Some Projective Modules Determined by Sporadic Geometries

A.J.E. Ryba^{∗}, Stephen D. Smith^{†}, and Satoshi Yoshiara Department of Mathematics University of Illinois at Chicago Chicago Illinois 60680 USA

(drafted June 1987; version of 5 August 1988)

Abstract

We determine which known sporadic geometries have projective Lefschetz modules. An elementary lemma on local collapsibility greatly simplifies the task of verifying projectivity. The modules are analyzed when possible in terms of projective covers of individual irreducibles.

Introduction.

For a finite simple group of Lie type in characteristic p, the Solomon-Tits theorem [Sol] shows that the reduced Lefschetz module of the building geometry is the Steinberg module, a projective module in that characteristic. This paper investigates when a projective module arises in this way for a sporadic geometry. We state the main result as follows, explaining terminology afterwards:

Theorem. Let Δ be a catalogued sporadic geometry in characteristic p, with p^{2} dividing the order of the group G. Then the reduced Lefschetz module L̃(Δ) is projective just in the cases listed in Table 1 below.

Here by “sporadic geometry” we mean a discrete incidence geometry admitting flagtransitive action by a sporadic simple group, or such a geometry other than the building for an alternating or Lie-type group. Many examples have been discovered in recent years, and we take as the “catalogue” in the above statement the set of geometries described in the standard references [RoSm80] [As] [AsSm] [RoSt] [Ka81] [Bu] and the GAB examples in [Ka86]. (We expect the methods presented here will handle geometries from other sources, and interesting geometries which may remain to be discovered). Because of our focus on p-modular representations, we restrict attention to geometries “in characteristic p”: those in which the geometric parameters [Bu] are powers of a single prime p (including the power p^{0} = 1). To avoid trivialities, we have neglected cases where p divides the group order to the first power only.

∗Ryba’s current address: Department of Mathematics, University of Michigan, Ann Arbor MI 48109.

Yoshiara’s current address: Department of Mathematics, Tufts University, Medford MA 02155.

†Smith and Yoshiara were partly supported by NSF grant DMS 86-01791. Ryba and Yoshiara held postdoctoral positions funded by UIC matching on Smith’s NSA grant MDA 904-87-H-2006.

Our preliminary analysis involved ruling out geometries whose Lefschetz modules could not be projective; some typical eliminations will be described shortly, but most such details will be omitted as routine. The main work of the paper consists of showing that each of the remaining candidates listed in Table 1 below actually provides a projective module, and decomposing that module as far as possible into individual projective covers P(I) of modular irrreducibles I for G. In the Table, irreducibles are identified simply by their degree. An asterisk in the column for the group G indicates a subgeometry obtained by truncation from the geometry Δ for G in the previous line (see Lemma 2.6).

Table 1: Some projective Lefschetz modules

G  p χ̃(Δ) …. character of L̃(Δ)  (defect-0 part)
A_{7}  2 2^{3}.7  P(20)  —
∗  2 2^{6}  P(14)  —
M_{24} 2 2^{10}.21  P(1792)  —
M_{22} 2 2^{8}.7  P(70) ⊕ P( \overline{70})  —
U_{4}(3) 2 2^{7}.31  —  640 ⊕ \overline{640} ⊕ 3(896)
G_{2}(3) 2 2^{7}.85  P(90) ⊕ P(378) ⊕  64 ⊕ \overline{64} ⊕ 448 ⊕ \overline{448} ⊕ 2(832)
∗  2 2^{6}.181  2P(some I =?) ⊕  448 ⊕ \overline{448} ⊕ 3(832)
McL 2 2^{8}.521  P(2124) ⊕ P( \overline{2124}) ⊕  896 ⊕ \overline{896} ⊕ 9856 ⊕ \overline{9856}
∗  2 2^{7}.7483  3P(2124) ⊕ 3P(\overline{2124}) …
2P(3520) ⊕ 5P(3584) ⊕ 2(896 ⊕ \overline{896}) ⊕ 12(9856 ⊕ \overline{9856})
J_{3}  2 2^{8}.511  ? projectives not known 5(1920 ⊕ 1920′ ⊕ 1920′′ ⊕ 2432)
M_{11} 3 3^{2}.5  —  45
L_{3}(4) 3 3^{2}.25  P(15) ⊕  63 ⊕ \overline{63}
U_{5}(2) 3 3^{6}.7  P(891) ⊕  1215 ⊕ \overline{1215}
McL 3 3^{6}.106  P(some I =?) ⊕  5103
∗  3 3^{6}.169  P(some I′ =?) ⊕  5103 ⊕ 8019 ⊕ \overline{8019}
Ly  5 5^{6}.7065863 ? projectives not known huge–see Table 4.12
∗  5 5^{6}.1769293 ? projectives not known  –see Table 4.12
O′N 7 7^{3}162487 ? projectives not known  –see Table 4.13

The paper is organized as follows: Section 0 indicates briefly how failure of projectivity is established for most catalogued geometries. The first section then surveys some results on projectivity in the literature, leading up to a sufficient condition formulated by Peter Webb [We87]. In the second section, we give an elementary but fundamental lemma on local toplogical collapsibility, which greatly simplifies the verification of Webb’s condition, making it practical even in large examples. Further lemmas indicate how information on subgroups of G can be translated into topological terms. Then the actual verification is carried out in Section 3, establishing projectivity of the modules listed in the Table. Finally Section 4 discusses the decomposition of the projective examples into projective covers of individual irreducibles.

