The Universal Embedding for the Involution Geometry of Co1

Journal of Algebra 217, 555᎐572 Ž1999. Article ID jabr.1997.7058, available online at http:rrwww.idealibrary.com on

The Universal Embedding for the Involution Geometry of Co1 Matthew Kirby Bardoe* Department of Mathematics, Imperial College, 180 Queen’s Gate, London, SW7 2BZ, United Kingdom Communicated by Gernot Stroth Received April 10, 1996

A proof that the universal embedding for the 2-local involution geometry for Co1 over F2 is 300 dimensional. 䊚 1999 Academic Press

INTRODUCTION We conclude in this paper the work started in wBar96b, Bar96ax determining the universal embeddings for groups U4 Ž3. ; Suz ; Co1. These geometries are contained in the dual parapolar space for the Monster simple group, and work on the embeddings for that space has led to a conjecture for the universal embedding of involution geometry for Co1 , see wSmi94x. This work answers this conjecture. An embedding is a mapping of an abstract geometry into some vector space so that the ‘‘points’’ and ‘‘lines’’ of the geometry are mapped to 1-spaces and 2-spaces. By a universal embedding we mean an embedding of the geometry which can be mapped surjectively onto every other embedding for that geometry. The geometry we deal with here is the 2-local geometry described in wRS80x, where the ‘‘parabolic’’ 2 1q 8 Oq Ž . 8 2 will be considered a point stabilizer. This geometry is termed the involution geometry because the point stabilizer corresponds to the centralizer of the 2-central involution and the line stabilizer normalizes a Klein four group of these involutions. Also the universal embedding of the 2-local geometry for Co1 , where the * Partially supported by NSA Grant MDA 904-93-H-3039. E-mail address: m.bardoe@ ic.ac.uk. 555 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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point stabilizer is in a 2 11 : M24 , has already been determined, so our terminology of involution geometry should help to reduce confusion. Notation and Definitions. Let ⌫ be the involution geometry for Co1; this geometry is over F2 in the sense that there are 3 points on each line Žas in the projective line over F2 .. So it will be natural to look for an embedding in the projective geometry of a vector space over F2 . Let V⌫ be the uni¨ ersal embedding of ⌫. Then V⌫ is characterized by the next two definitions. DEFINITION 1 ŽEmbedding.. An embedding of a geometry ⌫ is a vector space V, over the appropriate field Žas explained above., and an injective incidence preserving map, ␲ , from ⌫ to PGŽ V ., the projective geometry of a vector space, such that the points are mapped into the 1-spaces of V and the lines are mapped into the 2-spaces of V. To avoid degenerate situations, we assume that V is generated by images of the points of ⌫. DEFINITION 2 ŽUniversal Embedding.. V⌫ is the universal embedding if V⌫ maps onto every other embedding of ⌫; that is, if Ž V⬘, ␳ . is another embedding of ⌫, then there exists a map f mapping V⌫ onto V⬘, such that f (␲ s ␳ . To determine an upper bound for the dimension of V⌫ we use the facts that the points of ⌫ are mapped to 1-spaces and the lines of ⌫ are mapped to 2-spaces. The primary formulation of these facts will be the line relations. If p is a point of ⌫, then Vp is a 1-space of V⌫ . Let ¨ p denote the unique nonzero vector of Vp . Let l be a line l s  p, q, r 4 where  p, q, r 4 are the points on l. Mapping lines to 2-spaces over F2 implies the following relation in V⌫ : ¨ p q ¨ r q ¨ q s 0. This equation will be called the line relation associated with l. For S ; ⌫ let VS be the subspace spanned by the image of S in V⌫ . Then it follows from the definition that VS q VT s VS j T . During most of the geometrical arguments of this paper we will be working with respect to a fixed point, ␻ ; and at points will use the natural metric, where collinearity is distance 1. If S ; ⌫ then S F n refers to all the points of S at distance less than or equal to n from ␻ . Our proof uses information about several types of subgeometries in ⌫: the involution geometry for G 2 Ž2., also known as the dual hexagon; the involution geometry for J 2 , also known as its near-octagon geometry; and the Suzuki involution geometry related to the GAB geometry of wKan81x. Technical information about these geometries and their universal embeddings is crucial in what follows, and the reader is referred to wFS92, pp. 458᎐459; wBar96ax which contain all necessary information about these geometries.

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It will be necessary later to talk of what we term a ‘‘⌬ base,’’ where ⌬ is isomorphic to one of the geometries discussed above. This configuration is a set of lines, all incident to a single point, such that there exists a subgeometry ⌬ containing these lines. By studying for a fixed involution, z, 1q 8 q Ž . the inclusion of centralizers C Ž AutŽ⌬ .Ž z . ; CC o 1Ž z . ( 2q O 8 2 , we are able to describe the nature of the arrangement of ⌬ bases over a single point. An important feature of our special situation is that the structure of CAutŽ⌬ .Ž z ., in all cases for ⌬ as well as with ⌫ itself, is an extra-special group extended by the corresponding orthogonal group. This extra-special structure of the subgroup O 2 Ž CAutŽ⌬ .Ž z .. allows us to make a correspondence between the lines on a point z and the singular vectors of the corresponding orthogonal space.

