Symmetric Representation of the Elements of the Conway Group ·0 more

Symmetric Representation of the Elements of the Conway Group ·0 R.T. CURTIS and B.T. FAIRBAIRN School of Mathematics, University of Birmingham, Edgbaston, Birmingham B29 2TT, UK Abstract In this paper we represent each element of the Conway group ·0 as a permutation on 24 letters from the Mathieu group M24 , followed by a signchange on a codeword of the binary Golay code (multiplication by a diagonal matrix taking the value -1 on the positions of a codeword and 1 otherwise), followed by a word of length at most four in a highly symmetric generating set. We describe an algorithm for multiplying elements represented in this way, that we have implemented ¯ in Magma. We include a detailed description of Λ4 , the sets of 24 mutually orthogonal 4-vectors in the Leech lattice Λ often referred to as frames of reference or crosses, as they are instrumental to our procedure. In particular we describe the 19 orbits of M24 on these crosses. Key words: Conway Group, Leech Lattice, Symmetric Generation 1. Introduction The Leech lattice, Λ, was discovered by Leech in 1965 in connection with the packing of spheres into 24-dimensional space R24 , so that their centres form a lattice. Its construction relies heavily on the rich combinatorial structure of the Mathieu group M24 . Leech himself considered the group of symmetries fixing the origin 0; he had enough geometric evidence to predict the order of this group to within a factor of two, but could not prove the existence of all the symmetries he anticipated. John Conway subsequently produced a beautifully simple additional symmetry of Λ and in doing so determined the order of the group it generated together with the monomial subgroup used in the construction of Λ (see (6)). He proved that this is the full group of symmetries of Λ fixing the origin; that it is perfect; has centre of order two and that the quotient by its centre is simple. An element of the group may be represented as a permutation on the 2-vectors of the Leech lattice, of which there are 196560, or as a 24 × 24 matrix. Now Magma, (5), and This research was partly supported by The EPSRC Email address: fairbaib@maths.bham.ac.uk, rtc@bham.ac.uk (R.T. CURTIS and B.T. FAIRBAIRN). Preprint submitted to Elsevier 30 January 2009 other group theoretic packages, handle permutations and matrices of these sizes with ease. Even so, recording and transmitting particular elements (other than electronically) is rather inconvenient. In this paper we describe a black box algorithm using the techniques of symmetric generation to express the elements in an extremely concise manner, namely as a permutation on 24 letters from the Mathieu group M24 , followed by a signchange on a codeword of the binary Golay code, followed by a word of length at most four in a highly symmetric set of generators, as will be defined in Section 2. Our approach thus records an element of ·0 with at most 64 (= 24+24+4×4) pieces of information, and typically much less. This compact representation, which we refer to as symmetrical representation neatly restricts to many of the subgroups of ·0, and often illuminates the geometric structure of the Leech lattice. However, the authors are aware that manipulation with elements represented in this manner is unlikely to compete for speed with 24 × 24 matrices, and so we have written programs which translate between the two forms. Thus symmetrically represented elements may be converted into matrices, whatever operation is required is carried out using the matrices, and they are then converted back into their symmetrically represented form. The program itself provides a good example of the power and effectiveness of Magma in performing complicated mathematical operations. For further information about ·0 the reader is referred to the Atlas (9) and Conway and Sloane (8). The symmetric generation approach to groups has provided a practical means of computing inside some quite complicated groups. For instance, in (12), Curtis symmetrically generates the sporadic group J1 . Specifically, he proved that: 2 11 : L2 (11) ∼ J1 , = ((0, 8, 1)(2, 7, 9, X, 6, 5)(3, 4)t0 )5 where the notation is explained in the next section. Curtis and Hasan were subsequently able to use this expression to write programs giving the elements of J1 in the form πw where π ∈ L2 (11) ≤ S11 is a permutation of degree 11 and w is a word in the 11 symmetric generators of length at most four (see (13)). Their ‘Program B’ used the relations obtained when performing an enumeration of the double cosets of form N wN , where N ∼ L2 (11), in J1 to shorten words in the symmetric generators to give words in = canonically shortest form. In Section 2 we describe the techniques of symmetric generation and their application to constructing the Conway group ·0. In Section 3 we describe the geometric features of the Leech lattice used by our algorithm that is described in Section 4. 2. 2.1. Preliminaries Involutory Symmetric Generation We shall describe here only the case when the symmetric generators are involutions as originally discussed by Curtis, Hammas and Bray, (2). For a discussion of the more general case see (1). We let 2 n denote the free product of n cyclic groups of order 2 and write {t1 , t2 , . . . , tn } for a set of involutory generators of this free product. A permutation π ∈ Sn induces an 2 ˆ automorphism, π , of this free product by permuting its generators, ie tπ = tπ(i) . Given a ˆ i n subgroup N ≤ Sn we can form a semi-direct product P= 2 : N where, for π ∈ N , π −1 ti π = tπ(i) . When N is transitive we call P a progenitor. (Note that some of the early papers on symmetric generation insisted that N acts at least two transitively.) We call N the control group of P and the ti ’s the symmetric generators. Elements of P can be written in the form πw with π ∈ N and w is a word in the symmetric generators, so any homomorphic image of the progenitor can be obtained by factoring out relations of the form πw = 1. The desired homomorphic image of P, often a finite group, is called the target group. If G is the target group obtained by factoring the progenitor 2 n : N by the relations π1 w1 , π2 w2 , . . . we write 2 n: N ∼ G. = π1 w1 , π2 w2 , . . . Consideration of the structure of N often leads us to suitable relations by which to factor P, without causing total collapse. More specifically, Lemma 1 of (11) states that Lemma 1. ti1 , . . . , tjm ∩ N ≤ CN (N i1 i2 ...im ) In the above N i1 i2 ...im stands for the stabilizer in N of i1 , i2 . . . , im . In other words an element of N that can be written in terms of a given set of symmetric generators without causing collapse must centralize the pointwise stabilizer in N of those generators. So, in particular, an element of the control subgroup N can only be written in terms of two of the symmetric generators if it lies in the centralizer in N of the corresponding 2-point stabilizer. To decide whether a given target group of the progenitor is finite, we use the orbit-stabilizer theorem. Given a word in the symmetric generators, w, we define the coset stabilizing subgroup N (w) := {π ∈ N |N wπ = N w}. This is clearly a subgroup of N and there are |N : N (w) | right cosets of N (w) in the double coset N wN . We can often count by hand the number of single cosets in each of these double cosets, and sum these numbers to obtain the index of N in G. However, the double coset enumerator of John Bray and the first author, see (3), effectively performs this operation mechanically for very large indices. A family of results suggests that this approach lends itself particularly well to the construction of simple groups. For instance we have: Lemma 2. If N is perfect and transitive, then |P: P |=2 and P = P . See (15, Theorem 3.1, page 61). Corollary 3. Let G= 2 n:N , π1 w1 , π2 w2 , . . . 