0. Methods for establishing non-projectivity.

We establish some general notation holding throughout the paper. We use G to denote some finite group, and p a prime dividing its order. All representations will be defined over a splitting field k for G in characteristic p; in most cases k = F_{p} will suffice. For geometries and their diagrams, we will follow the standard conventions of [Tits]: we let Δ denote a simplicial complex with types of vertices (such as points, lines, …) admitting a type-preserving simplicial action by G, transitive on maximal flags. From the complex, we define the corresponding chain groups C_{i}(Δ) and homology groups H_{i}(Δ); we take the coefficients in the field k, and these groups afford kG-modules. Of particular interest is the virtual module given by the alternating sum of the chain groups:

L̃(Δ) := ∑_{i=−1}^{dim Δ}(−1)^{dim Δ−i}C_{i}(Δ);

we refer to this as the (reduced) Lefschetz module. ^{1} Its degree term

χ̃(Δ) := ∑_{i=−1}^{dim Δ}(−1)^{dim Δ−i} dim C_{i}(Δ)

is the (reduced) Euler characteristic.

Among the geometries Δ catalogued in the references, the vast majority can be shown to afford non-projective L̃(Δ) by the well-known necessary condition on the degree-term χ̃(Δ):

0.1 Lemma. If L̃(Δ) is projective, then |G|_{p} divides χ̃(Δ). In particular, if |G_{v}|_{p} < |G|_{p} for all vertices v, then L̃(Δ) is not projective. □

However, in the largest examples, or in the infinite families of GABs listed in [Ka86], application of this lemma may not be convenient or even feasible. To deal with these cases, another lemma (still quite elementary) is useful:

0.2 Lemma. If L̃(Δ) is projective, then for any element z of order p in G, the order of the centralizer |C_{G}(z)| must divide the least common multiple of the orders |C_{G_{v}}(z)|, where v ranges over the vertices fixed by z. In particular, |C_{G}(z)| must divide the least common multiple of the vertex-stabilizer orders. Proof: By the universal coefficient theorem, we may regard L̃(Δ) as a module in characteristic 0; and if it is projective, its complex character must vanish at the p-singular element z. But the Hopf trace formula shows that this character value is just χ̃(Δ^{z}). On the other hand, this value is by definition the alternating sum of -1 and the numbers of flags of various types fixed by z. Now the flags of each type break up into orbits under C_{G}(z), and each such orbit has length of form |C_{G}(z)|/|C_{G_{v}}(z)| where v is a flag type. Consequently these terms determine a common denominator l dividing the least common multiple of the orders |C_{G_{v}}(z)|. And the equation χ̃(Δ^{z}) = 0 has form −1 + k|C_{G}(z)|/l = 0 for some integer k, or l = k|C_{G}(z)|. So |C_{G}(z)| divides l, as asserted. □

1The more usual definition of L̃(Δ) uses the homology groups in place of the chain groups. By the Hopf trace formula, the two different definitions at least afford the same character. Furthermore most of our examples have the Cohen-Macaulay property, implying the vanishing of reduced homology outside the top dimension, in which case the two definitions coincide as modules.

To demonstrate the use of this technique, we will indicate how to eliminate from the 15 GABs listed in [Ka86, pp.131-133] all but those for G = Ω_{6}^{−}(3) ≅ U_{4}(3), G_{2}(3), A_{7}, Ly, and McL (for p = 2); as required by Table 1. Various other cases can also be treated in this way.

Among these 15 cases are 7 corresponding to individual groups G. It is easy to compute χ̃(Δ) for these cases, and then by 0.1 there remain only the three corresponding to A_{7}, McL, and Ly.

Each of the other 8 cases corresponds to an infinite family of groups G = G_{q} parametrized by an odd prime q. Among these, the possibility numbered (20) in [Ka86] is also eliminated by an Euler characteristic computation; while (4) and (5) of [Ka86] are also ruled out by 0.1, since we find that the indices of vertex stabilizers |Ω_{5}(q) : Sp_{4}(2)| and |Ω_{6}(q) : A_{7}| are even.

In four of the remaining five cases, the stabilizers are fixed groups not depending on q, and can be treated similarly. Take for example the largest case (1) of [Ka86] with G_{q} = Ω_{8}^{+}(q), where the vertex stabilizers are either Ω_{8}^{+}(2) or a solvable group of order 2^{12}3^{4}. We choose an involution z whose centralizer contains a central product of four copies of SL_{2}(q). The order of the centralizer is divisible by q^{4}, and so by 0.2 we get q^{4} dividing |Ω_{8}^{+}(2)| = 2^{12}3^{4}5·7. Thus q = 3; but this single case is eliminated via 0.1 with an Euler characteristic computation. Similar reasoning applied to the cases (2), (3), and (6) of [Ka86] again reduces quickly to q = 3; and then (2) is finally eliminated by Euler characteristic, while (3) and (6) give rise to the cases Ω_{6}^{−}(3) ≅ U_{4}(3) and G_{2}(3) of Table 1.