1. THE LOWER BOUND Let us start by establishing some notation, then outlining the argument for the main result of this section: THEOREM 3. There exists a 300 dimensional embedding, V300 , for ⌫. Let 24 denote the irreducible F2 module for Co1 which is given by the Leech Lattice mod 2. Let V be a F2 module for some group G. Then S 2 Ž V . denotes the subspace of V m V spanned by the vectors  ¨ m w q w m ¨ , ¨ m ¨ 4 , and let H2 Ž V . be the subspace of V m V spanned by the vectors  ¨ m w q w m ¨ 4 . It is standard that V 2rS 2 Ž V . ( H2 Ž V .. Note that because we are working over F2 , we have that S 2 Ž V . > H2 Ž V .; this is different from the case in odd characteristic. Thus it is somewhat natural ˜ 2 Ž V . ( V 2rH2 Ž V . to add another player to the game, what we call H 2 Žsome authors prefer to denote this quotient by S Ž V ... This is similar to ˜ wswn ˜ ¨ , but is different because the normal wedge product as ¨ n ˜ ¨ n ¨ / 0. Throughout this paper we will make use of the commutator submodule of a G-module M, denoted w G, M x. This is the minimal submodule of M spanned by vectors of the form ¨ y g¨ , g g G and ¨ g M. It is characterized by beginning with the smallest submodule of M such that the G action on the quotient Mrw G, M x is trivial. We prove Theorem 3 in 2 steps: first, we show that there is an embedding in a 299 dimensional section of 24 m 24 for which Ext 1G Ž V, 1. / 0. Then we apply knowledge of the structure of 24 m 24 to show that this module has a nonsplit extension by a trivial module. Applying the following proposition, this will show there exists a 300 dimensional embedding of ⌫.

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PROPOSITION 4. Let ⌫ s Ž P, L . with 3 points per line and G s AutŽ ⌫ . a flag-transiti¨ e automorphism group, such that Gp has no subgroup of index 2. Let 1 be the tri¨ ial F2 G-module and let E s ² ¨ p :p g P be an embedding of ⌫ and X a F2 G-module such that there exists an exact nonsplit sequence of F2 G-modules: ␺

␾

0 ª 1 ª X ª E ª 0. Then X is also an embedding of ⌫. Proof. Let P s  ¨ p g E4 the nonzero vectors in E which generate the 1-space associated to each point p in ⌫ and let the trivial submodule of X be denoted by ImŽ ␺ . s ² u G :. Note that ␾y1 Ž P . can be broken into exactly two orbits of length w G : Gp x as Gp has no subgroup of index 2. Fix p g P and let ¨˜p be an inverse image of ¨ p under ␾ . Label all elements of the G-orbit of ¨˜p to be ¨˜g Ž p. , so that those in the other orbit of ␾y1 Ž P . may be labeled ¨˜p q u G . Let l g L and let  p, q, r 4 be the points on l. As E is an embedding ¨ p q ¨ q q ¨ r s 0, hence either ¨˜p q ¨˜q q ¨˜r s 0 or ¨˜p q ¨˜q q ¨˜r s u G . By transitivity on lines, one of these equations holds independently of the line chosen. In the first case we would have shown that ²Ž ¨˜p .G : ; X is an embedding module of codimension of at most 1 in X. If ²Ž ¨˜p .G : were of codimension 1 then the map ␪ : E ª X given by ␪ Ž ¨ p . ¬ ¨˜p would define a splitting, hence ²Ž ¨˜p .G : s X. The second case easily reduces to the first, by considering ¨˜p q u G instead of ¨˜p . Again let 24 denote the Leech lattice mod 2. Note that this module is self-dual as a Co1 module. We wish to understand the structure of 24 m 24. It is well known that 24 m 24 > S 2 Ž24. > H2 Ž24., and that 24 m 24rS 2 Ž24. ( H2 Ž24. and S 2 Ž24.rH2 Ž24. ( 24. So to understand the structure it is first important to understand the structure of H2 Ž24.: this is a module of dimension 276, and may easily be determined using the known mod-2 characters of Co1 wSq 92x and the self-duality of H2 Ž24.. These considerations show H2 Ž24. is uniserial and 1

. H 2 Ž 24. ( 274 1 Therefore 24 m 24 has a composition series such that the composition factors are Žin order. 1, 274, 1, 24, 1, 274, 1. We now show that 24 cannot be a submodule Žby self-duality, equivalently a quotient. of 24 m 24. Suppose that 24 is a submodule of 24 m 24; then by the description of the 24 given in wSmi95x we know that the maximal subgroup 2 11 : M 24 of Co1 must fix a 1-space of 24 m 24 which is not fixed under Co1. By the adjoint property of Hom, as well as the 2 modular characters of M24

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wSq 92x, we know that the fixed point space of M24 on 24 m 24 is 2 dimensional, dim Hom Ž 24 m 24x CMo241 , 1 . s dim Hom Ž 24x CMo241 , 24x CMo241 . s dim Hom