3 where N is a perfect permutation group acting transitively on n letters, and one of the words w1 , w2 , . . . is of odd length. Then G itself is perfect. 2.2. The Conway group ·0 A Steiner system S(5,8,24) is a collection of 759 8-element subsets known as octads of a 24-element set, Ω, such that any 5-element subset is contained in precisely one octad. It turns out that such a system is unique up to relabelling, and the group of permutations of Ω that preserve such a system is a copy of the sporadic simple Mathieu group M24 which acts 5-transitively on the 24 points of Ω. A much used approach to this Steiner system is the first author’s Miracle Octad Generator (MOG) first appearing in (14; 10). A more recent account is given in Chapter 11 of (8). 1 The MOG is an arrangement of the 24 points of Ω into a 4×6 array in which the octads assume a particularly recognisable form; so it is easy to read them off and to write down elements of M24 . The 24 points may be partitioned into six tetrads (subsets of size 4) so that the union of any two of these tetrads is a (special) octad; such a partition is known as a sextet. Naturally the MOG arrangement is chosen so that its six columns form a sextet. A partition of the 24 points into 3 disjoint octads, such as is achieved by a pairing of the columns of the MOG, is called a trio. Indeed, the pairing of the columns 12 · 34 · 56 is known as the standard trio and its threee octads are the bricks of the MOG. The Hexacode, H, is a 3-dimensional quaternary code of length six whose codewords give an algebraic notation for the binary codewords of C as given in the MOG. Explicitly if {0, 1, ω, ω } = K ∼ GF4 , then ¯ = H = = (1, 1, 1, 1, 0, 0), (0, 0, 1, 1, 1, 1), (¯ , ω, ω , ω, ω , ω) ω ¯ ¯ {(0, 0, 0, 0, 0, 0) (1 such), (0, 0, 1, 1, 1, 1) (9 such), (¯ , ω, ω , ω, ω , ω) (12 such), ω ¯ ¯ (¯ , ω, 0, 1, 0, 1) (36 such), (1, 1, ω, ω, ω , ω ) (6 such)} ω ¯¯ where multiplication by powers of ω are of course allowed, as is an S4 of permutations of the coordinates corresponding to S4 ∼ (135)(246), (12)(34), (13)(24) (the even permu= tations of the wreath product of shape 2 S3 fixing the pairing 12 · 34 · 56). As is explained below, each hexacode word has an even and odd interpretation and each interpretation corresponds to 25 binary codewords in C, giving the 43 × 2 × 25 = 212 binary codewords of C. The rows of the MOG are labelled in descending order with the elements of K as shown in Figure 1, thus the top row is labelled 0. Let (h1 , . . . , h6 ) ∈ H. Then in the odd interpretation if hi = λ ∈ K we place a 1 in the λ position in the ith column and zeros in the other three positions, or we may complement this and place 0 in the λth position and 1s in the other three positions. We do this for each of the 6 values of i and may complement freely so long as the the number of 1s in the top row is odd. So there are 25 choices. In the even interpretation if hi = λ = 0 we place 1 in the 0th and λth positions and zeros in the other two, and as before we may complement. If hi = 0 then we place 0 in 1 Warning: there are two versions of the MOG appearing in the literature. They are mirror images of one another, and may also be obtained from one another by interchanging the last two columns. 4 0 1 ω ω ¯ × 0∼ or × × × × × × × 1∼ × × or × × ω∼ × or × × × ω∼ ¯ or × × ω∼ ¯ or × × × The Odd Interpretation 0 1 ω ω ¯ 0∼ or 1∼ × × × or × × ω∼ × or × × × × × The Even Interpretation Fig. 1. The odd and even interpretations of hexacode words all four positions or 1 in all four positions. This time we may complement freely so long as the number of 1s in the top row is even. Thus for instance × (0, 1, ω , ω, 0, 1) ∼ ¯ ×× ×× × ×× or ××× × × × × × 24 23 11 1 22 2 3 19 4 20 18 10 6 15 1614 8 17 9 5 13 21 12 7 in the odd and even interpretations respectively, where evenly many complementations are allowed in each case. The last figure above shows the standard labelling of the 24 points of Ω used when entering information into a computer. (One usually finds the ‘23’ and ‘24’ replaced with the symbols ‘0’ and ‘∞’ so that the 24 points are labelled using the projective line P1 (23) such that all the permutations of L2 (23) are in M24 .) Let P(Ω) denote the power set of Ω regarded as a 24-dimensional vector space over Z2 . The octads span a 12-dimensional subspace of P(Ω) known as the the binary Golay code, C. This contains the empty set ∅; the 759 octads; 2576 12-element subsets known as dodecads; the 759 complements of the octads known as 16-ads and the whole set, Ω. Since the degree 24 permutation representation of M24 described above is five transitive, M24 acts transitively on tetrads. We can thus form the progenitor 24 P = 2 ( 4 ) : M24 . John Bray used the double coset enumerator (3) to show that a single short relation involving what are essentially Conway’s elements is sufficient to define the group ·0 directly from M24 , obtaining the Leech lattice as a by-product. In (4) he and the first author use this to construct ·0. We describe this construction in more detail. After eliminating shorter relations and other length three relations we are naturally led by Lemma 1 to consider the relation shown diagrammatically in Figure 2. ×× ×× ×× ×× ×× = ×× Fig. 2. The defining relation for ·0 which is also written by the authors more concisely as tab tac tad = ν where {a, b, c, d} is a partition of an octad into four pairs and ν is the permutation uniquely determined by Lemma 1. That is to say CM24 (StabM24 ([a, b, c, d])) = ν . 5 Here, symmetric generators (tetrads) are displayed by crosses placed in the MOG-diagram in the positions corresponding to the four points. To show that the resulting group is finite, we need to enumerate the double cosets of the form N wN where N ∼ M24 and w is a word in the symmetric generators. Unfortunately = the index we hope for is too large to be computed by hand. The authors of (4) had previously described in (3) a double coset enumerator for symmetrically presented groups which resembles the famous Todd-Coxeter single coset enumeration algorithm (17); it is written in the Magma language (5). The index of M24 inside ·0 is still too large for this program to be used directly to enumerate double cosets N wN , where N is the control subroup M24 . To get round this the authors consider the product of two symmetric generators corresponding to a pair of tetrads whose union is an octad. Indeed they prove by hand the following. Lemma 4. If A and A are two tetrads such that A ∪ A = O, an octad, then tA tA = tA tA = tX tX for any tetrads X and X such that X ∪ X = O. In particular tA tA has order two. 2 So tA tA depends only on the octad O and we may label this element O . They further show that any two elements O and O commute and that together the 759 elements so obtained generate an elementary group of order 212 . This group is plainly normalised by our M24 and so we may use a subgroup H ∼ 212 : M24 = in our coset enumeration. Indeed an easy application of symmetric generation shows that 2 759 : M24 ∼ 12 = 2 : M24 , tO tU tO U = 1 where O and U are octads intersecting in four points, and denotes symmetric difference. In particular this leads to the existence of an element Ω ∈ H that commutes with the whole of M24 and with all of the symmetric generators and will thus be central in the target group. This enables the authors to enumerate the double cosets of form HwN where H ∼ 212 : M24 , which is within range of computers. An adapted version of the = coset enumeration program is then used to prove that the target group is indeed finite and