter _later on_.

The ternary Golay code $\mathcal{C}_{12} $ is a six-dimenional subspace in $\mathbb{F}_3^{\oplus 12} $ and is spanned by its codewords of weight six (the Hamming distance of $\mathcal{C}_{12} $ whence it is a two-error correcting code). There are $264 = 2 \times 132 $ weight six codewords and they can be obtained from the 132 hexads, we encountered before as the winning positions of Mathieu’s blackjack, by replacing the stars by signs + or – using the following rules. By a tet (from tetracodeword) we mean a 3×4 array having 4 +-signs indicating the row-positions of a tetracodeword. For example

$~\begin{array}{|c|ccc|} \hline & + & & \\ + & & + & \\ & & & + \\ \hline + & 0 & + & – \end{array} $ is the tet corresponding to the bottom-tetracodeword. $\begin{array}{|c|ccc|} \hline & + & & \\ & + & & \\ & + & & \\ \hline & & & \end{array} $ A col is an array having +-signs along one of the four columns. The signed hexads will now be the hexads that can be written as $\mathbb{F}_3 $ vectors as (depending on the column-distributions of the stars in the hexad indicated between brackets)

$col-col~(3^20^2)\qquad \pm(col+tet)~(31^3) \qquad tet-tet~(2^30) \qquad \pm(col+col-tet)~(2^21^2) $

For example, the hexad on the right has column-distribution $2^30 $ so its signed versions are of the form tet-tet. The two tetracodewords must have the same digit (-) at place four (so that they cancel and leave an empty column). It is then easy to determine these two tetracodewords giving the signed hexad (together with its negative, obtained by replacing the order of the two codewords)

$\begin{array}{|c|ccc|} \hline \ast & \ast & & \\ \ast & & \ast & \\ & \ast & \ast & \\ \hline – & + & 0 & – \end{array} $ signed as
$\begin{array}{|c|ccc|} \hline + & & & \\ & & & \\ & + & + & + \\ \hline 0 & – & – & – \end{array} – \begin{array}{|c|ccc|} \hline & + & & \\ + & & + & \\ & & & + \\ \hline + & 0 & + & – \end{array} = \begin{array}{|c|ccc|} \hline + & – & & \\ – & & – & \\ & + & + & \\ \hline – & + & 0 & – \end{array} $

and similarly for the other cases. As Conway&Sloane remark ‘This is one of many cases when the process is easier performed than described’.

We have an order two operation mapping a signed hexad to its negative and as these codewords span the Golay code, this determines an order two automorphism of $\mathcal{C}_{12} $. Further, forgetting about signs, we get the Steiner-system S(5,6,12) of hexads for which the automorphism group is $M_{12} $ hence the automorphism group op the ternary Golay code is $2.M_{12} $, the unique nonsplit central extension of $M_{12} $.

Right, but what is the connection between the Golay code and Conway’s M(13)-puzzle which is played with points and lines in the projective plane $\mathbb{P}^2(\mathbb{F}_3) $? There are 13 points $\mathcal{P} $ so let us consider a 13-dimensional vectorspace $X=\mathbb{F}_3^{\oplus 13} $ with basis $x_p~:~p \in \mathcal{P} $. That is a vector in X is of the form $\vec{v}=\sum_p v_px_p $ and consider the ‘usual’ scalar product $\vec{v}.\vec{w} = \sum_p v_pw_p $ on X. Next, we bring in the lines in $\mathbb{P}^2(\mathbb{F}_3) $.

For each of the 13 lines l consider the vector $\vec{l} = \sum_{p \in l} x_p $ with support the four points lying on l and let $\mathcal{C} $ be the subspace (code) of X spanned by the thirteen vectors $\vec{l} $. Vectors $\vec{c},\vec{d} \in \mathcal{C} $ satisfy the remarkable identity $\vec{c}.\vec{d} = (\sum_p c_p)(\sum_p d_p) $. Indeed, both sides are bilinear in $\vec{c},\vec{d} $ so it suffices to check teh identity for two line-vectors $\vec{l},\vec{m} $. The right hand side is then 4.4=16=1 mod 3 which equals the left hand side as two lines either intersect in one point or are equal (and hence have 4 points in common). The identity applied to $\vec{c}=\vec{d} $ gives us (note that the squares in $\mathbb{F}_3 $ are {0,1}) information about the weight (that is, the number of non-zero digits) of codewords in $\mathcal{C} $

$wt(\vec{c})~mod(3) = \sum_p c_p^2 = (\sum_p c_p)^2 \in \{ 0,1 \} $

Let $\mathcal{C}’ $ be the collection of $\vec{c} \in \mathcal{C} $ of weight zero (modulo 3) then one can verify that $\mathcal{C}’ $ is the orthogonal complement of $\mathcal{C} $ with respect to the scalar product and that the dimension of $\mathcal{C} $ is seven whereas that of $\mathcal{C}’ $ is six.
Now, let for a point p be $\mathcal{G}_p $ the restriction of

$\mathcal{C}_p = \{ c \in \mathcal{C}~|~c_p = – \sum_{q \in \mathcal{P}} c_q \} $

to the coordinates of $\mathcal{P} – \{ p \} $, then $\mathcal{G}_p $ is clearly a six dimensional code in a 12-dimensional space. A bit more work shows that $\mathcal{G}_p $ is a self-dual code with minimal weight greater or equal to six, whence it must be the ternary Golay code! Now we are nearly done. _Next time_ we will introduce a reversi-version of M(13) and use the above facts to deduce that the basic group of the Mathieu-groupoid indeed is the sporadic simple group $M_{12} $.

References

Robert L. Griess, “Twelve sporadic groups” chp. 7 ‘The ternary Golay code and $2.M_{12} $’

John H. Conway and N. J.A. Sloane, “Sphere packings, lattices and groups” chp 11 ‘The Golay codes and the Mathieu groups’

John H. Conway, Noam D. Elkies and Jeremy L. Martin, ‘The Mathieu group $M_{12} $ and its pseudogroup extension $M_{13} $’ arXiv:math.GR/0508630