ž

1 [ 11 1 [ 11 , s 2. 1 [ 11 1 [ 11

/

One of these dimensions is the fixed point space for Co1; by the composition series of 24 m 24 another is easily seen to be ¨ m ¨ where ¨ is a unique nontrivial subspace of 24 fixed by 2 11 : M24 Žthe reduction mod 2 of a coordinate frame.. Let  ¨ , w, ¨ q w4 be the 3 coordinate frame vectors setwise stabilized by 2 4q 12 Ž3.Sp4Ž2. = S3 .. Then ¨ m ¨ q w m w q Ž ¨ q w . m Ž ¨ q w . s ¨ m w q w m ¨ / 0. So the 3 translates of ¨ m ¨ under the S3 in 2 4q 12 Ž3.Sp4Ž2. = S3 . do not span a 2-space, as they would if 24 were a submodule of 24 m 24, similarly for the sum of ¨ m ¨ with the fixed vector for Co1. Therefore 24 is not a submodule Žequivalently a quotient. of 24 m 24. The last information we need to have about this module is supplied by Griess in his Friendly Giant paper wGri82x. LEMMA 5. Ext 1C o 1Ž24, 1. s 0. Proof. In wGri82, Lemma 2.11, p. 10x it is shown that Ext 1C o 1Ž1, 24. s 0, whence the result. With this in hand we now have enough information to explicitly give the structure of 24 m 24. This was also verified by the MeatAxe in GAPwSq 92x, 274 1 24 m 24 ( 1 [ 1 [ 24. 274 1 Proof of Theorem 3. We start by looking at the structure of 24 on Ž . Ž . restriction to our point stabilizer, 2 1q 8 : Oq 8 2 . It is known that O 2 Gp Ž . centralizes an 8-space of 24 on which Oq 2 acts as the natural module, 8 with a quadratic form which is determined by, but not equal to, the standard quadratic form on 24. By the information contained in wRS80, p. 285x three points are on a line l if the three 8-spaces associated to the points meet in a 4-space singular under all three quadratic forms for the three points. This 4-space is stabilized by Gl . The three 8-spaces are permuted by the S3 in Gl . Let 8, 8⬘, 8⬙ be 8-spaces in 24 corresponding to three points on a line. Let  e i 4 be a basis of 8 such that  e1 , . . . , e4 4 is a basis of the singular 4-space equal to the intersection of any 2 of 8, 8⬘, or

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8⬙ and so that e9y i is identified with the dual basis for 8 under the isomorphism provided by the quadratic form. From the structure of Gl ( 2 2q 12 .Ž S3 = L4 Ž2.. we see that there exists an element ␴ of Gl of order 3, such that ␴ cyclically permutes 8, 8⬘, 8⬙ while centralizing their intersection. As ␴ centralizes ² e1 , . . . , e4 : we may take the images of  e i 4 under ␴ to be bases for 8⬘ and 8⬙, and alter e i for 5 F i F 8 so that e i q ␴ Ž e i . q ␴ 2 Ž e i . s 0. ˜ e9yi is fixed under By wGri82, Lemma 2.39Žiii.x the vector ¨ s Ý4is1 e i n ˜ Ž e9yi . the stabilizer of our original 8-space. Similarly ¨ ⬘ s Ý4is1 e i nrsigma 2Ž ˜ and ¨ ⬙ s Ý4is1 e i nrsigma e9yi . are fixed vectors for the point stabilizer of the other points associated to 8⬘ and 8⬙. One easily sees that ¨ q ¨ ⬘ q ¨ ⬙ s 0 by construction. Therefore M s ² ¨ C o 1 : is an embedding module ˜ 24, Co1 x, as embedding modules have no trivial for ⌫; we have M ; w24 n quotients and as ¨ is not symmetric; so M(

274 . 1 [ 24

This shows that V⌫ has dimension at least 299; but by the structure of ˜ 24 has a nonsplit extension by a trivial module. 24 m 24 we know that 24 n Therefore by Proposition 4 there exists a 300 dimensional embedding for ⌫, and it can be constructed as a section of 24 m 24.

2. THE UPPER BOUND Our primary geometric tool in this part of the proof is the use of similar techniques to the ones found in Frohardt and Smith wFS92x in their proof of the universal embedding for J 2 . These techniques have already been generalized in wBar96b, Bar96ax. In this situation we will use a set of well places Suz involution geometries, show that they span almost all points of distance up to 2, then use connected component arguments and arguments akin to those of wBar96ax to complete spanning. The first step is to determine the suborbit diagram. This diagram describes the incidence relations between orbits of the point stabilizer for a fixed involution, ␻ . LEMMA 6. The in¨ olution geometry for Co1 has the suborbit diagram gi¨ en in Fig. 1. Proof. This is presumably well known. The sizes of suborbits have been computed in wILLq 95x. The details for adjacency can be verified exactly as for U4Ž3. in wBar96b, Lemma 2.1x.

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FIG. 1. Suborbit diagram of Co1 involution geometry.

Again, we have labelled each suborbit of the diagram by the ATLAS name of that conjugacy class of the product of an element of that orbit with our fixed involution, ␻ . We add subscripts to our notation when the point stabilizer is not transitive on the set so determined. LEMMA 7. There are 4480 Suz in¨ olution geometries o¨ er a point of ⌫. Proof. The normalizer of a Suz geometry in Co1 is a subgroup 3.Suz : 2 of index 1,545,600. This implies that there are 1,545,600 Suz geometries in ⌫. There are 46,621,575 points in ⌫ and 135,135 points in the Suz geometry in ⌫. Then there are 1545600 = 135135r46621575 s 4480 Suz geometries over a point in ⌫. Recall that ␻ is a fixed point Žinvolution. of ⌫. Now we need a calculation giving us information similar to that of wFS92, 3.5x, about the number of Suz geometries over the appropriate 27 lines. Let ⌬ be a ‘‘Suz base,’’ i.e., the union of 27 lines on ␻ such that there exists a Suz geometry containing ⌬.