of order |·0|. In particular the output of the program gives the sizes of the orbits of the action of M24 on the right cosets of H. This can be realised geometrically in the Leech lattice in terms of the ‘crosses’ - see the next section for details. To actually prove that the target is indeed ·0 it is necessary to exhibit elements of ·0 that satisfy the relation of Figure 2, i.e. to find elements of ·0 to act as symmetric generators. To this end the authors of (4) construct a representation of the target group as follows. The lowest dimension in which M24 can act faithfully is 23. By Corollary 3, and the fact that the above relation has odd length, we know our target group is perfect. Consequently all elements of the group must have determinant 1, since all commutators have determinant 1. Since Ω ∈ H and the determinant of this element when represented in n dimensions is (-1)n any representation of our target group must be of even dimension. The lowest possible dimension for a representation of the target group is therefore 24. Furthermore the image of the symmetric generator, tT for some tetrad T , in any representation must also: • commute with the tetrad stabilizer in M24 ; 6 • commute with the elements O where O is an octad that is the union of two tetrads in the sextet defined by T ; • have order 2; • satisfy the relation of Figure 2. With the above conditions the authors of (4) show not only that such a representation exists in 24 dimensions, but that the action of the symmetric generators admits a beautiful and startlingly simple description. Let T be a tetrad and let {T1 = T, T2 , . . . , T6 } be the sextet defined by T . Then to apply tT to a vector in R24 one need only compute as follows: for each tetrad Ti work out one half the sum of the entries in Ti and subtract it from each of the four entries; then negate on every entry except those in T = T1 . (This element is equal to the central involution of ·0 multiplied by the original involution ξT discovered by Conway that corresponds to the tetrad T in (6).) For instance taking T to be the leftmost column of the MOG (so the sextet of T is simply the columns of the MOG, the standard sextet) we have the action shown in Figure 3. 8 4 → -4 -4 -4 22222 -3 -1 -1 -1 -1 -1 → -1 1 1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 1 tT : and tT : 2 2 2 Fig. 3. The tetrad element tT acting on vectors of Λ Furthermore, if T1 and T2 are two tetrads of a sextet and T1 ∪ T2 = O, then it is readily checked that tT1 tT2 = O . Moreover these elements act in this representation as follows:   −e for i ∈ O; i O (ei ) =  e for i ∈ O, i where {e1 . . . e24 } is the standard basis for R24 . It is for this reason the elements S for S ⊂ Ω are known as signchanges. From this the Leech lattice, Λ, simply ‘falls out’ by considering the Z-span of the images of the basis vectors, normalised to avoid fractions. In particular we have that Theorem 5. (Conway) The integral vector v = (x1 , x2 , . . . , x24 ) is in Λ if, and only if, (i) the xi all have the same parity; (ii) the set of i where xi takes any given value (modulo 4) is contained in C, the binary Golay code as defined in Section 2.2; and (iii) Σxi ≡ 0 or 4 (modulo 8) according as xi ≡ 0 or 1 (modulo 2). as first proved in (7). The conditions (i)–(iii) are clearly satisfied by the vectors 8ei and clearly extends linearly. Moreover these conditions are preserved by the elements tT for any tetrad T . Thus the lattice just obtained is the Leech lattice. 7 With this information the authors prove: a b c d a b c d 24 2 ( 4 ) : M24 ∼ = ·0, where a, b, c, d are as shown in tab tac tad = ν . (6) and ν is the involution shown in Figure 2. 3. The Frames of Reference or Crosses of the Leech Lattice Let Λ be the Leech lattice. For n ∈ Z+ set Λn := {x ∈ Λ | x · x = 16n} and call the members of Λn n-vectors. Factoring out by 2Λ we get the vector space Λ/2Λ of dimension 24 over the field Z2 , as follows. Suppose u, v ∈ Λ2 ∪ Λ3 ∪ Λ4 , that u = ±v, and that u ≡ v modulo 2Λ, then u − v = 2w, say, for some w ∈ Λ and u + v = 2(w + v) = 2w , w ∈ Λ. Without loss of generality, replacing v by −v if necessary, we have 8.16 ≤ 4||w|| = ||u − v|| ≤ ||u|| + ||v|| ≤ 2 × 4.16 = 8.16. Thus equality holds throughout, and so u, v ∈ Λ4 , u ⊥ v, and w, w ∈ Λ2 . In particular, the cosets u+2Λ for u ∈ Λ2 ∪Λ3 are all distinct and contain no vector of Λ4 . Moreover, u = w + w and v = w − w . Thus if, for u ∈ Λ4 , we wish to find other vectors v ∈ Λ4 such that u ≡ v modulo 2Λ, then we write u as a sum of two (perpendicular) vectors of Λ2 , w and w say, when v = w − w is such a vector. Now the maximum number of mutually perpendicular vectors we can have is 24 (together with their negatives), and we see that this maximum is achieved by the set X = {(±8, 0, . . . , 0), (0, ±8, 0, . . . , 0), . . . , (0, . . . , 0, ±8)} , the set of all vectors with a single non-zero entry of ±8. Since this set of vectors corresponds to a scaled version of the standard basis we refer to it as a frame of reference or, more concisely, as a cross, this one being the standard cross. Now the group ·0 acts transitively on vectors in Λ4 and, since congruence modulo 2Λ is an equivalence relation, we see that the 4−vectors may be partitioned into a set of disjoint crosses which we ¯ denote by Λ4 . Note that ¯ 1 + |Λ2 | + |Λ3 | + |Λ4 | = 1 + 98280 + 8386560 + 8292375 = 16777216 = 224 , so this is a complete set of coset representatives of 2Λ in the 24-dimensional Z2 −vector space Λ/2Λ. Given a cross Ξ, the 48 vectors that make up Ξ will be referred to as the branches of Ξ and the 24 pairings of a 4-vector in Ξ and its negative will be referred to as the diameters of Ξ. The stabilizer of a cross is the monomial maximal subgroup H ∼ 212 :M24 ≤ ·0, which has just six orbits on the crosses. These were tabulated in = matrix form in (14) and for convenience are reproduced in Figure 8. It should be noted that, since the standard cross is simply a scaled version of the standard basis of our 24-dimensional vector space, the rows of each element of ·0 written as a 24 × 24 matrix will be 24 mutually orthogonal vectors of a cross. In this sense the crosses are fundamental to an understanding of ·0, and so we shall give a more detailed description of them. Explicitly, for each type we shall give a canonical representative, the other crosses of this type being obtained from this one by applying a permutation of M24 followed by sign changes on a C-set. 8 3.1. The Types of Cross The types of cross are simply the orbits of crosses under the action of the monomial subgroup isomorphic to 212 : M24 stabilizing the standard cross. However it should be noted that the type of a cross determines it up to sign changes, in the sense that for instance all crosses corresponding to a particular involution can be obtained from one another by sign changes. (i) The Standard Cross. The vectors of this cross consist of the 48 vectors with ±8 in one coordinate position, and zeros elsewhere as mentioned on the previous page. Thus they are the standard basis vectors multiplied by 8, so that their norm is 64 = 4 × 16. This cross is fixed by all 212 sign changes. (ii) Sextet Type. Each sextet has two crosses associated with it, each of which consists of 4-vectors with ±4 in the positions of a tetrad of the sextet; in one every 4-vector has an even number of +4s and in the other every 4-vector has an odd number of +4s. Such a cross is fixed by a subgroup of the group of sign changes of order 211 , with the remaining coset interchanging odd and even sextet crosses. (iii) Octad Type. Subtracting a suitable 4-vector in the standard cross from a 2-vector of shape (28 016 ) we obtain the 4-vector v = (−6 27 016 ). We shall now apply the process outlined above to write down the other vectors in the cross defined by v; explicitly we shall write v as the sum of two 2-vectors v = u1 + u2 in all possible ways, then the required vectors will be given by u1 − u2 and their negatives. v (−6 27 016 ) 016 ) (2 −6 26 u2 (−4 u1 (−3 u2 (−3 u1 (−3 u2 (−3 40 6 u1 − u2 Number u1 (−2 −2 26 016 ) 7 0) 18 −18 ) (0 07 28 −28 ) 15 18 ) 116 ) (0 07 216 ) 1 16 16 17 17 −18 17 7 1 −1 ) Total 23 Note that the coordinate −6 can appear in any of the eight positions of the octad, and so this cross depends only on the octad chosen. The only sign changes that fix this cross are a sign change on that octad itself, or a sign change on a set which does not intersect that octad. There is a space of dimension 5 of C-sets disjoint from an octad (30 octads, the complementary 16-ad and the empty set), so altogether this cross is fixed by 21+5 = 26 sign changes. Thus each octad yields 212−6 = 26 crosses. 9 (iv) Triad Type. Given a triad of the 24 points of Ω there is a 4-vector of form v = (5, −3, −3, 121 ). We shall now decompose this 4-vector in the standard manner. v (5 −3 −3 u1 (1 1 −3 121 ) 121 ) (−3 5 −3 u2 (4 −4 0 021 ) 1 −35 116 ) u1 (3 −1 −1 −15 116 ) (1 1 u2 (2 −2 −2 25 016 ) Total 23 21 121 ) 2 u1 − u2 Number Note that the first type of decomposition into a sum of 2-vectors demonstrates that this cross depends only on the triad (in this case the first three coordinate positions). In the second type of decomposition the 21 vectors obtained correspond to the 21 octads containing our invariant triad. Note that no sign change apart from negation preserves this cross, so there are 211 crosses corresponding to a given triad. (v) Involution Type. The sum of two 2-vectors, one of shape 42 022 and the other of shape 016 28 in which the non-zero entries are disjoint, clearly gives rise to a 4-vector of shape 42 014 28 . We shall decompose such a vector as above. v (42 014 014 14 28 ) 28 ) (−42 u1 − u2 Number u1 (02 u2 (4 2 014 28 ) 1 0 0) (02 42 24 − 24 ) 8 u1 (22 u2 (22 u1 (3 1 u2 (1 3 22 012 24 04 ) 14 −22 012 04 24 ) 17 −17 7 7 18 ) (2 −2 27 −27 08 ) 8 8 −1 1 1) Total 23 Let us refer to the octad on which the 2s of v lie as the fixed octad, to its complement as the fixed 16-ad and to the positions on which the two 4s of v lie as the initial pair. Then the first decomposition shows that the 4-vector obtained by negating on the initial pair belongs to the same cross. The second type of decomposition shows that certain vectors having two 4s in positions which, together with the initial pair, form a special tetrad of 10 the fixed 16-ad (that is to say a tetrad of the 16-ad which can be completed to an octad with four points of the fixed octad) also belong to the same cross. However, the pairs of the fixed 16-ad defined in this manner are precisely the transpositions of an involution of M24 of cycle shape 18 .28 , and so the stabilizer of this cross must lie in the centralizer of the corresponding involution. The third type of decomposition shows that the eight 4-vectors of shape (±2)16 08 in which the non-zero entries lie in the fixed 16-ad and have precisely eight +2s lying on an octad containing one but not both points of the initial pair also belong to this cross. Now sign changes on a C-set which lies entirely in the fixed 16-ad and either contains both points of the initial pair, or neither of them, will fix this cross; as will, of course, negation on the whole set Ω. So the subgroup of sign changes fixing this cross is of order 25 . There are thus 27 crosses corresponding to each involution. (vi) Duum Type. Finally we consider a 4-vector which has one coordinate equal to 4 and twelve equal to ±2 lying, of course, on a docecad; parity considerations require there to be an odd number of +2s and so we let v = (4, 011 , −2, 211 ). The partition into two dodecads determined by this vector will be referred to as the fixed duum. v (4 u1 (1 u2 (3 u1 (2 u2 (2 u1 (1 u2 (3 011 −2 111 −3 −111 2 010 10 211 ) 111 ) (−2 u1 − u2 Number 211 −4 011 ) 1 1 111 ) 4 010 2 −25 26 ) 0 05 26 ) (0 11 5 6 −2 0 −2 2 0 ) 3 110 ) (−2 25 − 26 0 4 010 ) 11 15 −16 −1 −15 16 −1 −1 110 ) Total 23 Notice that the 4 appears once in each of the 24 cooordinate positions and so any sign change which fixes this cross must fix (or negate) each of its 4-vectors. Negation on a dodecad of the fixed duum does not fix this cross and so the one sign change that does fix it is negation on the whole set Ω. There are thus 211 crosses of this type corresponding to each duum. Note: In all but one case, the monomial subgroup 212 : M24 acts transitively on 4-vectors of a particular shape, and so all 4-vectors of that shape lie in the same type of cross. The exception is 4-vectors having shape 08 .(±2)16 which, as can be observed from Figure 8, appear in both octad and involution type crosses. Call the 4-element subsets of a 16-ad which can be completed to an octad by adjoining four points of the complementary octad special tetrads. Then the special tetrads of a 16-ad form a Steiner system S(3, 4, 16). So if a 08 .(±2)16 type 4-vector (which must by parity have an even number of -2s) has more than three entries -2, then by sign changing on a suitable octad we can reduce the number 11 of -2s. In this way we may reduce the number of -2s to two or zero, but cannot necessarily reduce them all to zero as no C-set intersects a 16-ad in two points. The subgroup of M24 fixing an octad, which is isomorphic to 24 : A8 , acts doubly transitively on the complementary 16-ad and so the 4-vectors of the first kind are all in the same orbit; they lie in involution type crosses, as can be seen by changing sign on the designated octad in the 8-orbit of the table in (v) above. However, 08 .(±2)16 type 4-vectors of the second kind, that is to say those that can be reduced to 08 .216 by sign changes, lie in octad type crosses. Since our control subgroup here is N ∼ M24 rather than H ∼ 212 : M24 , we shall also = = describe the nineteen orbits under the action of M24 . Thus the number of orbits here will be the number of double cosets of form HwN . Finally in Figure 5 we exhibit a convenient notation for the crosses in each of the nineteen orbits. The standard cross is, as usual denoted by X. The type of a cross is indicated by the subscript: S (sextet), O (octad), T (triad), I (involution), D (duum). Sextet type crosses are denoted by the fixed sextet with a superscript of + (even type) and − (odd type). The canonical octad type cross is denoted by (white) circles in the positions of the octad; a cross obtained from this one by negating on a C−set shows in black the points of intersection of this C−set with the fixed octad (note that this defines the cross uniquely). The triad type crosses are all obtained from the canonical triad cross by negation on a C−set (or its complement) so the orbits of M24 on such crosses are just the orbits of the stabilizer of a triad, namely a subgroup isomorphic to M21:S3 , on the C−sets (modulo the whole set Ω). They are not drawn in full in Figure 5 as they are all denoted by the fixed triad, shown by asterisks, and a C−set shown by circles. The involution type crosses fall into four orbits under the action of M24 . Members of the first two types possess a unique 4−vector with +2 in the positions of an octad and ±4 in two positions outside this octad; they are represented in Figure 5 by circles in the positions of the octad and either plus signs indicating two 4s of the same sign or two minus signs indicating two 4s of different signs. The canonical such cross has +2 on each coordinate corresponding to the fixed octad of the 18 .28 shaped involution; the 8 canonical crosses associated with this involution