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LEMMA 8. There are 4 Suz geometries o¨ er the Suz base, ⌬. Proof. We prove this by counting the number of possible Suz bases. This follows from the correspondence between the lines over ␻ and the Ž . points Ži.e., singular vectors. of Oq 8 2 space. Twenty-seven lines over ␻ will have a Suz geometry containing them precisely when the 27 correŽ . sponding vectors span a O6yŽ 2. space in the Oq 8 2 space. Therefore it is enough to find the number of O6yŽ 2. spaces. From the ATLAS wCq 85x we see that there are 1120 O6yŽ 2. spaces inside Ž . a Oq 8 2 space. Therefore there are 4480r1120 s 4 Suz geometries over ⌬. As explained in the Introduction there is a natural notion of distance in the collinearity graph of ⌫. Any path in the graph whose number of edges is equal to the distance between the endpoints is called a geodesic path. There is a natural notion of distance given by the minimum number of edges in a path. Note from the suborbit diagram that the points of suborbits 4 A and 3 D have a unique point adjacent to them in the collinearity graph which is closer to ␻ . We call the unique path determined in this way a geodesic path Žwith respect to ␻ .. LEMMA 9. Fix a Suz base ⌬ o¨ er ␻ . Let ␥ be a point in any of the 4 Suz geometries o¨ er ⌬, and in suborbits 4 A or 3 D. Then all geodesic paths between ␥ and omega pass through ⌬. Proof. Suborbit 4 A. Consider a point ␥ in suborbit 4 A; from the suborbit diagram it is clear that ␥ determines a unique line back to ␻ . We now count the number of pairs Ž S, L. where S is a Suz geometry over ␻ and L is a line spanning points of S l 2 A1 and S l 4 A. Then there are 4480 possible S’s over ␻ , and in each S, 864 lines spanning points in suborbits 2 A1 and 4 A, so 4480)864 s 3870720 ways of choosing Ž S, L.. On the other hand we have as above 1120 choices of the Suz base, so each of these bases has 54 points adjacent to ␻ and each of these points has 64 ⌫-lines into suborbit 4 A. Therefore, there are 1120)54)64 s 3870720 ways of choosing a Suz base and then a line. This equality shows there is only one way to choose S. Therefore we determine a Suz geometry by choosing a base, then a line into 4 A with a point in that base. This shows that each point of suborbit 4 A with geodesic paths through ⌬ is contained in Suz geometry over ⌬. Suborbit 3 D. Consider a point ␥ in suborbit 3 D; there are 4480 choices for the Suz geometry over ␻ and 9216 points in suborbit 3 D in each Suz geometry. This makes 4480)9216 s 41287680 ways of choosing a S and then ␥ . On the other hand, since the two point stabilizer G␻ , ␥ s A 9 acts 6-transitively on the lines on ␥ back to ␻ , any six lines lie in some Suz geometry. Also, inside each such geometry each set of six lines determines

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a unique set of 27 lines on ␻ , as the vectors associated to these six lines 9 Ž . span a O6y Ž2. in the Oq 8 2 space. Therefore there are 491520) 6 s 41287680 ways of choosing a point and then determining a base, determining a unique Suz geometry on that base. As before, this equality shows that each point of suborbit 3 D with geodesic paths through ⌬ is contained in the Suz geometry over ⌬.

ž/

Let T and U be suborbits, and let lines in T to U mean the lines with 2 points in T and the third in U. We then can form a graph on T by considering the points in T joined by an edge if they are colinear via some line in T to U. T will typically be the disjoint union of connected components with respect to lines in T to U Žoften we will drop the reference to U when it is clear.. PROPOSITION 10. The following hold for ⌫. Ži.

Suborbit 3 D has at most 960 connected components back to 4 A.

Žii. to 3 D.

Suborbit 5B has at most 12096 connected components back

Žiii.

Suborbit 6 E has 1 connected component back to 5B.

Proof. Žcompare with wBar96b, Lemma 2.6; Bar96a, Proposition 2.6x. The methods of proof are similar in all the above situations. First, determine the structure of the two point stabilizer then use the fact that if l s  p, q, r 4 is a line back toward ␻ with two points p and q in the desired suborbit, then the stabilizer of a connected component containing p and q contains the subgroups G␻ , p and G␻ , r, l . Therefore the number of connected components is F the index of the group ² G␻ , p , G␻ , r, l : in G␻ . Case Ži.. Fix notation as above with ␻ a fixed point of ⌫ and p and q points of suborbit 3 D collinear via a line with a third point r in suborbit 4 A. We wish to calculate first G␻ , p s G␻ l Gp . From the suborbit diagram we see that no point adjacent to p is adjacent to ␻ ; therefore O 2 Ž G␻ . l O 2 Ž Gp . s 1. This shows that under the quotient map from G␻ Ž . to G␻rO 2 Ž G␻ . ( Oq 8 2 . G␻ , p is mapped injectively to some subgroup of qŽ . O 8 2 . From the number of points we see that this subgroup must have index 491520r2 9 s 960. From this information, the orbit information given by the suborbit diagram for ⌫, as well as the table of maximal subgroups in the ATLAS wCq 85x, we easily identify G␻ , p ( A 9 .