correspond to the 8 tranpositions of the involution, the positions in which the two 4s may be placed. We repeat that such a canonical involution type cross contains just one 4-vector of this form; its other 4−vectors of this shape have four +2s and four −2s. All other crosses of this type contain a 4-vector with precisely two minus signs on the fixed octad. However, such a 4-vector is no longer unique and a cross in this orbit contains four such 4-vectors. Thus the two black points for such a cross can be chosen in four ways and the total number (for a particular involution) is 8 × 8/4 = 56, giving a total of 759 × 15 × 56 = 637560 2 crosses in this orbit. A cross of this type can be denoted diagrammatically by a sextet (whose tetrads are labeled 1, . . . ,6) together with an octad cutting the six tetrads of the sextet as 24 .02 . Thus the diagram shown in Figure 5, namely; ffff 12345 ffff 12345 12345 12345 represents the cross containing the 4−vectors: + 6 6 6 6I 12 -2 2 2 2 -2 2 2 2 4 4 2 -2 2 2 2 -2 2 2 4 4 2 2 -2 2 2 2 -2 2 4 4 2 2 2 -2 2 2 2 -2 4 4 Note that this notation readily allows us to count the number of crosses in this orbit: the number of sextets (1771) times the number of octads which intersect it 24 .02 (360) which equals 637560. Generators for the subgroup of M24 fixing this cross are given in Figure 3.1. A similar diagram but with the superscript + replaced by − corresponds to crosses which are identical to this one except that the two +4s are replaced by one +4 and one −4. Finally, in the notation for a duum type cross the white circles denote +2, the black circles denote −2 and the asterisk denotes +4. In the case with just one black point the cross contains a unique 4-vector of this form, so there are 2 × 12 × 12 = 288 such crosses for each duum, making a total of 1288 × 288 = 370944 crosses in this orbit; however, when there are three black points the corresponding cross contains three such 4-vectors and so there are 2 × 12 × 12 /3 = 1760 such crosses for each duum making a total of 3 1288 × 1760 = 2266880. Such a cross also contains three more 4-vectors with three -2s, nine +2s and a -4 (rather than a +4). For this reason we prefer the alternative notation for this orbit of crosses in which we identify a set of 6 points which are not contained in an octad (these are known as umbral hexads, as opposed to special hexads which are contained in an octad) and label three of its points + and three −. The stabilizer in M24 of an umbral hexad (the example chosen in Figure 5 is the top row of the MOG)is a subgroup isomorphic to the triple cover of the symmetric group S6 ; the subgroup fixing the two threes has shape 3 · (S3 × S3 ) and generators for it are shown in Figure 3.1. The six 4-vectors of shape ±4.011 . − 23 .29 are obtained from this symbol as follows. The fixed hexad is congruent modulo the Golay code to a sextet (in the example given this is the columns of the MOG) with one point of the hexad in each tetrad of this sextet. The associated duum consists of the symmetric difference of the hexad and the union of the three tetrads containing + signs (together with its complement). A +4 is placed in one of the positions labelled +, -2 is placed in the other three positions of its tetrad, and +2 in the remaining 9 positions of the dodecad. So the example shown in Figure 5, namely: +++−−− D represents the cross containing the 4−vectors: 13 4 222 -2 2 2 -2 2 2 -2 2 2 2 2 2 -4 -2 2 2 -2 2 2 -2 2 2 4 222 2 -2 2 2 -2 2 2 -2 2 222 -4 2 -2 2 2 -2 2 2 -2 2 4222 2 2 -2 2 2 -2 2 2 -2 222 -4 2 2 -2 2 2 -2 2 2 -2 3.2. The Double Cosets of form HwN , where H ∼ 212 : M24 and N ∼ M24 . = = Since H ∼ 212 : M24 fixes the standard cross, the orbits of N ∼ M24 on the various = = types of cross correspond to double cosets of form HwN . Below is a printout of the double coset enumeration of such cosets, obtained using (3), as it appears in (4). > g:=Sym(24); > m24:=sub<g| > g!(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), > g!(23,24)(19,3)(15,6)(5,9)(1,11)(20,4)(14,16)(21,13)>; > xx:=Stabilizer(m24,{23,24,19,3}); > f,nn,k:=CosetAction(m24,xx); > pi:=m24!(19,15,5)(3,6,9)(20,14,21)(4,16,13)(18,8,12)(10,17,7); > 1^f(pi); 1086 > 1086^f(pi); 1093 > sigma:=m24!(23,5)(19,15)(24,9)(3,6)(1,21)(20,14)(11,13)(4,16); > 1^f(sigma); 4192 > nu:=f(m24!(1,11)(20,4)(14,16)(21,13)(22,2)(18,10)(8,17)(12,7)); We first of all feed in two permutations of 24 letters which generate M24 . Here the tetrad {23, 24, 19, 3}, which corresponds to the first symmetric generator in Figure 2, is labelled ‘1’ by the computer. We have then found its images under the action of the elements: pppppp pp pp π= and σ = pp ?????? pp finding that the computer labels 1π = 1086 and 1π = 1093. Note that these numbers are nothing more than labels in the action of M24 on 24 letters. The relation then becomes 4 t1 t1086 t1093 = ν. Now 1 ∪ 1σ is an octad, O; so, since 1σ = 4192, t1 t4192 = O . We can now perform the coset enumeration over the group H = M24 , t1 t4192 ∼ 212 :M24 . = > RR:=[<[1,1086,1093],nu>]; 2 14 > HH:=[*nn,<[1,4192],Id(nn)>*]; > CT:=DCEnum(nn,RR,HH:Print:=5, Grain:=100); Index: 8292375 == Rank: 19 == Edges: 1043 == Time: 156.781 > CT[4]; [ [], [ 1 ], [ 1, 2 ], [ 1, 2, 24 ], [ 1, 4 ], [ 1, 17 ], [ 1, 7 ], [ 1, 2, 61 ], [ 1, 2, 8 ], [ 1, 2, 17 ], [ 1, 2, 59 ], [ 1, 2, 14 ], [ 1, 2, 117 ], [ 1, 2, 204 ], [ 1, 2, 7 ], [ 1, 2, 1 ], [ 1, 2, 259 ], [ 1, 2, 1212 ], [ 1, 2, 17, 1642 ] ] > CT[7]; [ 1, 1771, 637560, 2040192, 26565, 637560, 21252, 2266880, 370944, 728640, 91080, 566720, 91080, 425040, 42504, 759, 340032, 1771, 2024 ] The ordered sets of numbers labelled above as ‘CT[4]’ are representatives for the double (w) cosets HwN and ‘CT[7]’ are the indices of the coset stabilizers, M24 ≤ M24 . For example the 16th coset representative listed above corresponds to the double coset Ht1 t2 t1 N and contains 759 right single cosets of H. Comparing these numbers to the sizes of the orbits of the crosses under the action of H, we can see which cosets correspond to which crosses. We display these in Figure 9 in the order they appear in the above printout. We should emphasise that the program does not automatically shorten the coset representatives, and consequently the above words may not be of shortest possible length. In fact these are of shortest length, but we shall need to investigate the geometry of the lattice more closely before we are able to prove this. In order to investigate these double cosets further we note that Lemma 7. If T1 and T2 are tetrads such that T1 ∪ T2 = O, an octad, then HtT1 = HtT2 . Proof We have tT1 tT2 = O ∈ H. 2 We can see that there are at least four cosets corresponding to words in our tetrads of length two. By Lemma 7 a coset HtT can be represented by any tetrad in the sextet defined by T and so, since we are considering cosets of the form HwN , we need only 15 consider how the second tetrad of a word meets the whole of the standard sextet, rather than how the second tetrad of the word meets a specific tetrad of the standard sextet. In particular, when considering cosets of the form HwN where w has length two, there are just six cases to consider. In two of the cases the relation collapses the word to a word of length 0 or 1. These are shown in Figure 10. Consequently there are indeed only four cosets that can be represented by a word of length exactly two. Therefore, other than the two cosets with representatives of lengths zero and one, all cosets must have representatives of length at least three. The unique coset whose representative is of length four also cannot be indexed by a shorter word. We postpone the proof of this until the next section. Using this information we can begin to find explicit representatives for each double coset HwN , i.e. for each of the nineteen orbits of crosses under the action of M24 we can find a word in the symmetric generators w mapping the standard cross to a cross in that orbit. We consider each type of cross in turn, listing for each cross a coset representative (these are useful for testing the program.) 3.3. The Standard Cross The stabilizer of the standard cross is the group of monomial maps, namely the subgroup H. The coset representative corresponding to this cross is therefore the empty word. 3.4. The Sextet Crosses Figure 9 confirms that there are two cosets corresponding to sextet crosses, namely Ht1 N and Ht1 t2 t1212 N as they appear here. As we have seen, one coset corresponds to vectors containing an odd number of negative components, the other to vectors containing an even number of negative components. If B and C are two tetrads such that B ∪ C is an octad then tB tC = O and so if B ∪ C meets the sextet of another tetrad A oddly, then tA tB tC will map the standard cross to the even type cross corresponding to the sextet defined by A. There are 1771 crosses in each orbit (one for each sextet). From Figure 9, we see that no words of length two will map the standard cross to an even sextet cross, so these are the shortest possible coset representatives. In Figure 11 we give a representative for the double coset corresponding to the even sextet. 3.5. The Octad Crosses Recall that such a cross corresponds to an octad O and will contain vectors of the form (−6, ±27 , 016 ) taking non-zero values in O. There are three orbits of octad crosses under the action of M24 . One orbit contains crosses with vectors of the form v := (−6, 27 , 016 ). The other two orbits consist of crosses containing an images of v after a signchange on a Golay codeword that meets O in either two and four points. These orbits thus contain 759, 759 × 8 = 759 × 28 = 21252 and 2 759 × 8 /2 = 759 × 35 = 26565 crosses. From Figure 9 we see that two of these orbits 4 correspond to cosets represented by words of length two and the remaining one to words of length three. In Figure 12 we give coset representatives for these orbits. 16 3.6. The Triad Crosses Recall that each cross in this orbit will contain three vectors of the form (±5, ±32 , ±121 ). Under the action of M24 the triad crosses fall into seven different orbits depending on how the special triad meets an octad or a duum. These orbits are listed in Figure 13. The high transitivity of M24 ensures that none of these orbits split up further, and easy calculations show that they do not fuse. Since 211 =1+210+360+168+21+280+1008, together these account for all the triad crosses. To illustrate the sort of calculations needed to verify the orbit sizes appearing in Figure 13 we shall verify the third of these, the others being extremely similar in nature. Any cross in this orbit will contain a unique vector of the form (-5,-17 ,-32 ,114 ), the -5 and the -1s forming an octad. The number of such vectors, and thus the number of crosses, is 759 × 8 × 16 = 728640. 1 2 We now observe that the unique double coset represented by a length four word as shown, the final coset in Figure 9, has the size 2024= 24 and thus corresponds to the 3 first of these orbits. This sends the standard cross to the cross containing the vector (-32 ,5,121 ) (modulo the action of M24 ). In Lemma 8 of the next section we shall show that this is the shortest possible length of a coset representative. All the other orbits correspond to crosses at distance three from the standard cross X. In Figure 14 we give representatives for these orbits. 3.7. The Involution Crosses The involution crosses fall into four orbits under the action of M24 consisting of two pairs of orbits of equal sizes. One orbit will contain a unique vector of the form (28 , 42 , 014 ). It is thus labelled by the involution and one of its transpositions or, alternatively, by an octad and a pair of points outside that octad. There are therefore 759× 16 = 91080 crosses in this orbit. Similarly, there is another orbit of the same size 2 containing a vector of the form (28 , 4, −4, 014 ). Another orbit of involution crosses contains four 4-vectors of the form (−22 , 26 , 42 , 014 ), where the pairs of positions on which the -2s occur partition the octad into four pairs such that the union of any two defines a sextet one of whose tetrads contains both of the positions occupied by the ±4s. Now the centralizer in M24 of an involution of cycle shape 18 .28 is a subgroup isomorphic to 24 : (23 .L3 (2)), where the normal subgroup of order 16 consists of involutions fixing every point of the octad. The action on the 8 points of the octad itself is given by the homomorphic image 23 .L3 (2), and the pairings defined above correspond to the transpositions of the seven involutions in its normal subgroup of order 8. Turning this process around, we choose an octad and a pairing of its points. Each of the three partitions of these pairs into two tetrads determines a sextet, and these three sextets partition the complementary 16-ad into pairs. A cross in this orbit is determined uniquely by the pairing of the octad and any one of the 8 pairs in the complementary 16-ad. There are therefore 759×105 × 8 = 637560 crosses in this orbit. Similarly there is another orbit of crosses containing a vector of the form (−22 , 26 , 4, −4, 014 ). In Figure 15 we give representatives for these orbits. 17 3.8. The Duum Crosses There are two orbits of duum crosses under the action of M24 . Recall that a vector in this cross will have coefficients with value ±2 throughout some dodecad. An odd number of them will be negative. One of the coefficients outside the dodecad has value ±4. A cross from the first orbit will contain a unique vector of the form (4,011 ,-2,211 ) (and a unique vector of the form (-2,211 ,-4,011 )). All other vectors of such a cross will be of the form (25 ,-27 ,±4,011 ) where the octad defined by the ±2s contains the ±4s since: -2 -2 -2 -2 2 2 2 2 4 -2 2 2 2 2 2 2 2 2 -2 -4 2 2 2 2 2 2 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 3 -1 11 11 11 3 -1 11 11 11 -1 1 1 1 -1 1 1 1 -1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 1 1 1 1 1 3 -1 -1 -1 3 -1 -1 -1 = + = − We thus have an orbit of 1288 × 2 × 12 × 12 = 1288 × 288 = 370944 duum crosses. 1 1 We shall refer to these as monadic duum crosses. A cross from the second orbit will contain three vectors of the form (4,011 ,-23 ,29 ) and three vectors of the form (-23 ,29 ,-4,011 ). We thus have an orbit of 1288×2× 12 × 12 /3 = 1 3 2266880 duum crosses. We shall refer to these as triadic duum crosses. From Figure 9 we see that none of the words of length two can map to a duum cross. The shortest possible coset representative thus has length three. In Figure 16 we give representatives for these orbits. 