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Let ␴ be an element of G␻ , r, l which sends p to q. As A 9 is a maximal Ž . subgroup of Oq 8 2 which acts irreducibly on the natural 8 dimensional module, either ² G␻ , p , G␻ , r, l : s G␻ or ² G␻ , p , G␻ , r, l : ( 2 1q 8. A 9 , thus proving Case Ži.. Case Žii.. This works similarly. Case Žiii.. From wBar96a, Proposition 2.6x we know that the appropriate restriction of ² G␻ , p , G␻ , l : to 3.Suz : 2 is 2 1q 6 3.O6y Ž2..2. We can also tell from the sizes of the centralizer G␻ , l and G␻ , l l 3.Suz : 2 that G␻ , l is larger than G␻ , l l 3.Suz : 2. Therefore ² G␻ , p , G␻ , l : is larger than its intersection with 3.Suz : 2. But 3.O6yŽ 2..2 is a maximal subgroup of the O 8 Ž2. quotient of G␻ therefore ² G␻ , p , G␻ , l : s G␻ . Now we will mimic wFS92, Lemma 3.7x to span as many points of distance up to 2 as possible. THEOREM 11. dim V1 A j 2 A 1 j 4 A F 420. Proof. We show that the points in these three suborbits are in the span of the points of suborbits 1 A, 2 A1 , 4 A, and 3 D which are contained in the 4 Suzuki geometries over a fixed Suz base ⌬. The main tool in this theorem is that the points at distance F 2 from ␻ in a fixed J 2 near-octagon are spanned by the points of 2 G 2 Ž2. hexagons over a G 2 Ž2. base wFS92x. Note that also any 3 lines inside a J 2 base form a G 2 Ž2. base wFS92x. Clearly, ␻ is in the span of the points of the 4 Suzuki geometries over ⌬, as ␻ is contained in each of them. To span the points of suborbits 2 A1 , and 4 A it is enough to show that every line on ␻ and every line from 4 A to 2 A1 is in the span of points of the 4 Suzuki geometries. This is done by investigating the points at distance F 2 for any J 2 octagon over ␻ Žas every such line is contained in a J 2 octagon.. Let J be a J 2 octagon whose base intersects ⌬ in a G 2 Ž2. base. As we have added all the points in suborbits 1 A, 2 A1 , 4 A, and 3 D which have geodesic paths through ⌬, such a J 2 octagon intersects the points we have added in at least 2 of the G 2 Ž2. hexagons. Therefore we have spanned all points of J at distance F 2. Therefore if we can show that every line on ␻ outside of ⌬ is contained in some such J 2 octagon we will be done. We now phrase what we need to show in terms of orthogonal spaces: we must show that every singular vector is contained in a O4yŽ 2. space which intersects our fixed O6yŽ 2. space in at least 3 singular vectors. Ž . The stabilizer of a O6yŽ 2. has two orbits on singular vectors of an Oq 8 2 space, those inside the 6-space and an orbit of vectors outside the 6-space. Let V s ² ¨ 1 , ¨ 2 : H ² ¨ 3 , ¨ 4 : H ² ¨ 5 , ¨ 6 : H ² ¨ 7 , ¨ 8 : where ² ¨ 1 , ¨ 2 : and yŽ . ² ¨ 3 , ¨ 4 : are Oq Ž . ² : ² : 2 2 spaces and ¨ 5 , ¨ 6 and ¨ 7 , ¨ 8 are O 2 2 spaces. Let

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S s ² ¨ 1 , ¨ 2 : H ² ¨ 3 , ¨ 4 : H ² ¨ 5 , ¨ 6 : be our fixed O6yŽ 2. space associated to ⌬. We need to show that for any singular vector outside of S there exists a O4yŽ 2. space, J, containing that vector, such that J intersects S in 3 singular vectors. By transitivity of the stabilizer of S on singular vectors outside of S, it is enough to prove the claim for any singular vector outside of S. Let w s ¨ 1 q ¨ 2 q ¨ 7 be a fixed singular vector outside of S. Then w is in J s ² ¨ 1 , ¨ 1 q ¨ 2 q ¨ 7 , ¨ 2 q ¨ 3 , ¨ 2 q ¨ 4 :, a O4yŽ 2. space. J intersects S in the three singular vectors ¨ 2 q ¨ 3 , ¨ 2 q ¨ 4 , and ¨ 1. This shows that the points of suborbits 1 A, 2 A1 , and 4 A are in the span of the points of the four Suz geometries over a ⌬ contained in suborbits 1 A, 2 A1 , 4 A, and 3 D. Using the results of wBar96ax and the construction of the universal embedding of the Suz involution geometry and applying the tools of the computer algebra package GAP we compute that the points of a Suz involution geometry of suborbits 1 A, 2 A1 , 4 A, and 3 D span only 120 of the 143 dimensions of the universal embedding. Also, from the construction of the 300 dimensional embedding for Co1 we can see by characters that on Suz the points in a Suz involution geometry span 143 dimensions. The four Suz geometries over ⌬ all intersect in ⌬ and in the universal embedding for the Suz geometry the points ⌬ span 20 dimensions. Therefore the span of the points in 1 A, 2 A1 , 4 A, and 3 D inside a Suz geometry over ⌬ is at most 4 = 120 y 3 = 20 s 420. PROPOSITION 12. dimŽ V⌫ . F 566. Proof. From Proposition 10 and Theorem 11 we know that dim V1 A j 2 A 1 j 4 A j 3 D F 420 q 960 s 1380. This bound is clearly not yet good enough. From the suborbit diagram we observe that V1 A j 2 A 1 j 4 A j 3 D s V3 D and V1 A j 2 A 1 j 4 A s V4 A . Notice that V3 D rV4 A is a G␻ module. In fact, V3 D rV4 A is a quotient of ␻ the permutation module 1­ G G c o m p where Gc o m p is the stabilizer of a connected component of 3 D with lines back to 4 A. ␻ In a sense this is the source for our first bound, as dim 1­ G G c o m p s 960. What we must do is to find more information concerning the dimension of the kernel of the map from 1GG␻c o m p to V3 D rV4 A . We do this using the computation already completed in wBar96ax. Let W ( V143 be the universal embedding of the Suzuki involution geometry. From computations using the construction of W we see that W3 D l S rW4 A l S is a 35 dimensional quotient of a 72 dimensional permutation module. Using the MeatAxe contained in GAP we identified the kernel of that quotient map and a vector in that kernel. One can see from the size of the connected components of 3 D and it orbits on the lines incident to a point in 3 D, which may be identified with the points of orthogonal space, as well as its restrictions to 3.Suz, that the stabilizer of a connected component is the stabilizer of an oval in the

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orthogonal space. Then as the stabilizer of a connected component is the stabilizer of an oval in the orthogonal space we may identify each connected component with an oval. Therefore we may translate the relation which we computed in the Suz case and write it in terms of the permutation module of the Co1 case. This relation is necessarily satisfied by V3 D rV4 A from the inclusion of the Suz geometry in Co1. Therefore all relations which are conjugates of the G␻ are also satisfied in V3 D rV4 A . The submodule generated by these relations was calculated to be 866 dimensional by GAP. This shows that dim V1 A j 2 a1 j 4 A j 3 D F 420 q 960 y 866 s 514. Similarly for the suborbit 5B, we use the calculation of the universal embedding of the Suz involution geometry to generate a relation satisfied in V5 BrV3 D . We are then able to use GAP to show that this relation under the action of G␻ generates a 12045 dimensional submodule of the 12096 dimensional permutation module. Thus dim V1 A j 2 A 1 j 4 A j 3 D j 5 B F 514 q 12096 y 12045 s 565. Now the points of suborbit 6 E are connected by the lines going back to 5B by Lemma 10. Therefore to span these points costs 1 further dimension to our bound. This set of points clearly spans ⌫. This follows from the line relation and the suborbit diagram wBar96a, Lemma 2.1x. For instance, every point of suborbit 4 D is connected by a line to two points of suborbit 5B, and as every point 2 A 2 is connected by a line to 2 points of 4 D, the points of 2 A 2 are now also in the span. All remaining suborbits fall into the current span using this method. The construction of V300 and the universal property of V⌫ show that V⌫ has a 300 dimensional quotient. Now by the bound obtained above we see that the kernel of the map from V⌫ to V300 has dimension less than 274; and because the two smallest nontrivial irreducible modules for Co1 Žthe full automorphism group of ⌫ . have dimension 24 and 274 wSq 92x, we see that this kernel has composition factors of only modules isomorphic to the Leech lattice mod 2 and trivial modules. The rest of the section is devoted to using techniques of modular representation theory to show this kernel from V⌫ to V300 is zero. This is done by showing that the map from V⌫ to V300 is split and therefore that the kernel is zero because V⌫ has no trivial quotients and no quotients isomorphic to the Leech lattice mod 2. To show various splittings we use a slight extension of the Alperin᎐Gorenstein method wAG72x used in wBar96b, Bar96ax. As above, let 24 denote the Leech lattice taken mod 2. LEMMA 13. Ext 1C o 1Ž24, 24. s 0. Proof. The statement of the lemma is equivalent o Ext 1C o 1Ž24 m 24, 1. s 0, by self-duality of the Leech lattice. Let M be an extension of 24 m 24 by a trivial module T.

Co1

567

INVOLUTION GEOMETRY

First, it is necessary to understand the structure of 24 m 24. This has been completed earlier. From this we know that 274 1 24 m 24 s 1 [ 1 [ 24. 274 1 In Section 1 it is explained that this module is indecomposable and all splittings are displayed. Now we wish to determine the structure of M on restriction to the subgroups 3.Suz, 3 6 : 2. M12 , and 3 : 3 5 : M11. We start by analyzing the structure of 24 m 24 on restriction to the subgroup 3 6 : 2. M12 . This has the following structure as computed by the MeatAxe, 10 1

Ž 24 m 24 . x C3 6o2.1 M 12 s 1 [ 1 [ 264 [ 24. 264

10 1

But various pieces of this structure can be seen from knowledge of the Leech lattice mod 2. As the subgroup 2. M12 acts on 24 as it does on the permutation module on cosets of M11 and therefore in the tensor square, as 24 is self-dual and therefore the action of the 3 6 may be made trivial, the 3 6 trivial submodule is isomorphic to this permutation module. Therefore there does not exist a nonsplit extension. Also, we have dimw M, 3 6 : 2. M12 x s 575. On restriction to 3.Suz we have 1