4. The Program In this section we outline the algorithm of our program that multiplies two elements of ·0 together returning the result in the standard short form using the information of the previous section. The stabilizer of a cross has structure 212 : M24 and is maximal in ·0. Consequently we shall say an element of ·0 is expressed in near standard form if it is of the form π C w where π ∈ M24 ; C is a signchange on a binary Golay codeword and w is some word in our symmetric generators. We shall say that an element of ·0 expressed in near standard form is in standard form if the length of w is minimal. Our algorithm, when given two elements of ·0 in near standard form expresses their product in standard form. From the output of the coset enumerator that we saw in Section 3 we can see that if π C w is in standard form then w has length at most four, since permutations in M24 may be ‘pulled through’ any representative of the coset HwN to give a representative of the coset of the form Hw . Given our new understanding of the crosses of the Leech lattice we can now prove: Lemma 8. There exist double cosets of the form HwN with w a word in the symmetric generators such that the shortest possible length of a word representing it is four. Proof. As noted earlier, length two words can only map the standard cross either to certain types of octad cross or to certain types of involution cross. The diameter of the 18 Cayley graph in which a coset Hw is joined to cosets of form HwtT , for T a tetrad, is therefore at least three. Consider any cross containing a vector of the form u=(5,−32 , 121 ). The sum of any four of the components of u is 0, ±4 or ±8. Any symmetric generator will therefore map the cross corresponding to u to another type of triad cross. As we can see from Figure 9, all other triad crosses are distance three from the vertex corresponding to the standard cross. The longest shortest path in the Cayley graph must therefore have length four. 2 We note that crosses containing vectors of the form (−32 , 5, 121 ) are rare - of the 8292375 crosses there are only 24 =2024 of this form. From the printout appearing in 3 Section 2 we thus see that most crosses are distance three from the standard cross. Given a cross Ξ in the Leech lattice other than the standard cross, X, we define its distance, d(Ξ), to be the least n such that t1 . . . tn is a word of tetrads mapping the standard cross to Ξ. A retriever of Ξ is a tetrad, T , such that d(ΞtT ) = d(Ξ) − 1. (Note that this cannot be defined if we allow Ξ to be the standard cross X.) There are many possible choices for retrievers for any given non-standard cross. The coset representatives exhibited in the previous chapter give examples of retrievers - simply take the rightmost tetrad appearing in each coset representative. Given g ∈ ·0 our algorithm need only compute the image of the standard cross under the action of g. The problem then becomes little more than a question of finding a word in ·0 mapping a given cross back to the standard cross. We proceed to describe our algorithm. Once the type of the cross Xg has been identified we need only compute Xgt1 , where t1 is a retriever of Xg. This process may then be repeated until we have a word of retrievers such that gt1 . . . tn ∈ 212 :M24 . Lemma 8 tells us that if we choose the ti s to make n minimal then n ≤ 4. Next we are required to find a C-set, C, such that C tn . . . t1 g ∈ M24 . To find the C corresponding to tn . . . t1 g we need only determine the image of the vector v = (224 ) under the action of tn . . . t1 g. (Note that v ∈ Λ since v = (28 , 08 , 08 ) + (08 , 28 , 08 ) + (08 , 08 , 28 ).) The codeword C then corresponds to the -2s in tn , . . . , t1 gv. Finally, to determine the element of M24 for our element we use some of the general theory of permutation groups. Let G be a permutation group acting on a set S. A subset B ⊂ S is called a base if the point-wise stabilizer of B is trivial. Consequently the image of B under the action of some element of G will uniquely determine that element. In particular we have: Lemma 9. Any base of minimal size for the natural representation of M24 is of size seven and not contained in an octad of the Steiner system (we say it is umbral ). Proof. By the five transitivity of this action a base must have size at least five. The pointwise stabilizer of such a set has structure 24 : 3 ≤ 24 : A8 . To ‘kill off’ the 24 requires a point outside the octad defined by the five points (where this subgroup acts regularly). This forces our base to be an umbral set. Now the remaining cyclic group of order three acts regularly on the remaining eighteen points and thus requires only one further point to be added to create a base. 2 19 We remark that whilst there may be several ways to represent an element of ·0 in the form π C t1 ...tn with n minimal, testing the equivalence of two such words is easy since there is clearly only a unique such representation for any element acting monomially. In particular there is only a unique such representation of the identity element. Consequently, if π1 then id = −1 C2 π2 π1 C1 t11 −1 . . . t1n t2n . . . t21 = π2 π1 −1 π2 π1 C1 t11 C2 C1 t11 . . . t1n = π2 C2 t21 . . . t2n . . . t1n t2n . . . t21 which is in near standard form, so the algorithm to be applied, making equality easy to verify. We further remark that this algorithm is a black box algorithm since it may easily be used to determine inverses as follows. The inverse of the element π1 C1 t1 . . . tn is clearly equal to tn . . . t1 C π −1 . This is clearly equal to the product of two words written in near π −1 standard form namely (idM24 ∅ tn . . . t1 )(π −1 C ). The algorithm may now be applied to put this product into standard form. The above algorithm has been implemented in Magma (5). The program is freely available to the interested user on the second author’s website, which at the time of writing may be found at http://web.mat.bham.ac.uk/B.Fairbairn/. An additional program that converts elements of ·0 expressed in near standard form into 24×24 matrices with entries in the field Z3 may also be found there along with a program for expressing such a matrix in standard form. We illustrate the main program’s capabilities with an example. We generate random words of ·0 in our standard form by first choosing a random element of M24 then generating a random octad of the binary Golay code, and finally generating a random sequence of tetrads. > g:=Sym(24); > m24:=sub<g|g!(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23), > g!(23,24)(19,3)(15,6)(5,9)(1,11)(20,4)(14,16)(21,13)>; > pi1:=Random(m24); > pi2:=Random(m24); > codeword1:={23,24,19,3,15,6,5,9}^Random(m24); > codeword2:={23,24,19,3,15,6,5,9}^Random(m24); > pi1; (1, 5, 14, 19, 7)(3, 6, 20, 24, 16)(4, 13, 10, 15, 18)(9, 21, 11, 17, 22) > pi2; (1, 2)(3, 5, 24, 9, 17, 16, 21)(4, 13, 10, 15, 14, 8, 18, 20, 19, 11, 7, 23, 6, 22) > codeword1; { 5, 7, 9, 10, 11, 15, 16, 24 } > codeword2; { 7, 8, 10, 11, 16, 20, 21, 22 } 20 The set of numbers {23, 24, 19, 3, 15, 6, 5, 9} corresponds to the left brick of the MOG in the standard numbering given in Section 2, so its image under a random element of M24 will be another octad. (The program can handle more general codewords, though for illustrative purposes an octad is sufficient.) We next produce two random sequences of tetrads each of length 5. > > > > > > > [ tets1:=[{Integers()|0:i in [1..4]}:j in [1..5]]; tets2:=[{Integers()|0:i in [1..4]}:j in [1..5]]; for i in [1..5] do tets1[i]:={1,2,3,4}^Random(m24); tets2[i]:={1,2,3,4}^Random(m24); end for; tets1; { { { { { 4, 15, 19, 23 }, 2, 4, 8, 12 }, 7, 8, 10, 12 }, 6, 7, 16, 24 }, 11, 14, 18, 23 } ] > tets2; [ { 1, 2, 21, 23 }, { 9, 13, 15, 22 }, { 5, 17, 23, 24 }, { 6, 16, 19, 20 }, { 8, 10, 13, 20 } ] (The program can handle words of greater length but for illustrative purposes 5 is sufficient.) > dotto(pi1,codeword1,5,tets,pi2,codeword2,5,tets2); The image of the standard cross contains the vector: -1 -1 -1 -1 3 -1 1 1 -1 -1 -1 -1 1 -1 -1 -3 3 1 -1 1 3 1 3 1 The word of tetrads: 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 21 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 0 The Golay codeword (signchanges on the 1s): 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 and the M24 element: (1, 3, 6, 17, 21, 24, 2, 23, 14, 10, 20, 7, 5, 15, 19) (4, 13, 22, 8, 18)(11, 16, 12) References [1] JN Bray “Symmetric Generation of Sporadic Groups and Related Topics” PhD, Birmingham, 1997. [2] JN Bray, RT Curtis and AMA Hammas “A systematic approach to symmetric presentations. I. Involutory