1

132

1

1

132

Ž 24 m 24 . x C3.So 1u z s 142 [ 142 [ 24 . This can be easily seen by restricting 24 to 3.Suz and then considering the module over ⺖4 . The module can then be considered as the direct sum of the 12 dimensional irreducible and its dual. The structure of the 3-trivial part follows from the considerations in wBar96ax. By the proof of wBar96a, Theorem 2.14x, dim Ext 1S u z Ž142, 1. s 1 and as 3.Suz is perfect we see that any extension of a trivial module must split from this module. So our extension module must split under restriction to these two maximal subgroups. While for their intersection, 3 : 3 5 : M11 , we have the structure 1 1 132 10 [ 10 [ 132 [ 132 [ 24 . 1 1 132 5 Therefore, dimw M, 3 : 3 : M11 x s 574. Define M⬘ to be the complement to T under restriction to 3 6 : 2. M12 . This submodule is invariant under

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MATTHEW KIRBY BARDOE

3 : 3 5 : M11 ; 3 6 : 2. M12 . But this module must be invariant under 3.Suz as it contains w M, 3 : 3 5 : M11 x s w M.3.Suz x by comparing the two displays above, and every subspace containing w M.3.Suz x is invariant under 3.Suz. Therefore M⬘ is invariant under 3.Suz and 3 6 : 2. M12 , two subgroups which together generate Co1. This shows that M⬘ must be invariant under Co1 and therefore the extension is split. Next we show that there is no nonsplit extension of the form 274r24 [ 24. This will imply that any further extension of V300 by a Leech lattice mod 2 is a split extension. PROPOSITION 14. Ext 1C o 1Ž274r24, 24. s 0. Proof. The proof of this result is by reducing the situation many times to results on cohomology which can be easily proved by methods similar to those used above. So we will rely on the subgroup structure shown in Fig. 2, which is easily derived by the 3-local geometries for Co1 and Suz as well as a geometry for UAŽ3. described in wAS83x; this information is also easily obtained from the ATLAS wCq 85x. Start by analyzing the restriction of 274r24 to the subgroup 3.Suz. From the calculations above we know that restricting 274r24 to a 3.Suz has structure 142 [ 132r24. The center acts trivially on the submodule 142 while acting nontrivially on 132r24. The direct sum decomposition of these two submodules can be seen as an application of Maschke’s theorem to the center of 3.Suz. Restricting 274r24 to 3 6 : 2. M12 has the form 10 [ 264 [ 24. We may choose these subgroups so that they intersect in a subgroup of the form

FIGURE 2

Co1

INVOLUTION GEOMETRY

569

3 : 3 5 : M11 . Under restriction to 3 : 3 5 : M11 the module has the form 10 [ 132 [ 132 [ 24 by elementary weight theory for the normal 3-group. Let M be an extension of 274r24 by 24. By Lemma 13 we know that M has the form 274r24 [ 24. On restriction to 3.Suz by the action of the central 3 element either this 24 is nonsplit under 132 or it is a split extension for the whole module. If it splits under the action of 3.Suz then there is a complement, M⬘, to this 24 which is fixed under 3.Suz. Therefore M⬘ is fixed under 3 : 3 5 : M11 ; 3.Suz, and therefore split under 3 6 : 2. M12 , as every complement to a 24 under 3 : 3 5 : M11 is stabilized by 3 6 : 2. M12 . Therefore M⬘ is fixed under Co1 , and we are done in that case. We have reduced our problem to showing there is no nonsplit extension of the form 132r24 [ 24 for 3.Suz. The modules 132 and 24 are both best understood as F4 modules for 3.Suz. By wCR81x, dim Ext 1F2 w3, S u z xŽ132, 24. s dim Ext 1F4 w3.S u z x Ž66, 12. therefore we have reduced to calculating dim Ext 13. S u z Ž66, 12.. By a similar argument to the one above we can reduce the question of dim Ext 13. S u z Ž66, 12. to dim Ext 13 : 3.U4 Ž3.Ž15, 6.. We sketch that argument here. The module 66 is best understood as the exterior square of the complex Leech lattice, 66 ( 12 n 12. From wLin71x, the restriction of 12 to 3 : 3.U4 Ž3. is isomorphic to 6 [ 6 and therefore from that restriction to 3 : 3.U4Ž3. we have that uz Ž 66 . x 3.S 3 : 3 .U4 Ž3. ( 15 [ 15 [ 36,

where the action of the normal 3 subgroup is different from each irreducible summand. Only on one summand isomorphic to 15 does the 3 action of the central 3 element of 3.Suz agree with the action of this element on the 3 : 3U4Ž3. module 6. Therefore the only possible extension would be of the form 15r6. If it splits under the action of 3 : 3.U4 Ž3. then so would the 6 by duality and there is a complement, M⬘, to this 12 which is fixed under 3 : 3.U4 Ž3.. Similar considerations to the one above using 3 6 : A 6 and 3 6 : M11 in place of 3 6 : M11 and 3 6 : 2. M12 show that we can reduce to the case dim Ext 13 : 3.U4 Ž3.Ž15, 6.. The action of both the central three element of 3.Suz and the action of the 3 element of 3.U4Ž3. on 6 are the same so we may consider only 3.U4Ž3. in the next step. This is equivalent to calculating dim Ext 13.U4 Ž3.Ž15 m 6, 1.. Applying the MeatAxe we calculate that 15 m 6 ( 70 [ 20. We know that s 1 from wYos92x and Proposition 4. Therefore we have reduced the problem to showing that dim Ext 1U4 Ž3.Ž70, 1. s 0. To do this we again invoke Alperin and Gorenstein for 2 subgroups of U4Ž3., which are isomorphic to A 7 and which can be chosen to intersect in dimŽExt 1U4 Ž3.Ž20, 1..