generators” Math. Proc. Camb. Phil. Soc. 119 (1996), pp.2334. [3] JN Bray and RT Curtis “Double coset enumeration of symmetrically generated groups” J. Group Theory 7 (2004), 167-185. [4] JN Bray and RT Curtis “The Leech Lattice, Λ and the Conway Group ·0 Revisited” accepted by the Transactions of the AMS. [5] J Cannon and C Playoust, (1993). An Introduction to Magma. School of Mathematics and Statistics, University of Sydney. [6] JH Conway “A group of order 8,315,553,613,086,720,000” Bull. London Math. Soc., 1 (1969), 79-88. [7] JH Conway “A characterization of Leech’s lattice”, Invent. Math. 7 (1969), 137-142. [8] JH Conway and NJA Sloane “Sphere Packing, Lattices and Groups” third edition Springer-Verlag, New York, (1998). [9] JH Conway, RT Curtis, SP Norton, RA Parker and RA Wilson “An Atlas of Finite Groups”, OUP (1985). [10] RT Curtis “A new combinatorial approach to M24 ” Math. Proc. Cambridge Phil. Soc. (1976) 79, 25–42. [11] RT Curtis “Symmetric Presentations I: Introduction with particular reference to the Mathieu groups M12 and M24 ” in ‘Proceedings of the LMS Durham conference on ‘Groups and Combinatorics” (1990) 22 [12] RT Curtis “Symmetric presentations II: The Janko Group J1 ” J. London Math Soc. (2) 47 (1993), 294-308 [13] RT Curtis and Z Hasan “Symmetric representation of elements of the Janko group J1 ” J. Symbolic Comput. 22 (1996), 201-214. [14] RT Curtis “On the Mathieu Group M24 and Related Topics” PhD thesis, Cambridge (1972). [15] RT Curtis, Symmetric generation of groups, with applications to many of the sporadic finite simple groups. Cambridge University Press (2007). [16] J Leech “Notes on sphere packings” Canad. J. of Math. 19 (1967), pp.251-267. [17] JA Todd and HSM Coxeter “A practical method for ennumerating cosets of finite abstract groups” Proc. Edinburgh Math. Soc. 5 (1936), pp.26-34. 23 orbit type 4 − vectors |sign changes| = 2m |type| × 212−m Number (i) standard (8, 023 ) 212 1 1 (ii) sextet (44 , 020 ) 211 1771 × 2 3542 (iii) octad (−6, 27 , 016 ) (0 , 2 ) (5, −32 , 121 ) (13 , −35 , 116 ) 8 16 26 759 × 26 48576 (iv) triad 2 24 3 × 211 4145152 (v) involution (08 , −22 , 214 ) (28 , 42 , 014 ) (4, 011 , −2, 211 ) 25 759 × 15 × 27 1457280 (vi) duum 2 1288 × 211 2637824 Total Fig. 4. The six types of cross 8292375 24 + X (1) (1) S (1) S (1) O (28) O (35) O *** (1+21+168+360+210+280+1008) T ++ I I + (8) (8) +++ (56) I (56) I * D (288) * = D D (1760) Fig. 5. Concise notation for the 19 orbits of M24 on crosses 25 +/- I . . . . . . . . . . . . . . . . . . . . . . . . .... .... . . . . . . . . . . . . .. Fig. 6. The subgroup of shape 23 : (2 × S4 ) fixing an involution type cross. +++ .. .... .. . . . . . .. . . .. . ... . . . .... .... Fig. 7. The subgroup of shape 3 · (S3 × S3 ) fixing a duum type cross. 26               8 8 8 . . 8 (i) standard 1                             (4) (4) . . . (4) (ii) sextet 1771.2                       2J8 − 8I8                  2J16    (iii) octad 759.26  5 −3 −3        2J8×16        [4] [4]          [4]             −3 5 −3 J21×3       −3 5  −3          5 16   J3×21 −3 .1    (iv) triad 24 3 11      4I12 2J12                   2J12 4I12    (vi) duum 1288.211 2J16×8 (v) involution 759.15.2 7 .2 Fig. 8. The orbits of 212 : M24 on crosses exhibited as 24 × 24 matrices. The rows of each matrix (i)-(vi) are vectors in Λ4 which together with their negatives make up a cross. In each case Jn is the n × n all 1s matrix Jn×m is the n × m all 1s matrix and In is the n × n identity matrix. In case (ii), the partition of the matrix defines a sextet which corresponds to two crosses; the symbol (4) denotes a 4 × 4 matrix of ±4s. The branches of one have an even number of -4s in each row and the branches of the other have an odd number of -4s in each row. In case (iii) the partition defines an octad and a 16-ad. In case (iv), every row other than the first three contains five ±3s; the positions occupied by these make an octad with the fixed triad. All the other entries are ±1. In (v) the symbol [4] denotes a 2 × 2 matrix of ±4s in which the two rows are negatives of one another, whilst the columns are either equal or negatives of one another. Case (vi) has a distinguished duum of the Golay code. 27 Fig. 9. The Cosets/Cross correspondence. Here l(w) refers to the length of the word used to represent the coset and ‘×’ indicates which type of cross it corresponds to. Recall the names of our H-orbits: standard-(i), sextet-(ii), octad-(iii), triad-(iv), involution-(v) and duum-(vi) Type of cross l(w) 0 1 2 3 2 2 2 3 3 3 3 3 3 3 3 3 3 3 4 word [] [1] [1,2] [1,2,24] [1,4] [1,17] [1,7] [1,2,61] [1,2,8] [1,2,17] [1,2,59] [1,2,14] [1,2,117] [1,2,204] [1,2,7] [1,2,1] [1,2,259] [1,2,1212] [1,2,17,1642] |N : N 1 1771 637560 2040192 26565 637560 21252 2266880 370944 728640 91080 566720 91080 425040 42504 759 340032 1771 2024 × × × × × × × × × × × × × × × × (w) | (i) × (ii) (iii) (iv) (v) (vi) × × 28 l (w) representative tetrads, U , for the cosets HtT tU N × × × × × × × × × × × × ; × × × × × × × × × × × × 0 ; 1 ; 2 ; × × × × ××× × 2 ×××× 2 ××× 2 × ; ; ; ××× × ; ×××× × × ×× ××× × Fig. 10. Words of length two. We write l (w) for the length of the shortest word representing the coset HtT tU N . We fix T as the tetrad corresponding to the leftmost column of the MOG. Tetrads, U , appearing on the same row of the table will correspond to the same coset, HtT tU N , despite being in a different orbit under the action of StabM24 (T ). × × × × × × × × ×××× corresponds to 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6+ 6 6. 6 Fig. 11. Coset representatives for an even sextet cross. 29 × × × × × × × × × × × × × × × × corresponds to vf fv ff ff ff vf vf vf vf ff ff ff ff ×××× corresponds to ×××× × × × × corresponds to Fig. 12. Coset representatives for the octad crosses. vector (-3,-3,5,121 ) (-3,-3,5,1 ,-1 ) (3,-3,5,114 ,-17 ) (3,3,5,1 ,-1 ) (3,3,-5,116 ,-15 ) (-3,-3,5,1 ,-1 ) (-3,3,5,110 , −111 ) 9 12 15 6 13 8 number 1 210 360 168 21 280 1008 description Octad disjoint from triad Octad meeting one point of the triad Octad meeting two points of the triad Octad meeting the whole triad Duum meeting the triad in 0/3 points Duum meeting the triad in 1/2 points Fig. 13. The Triad Crosses. In the ‘vector’ column we give one vector from each cross. In the ‘number’ column we give the number of crosses appearing in that orbit divided by 2024(= 24 ) 3 thereby listing the number of different crosses modulo the action of M24 . In the ‘description’ column we give a condition that characterises the orbit in terms of the image of the vector v=(-32 ,5,121 ) under the action of a Golay codeword mapping v into the corresponding orbit. In each case the ‘triad’ is the three points whose coefficients have values ±3 or ±5 in an image of v. 30 × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×××× × × × × × × × × × corresponds to f f × + f × + f × + f f f × + × + × + ×××× f × corresponds to f f × f × × × × f+ × corresponds to f+ × f f+ f × × ff × f+ f × × f × × corresponds to f+ f f+ × × corresponds to × corresponds to fff f f f × × × × × × × × ×××× × ×××× × × × ××× ff f+ f × f+ f × f+ f × ×××× × × × × corresponds to f+ f × f+ f × f+ f × ff ff f fff f × + f × + f f+ × Fig. 14. Coset representatives for the triad crosses × × × × × × × × × × × × × × × × × × ×× 12345 corresponds to 1 2 3 4 5 ffff 12345 ffff 12345 ffff 12341 ffff 12342 corresponds to 65553 56664 66 6 6 1+ 2 3 4 + + ff ff f f- ff ff ff × ××× × × ×× × × × × corresponds to ff ff × × × × × × corresponds to × × Fig. 15. Coset representatives for the involution crosses. 31 × × × × × × × × × × ×× × × × × × × × × v corresponds to f f × + fffff ff ff −− + + × × ×× corresponds to + − Fig. 16. Coset representatives for the duum crosses 32