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an L3 Ž2. wAS83x. First let us introdeuce the notation that the projective cover of an irreducible module x will be denoted P Ž x .. Using the MeatAxe we can show that 70 on restriction to A 7 is the direct sum of the irreducible 6 and the projective cover of the irreducible 14 dimensional module for A 7 , P Ž14.. The irreducible 6 is in a non-principal block of A 7 , therefore any trivial module is split from it. On restriction to L3 Ž2. the module P Ž14. has the form 8 [ 8 [ 8 [ 8 [ P Ž3. [ P Ž3.. Therefore any extension of 70 by a trivial module splits on restriction to A 7 and on restriction to L3 Ž2. as well. Let X be an extension of 70 by a trivial module, and let T be the trivial submodule. Then by the above there exists a complement M to T, on restriction to the first A 7 . Note it is unique as the top composition factors of Ž70.x UA47Ž3. are single 1 and 14, and 6. Then M is also a complement to T for L3 Ž2. by the containment L3 Ž2. ; A 7 . Also it is the unique complement for L3 Ž2. as, similarly, the top composition factors are a unique 1, and a 3, 3, and 8 from P Ž14. and a 3 from the irreducible 6. Therefore M is also the unique complement for the other A 7 containing L3 Ž2. which stabilizes this complement as well. Therefore M is stable under the whole group U4Ž3.. This shows that dim Ext U4 Ž3.Ž70, 1. s 0, and completes the proof. LEMMA 15. dim Ext 1C o 1Ž V300 , 24. s 0. Proof. As a 24 splits from both 1 and 24, any 24 must be nonsplit over the 274, if we assume Ext 1C o 1Ž V300 , 24. / 0. But there is already one 24 nonsplit under 274. Therefore by Proposition 14, this new 24 splits from V300 . LEMMA 16. dim Ext 1Co 1Ž274r1 [ 1, 1. s 0. Proof. We apply the Alperin᎐Gorenstein method, choosing subgroups 3 6 : 2. M12 and 3.Suz that intersect in a 3 : 3 5 : M11. Let M be an extension of 274r1 [ 1 by a trivial module T. Our first step is to show that there exists a complement to T stable under the subgroup 3 6 : 2. M12 . First, we calculate the universal embedding of the involution geometry for M12 . This calculation and the calculation of its lattice of submodules are done through the presentation of the universal embedding given in the Introduction and using the MeatAxe and GAP. Applying Proposition 4 shows dim Ext 1M 12Ž10, 1. s 2. The three subgroups lemma to the central 2 subgroup, all of 2. M12 , and an extension of 10r1 [ 1 by any trivial 1 shows that the central 2 of 2. M12 is trivial on the whole extension, so that Ext 12. M 12Ž10, 1. ( Ext 1M 12Ž10, 1.. On restriction to 3 6 : 2. M12 the module 274r1 [ 1 ( Ž10r1 [ 1. [ Ž3-nontrivial part.. Now applying the calculation of Ext above we see that, in the extension of M by T, T splits on restriction to 3 6 : 2. M12 . Therefore dimw M, 3 6 : 2. M12 x s 276.

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INVOLUTION GEOMETRY

571

The trivial module T also splits under restriction to 3.Suz. The proof of wBar96a, Theorem 2.14x shows that dim Ext 13. S u z Ž142, 1. s 1. This result and the information contained above about the restriction of 274 to 3.Suz show that dimw M, 3.Suz x s 275. On the intersection of these two subgroups 3 : 3 5 : M11 we have that dim Ext 1M 11Ž10, 1. s 1 so that dimw M, 3 : 3 5 : M11 x s 275. Therefore the subgroup 3 6 : 2. M12 stabilizes a complement to the trivial module of the extension, namely w M, 3 6 : 2. M12 x. This submodule is also stabilized by 3 : 3 5 : M11 and contains w M, 3 : 3 5 : M11 x s w M, 3.Suz x. Therefore this complement is stable under 3.Suz as any space containing the commutator is stable. Therefore this complement is stable under the subgroup generated by the two subgroups, 3.Suz and 3 6 : 2. M12 . The subgroup 3 6 : 2. M12 is maximal. Therefore there is a complement to T stable under Co1. LEMMA 17. dim Ext 1C o 1Ž V300 , 1. s 0. Proof. As any further extensions split from the trivial submodules and 24’s must be nonsplit under 274, we assume Ext 1C o 1Ž V300 , 1. / 0. But Ext 1C o 1Ž274, 1. s 2 and therefore any further trivial modules split. LEMMA 18. V⌫ ( V300 . Proof. By Lemmas 17 and 15 we know that V⌫ is isomorphic to the direct sum of V300 and some number of modules isomorphic to the trivial module and 24. Neither the trivial module nor 24 is a quotient of V⌫ , as with the trivial module there is no 2-space for lines to map to, and for 24 there is no invariant 1-space for the point stabilizer. Therefore by Proposition 1.2 of wRS86, p. 139x, V⌫ ( V300 . ACKNOWLEDGMENTS The author thanks Professor Stephen Smith for many helpful conversations, as well as Professor George Glauberman whose comments and suggestions significantly improved the contents and coherence of this paper.

REFERENCES wAG72x wAS83x wBar96ax wBar96bx

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MATTHEW KIRBY BARDOE

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