It is only a little more complicated to eliminate the final case (15) for the group G_{q} = Ω_{7}(q); here the stabilizers are the fixed groups 2^{6}A_{7}, Sp_{6}(2), and a solvable group of order 2^{9}3^{3}, with the final stabilizer 2Ω_{6}^{ϵ}(q)2 depending on q (and determining the geometry (3) just considered). Now we can check that the centralizer of an involution in either of the first two stabilizers has order dividing 2^{9}3^{2}5, which for any q divides the order of the final stabilizer. Since the latter two centralize certain involutions, we conclude that the least common multiple determined in 0.2 is just 3 times the order of the last, namely:

\frac{4.3}{(4, q − ϵ)}q^{6}(q^{2} − 1)(q^{4} − 1)(q^{3} − ϵ).

Consequently for any involution z, the centralizer order |C_{G}(z)| should divide this order. So we choose z to be trivial on a 4-dimensional orthogonal subspace of plus-type in the natural module for G, and get |C_{G}(z)| = q^{3}(q^{2} − 1)^{3}. The quotient of the above order by this latter is:

\frac{4.3.q^{3}(q^{2} + 1)(q^{2} + ϵq + 1)}{(4, q − ϵ)(q + ϵ).}

Now q + ϵ is coprime to q^{3} and q^{2} + ϵq + 1, and (4, q − ϵ) = 2 or 4; so that q + ϵ must divide 2.3.(q^{2} + 1). Since (q + ϵ, q^{2} + 1) = 2, we have q + ϵ = 2^{a}3^{b}, where 0 ≤ a ≤ 2 and 0 ≤ b ≤ 1. Thus q = 3, 5, 7, or 13. However, in these cases we directly eliminate projectivity by an Euler characteristic calculation.

In this way, all GAB cases in [Ka86] are treated.

1. Background on homology modules and projectivity.

We begin our exposition with the theorem of Solomon-Tits [Sol], which shows that the unique projective and irreducible module for a finite Lie-type group (discovered by Steinberg and denoted St) can be constructed from the top homology of the building geometry. In our notation, their result includes the statement:

1.1 Theorem. (Solomon-Tits) Let G be a finite Lie-type group in characteristic p, and Δ its building. Then L̃(Δ) ≅ St. □

At roughly the same time, a different and very general homological construction of a projective module arose in algebraic topology. For an arbitrary finite group G, we let Δ_{p} denote the complex whose simplexes are chains (under inclusion) of non-trivial p-subgroups of G. K.S. Brown in [Br] had observed that χ̃(Δ_{p}) ≡ 0 (mod |G|_{p}); this result was explained at the level of Brauer characters by Quillen [Q] , and subsequently at the module level by Webb [We87]:

1.2 Theorem. (Webb) L̃(Δ_{p}) is a projective (virtual) kG-module. □

Quillen also indicated the connection of this module construction with that of Solomon-Tits:

1.3 Theorem. (Quillen) For G as in 1.1, the building Δ is homotopy-equivalent to the p-subgroups complex Δ_{p}. Thus L̃(Δ) ≅ L̃(Δ_{p}) ≅ St. □

Remarks: Of course 1.2 provides a projective module for any finite group, not just for a simple group. However, for a sporadic simple group G with |G|_{p} large, it often happens that the module L̃(Δ_{p}) is correspondingly unwieldy, in the sense of involving many different projective indecomposables, often with large multiplicities. By contrast, the apparently more natural geometries for these groups studied in recent years (see a survey such as [Sm]) usually lead to a complex Δ of smaller dimension, and a module L̃(Δ) which is easier to work with. In particular, it appears that homotopy-equivalence of such Δ with Δ_{p} is a fairly exceptional occurrence (as for example in the M_{24} case below).

Peter Webb [We87] (from motivation independent of the above comments) studied Quillen’s work with an eye to generalization; he abstracted from the particular complex Δ_{p} the properties required for the proof of projectivity, and his results include the following sufficient condition for more general complexes:

1.4 Theorem. (Webb) Suppose for all elements z of order p in G, the fixed subcomplex Δ^{z} is contractible. Then L̃(Δ) is a projective virtual module. □

We verify this contractibility condition in order to establish projectivity of the modules indicated in the Table. The following section discusses ways in which we can simplify the process of verification.

2. Lemmas on local collapsibility.

In a number of geometries, as reported in Section 5 of Webb [We88], it was possible to check directly the contractibility of the relevant Δ^{z} in 1.4. However, in the larger examples, the sheer size of the fixed subcomplexes prevented any transparent verification. (For example, in the case of M_{24} below, one fixed complex has as vertices 71 points, 99 lines, and 91 quads — and so is far from easy to visualize). Nonetheless, in many of these examples it was easy to compute that χ̃(Δ) ≡ 0 (mod |G|_{p}), which strongly suggested projectivity of L̃(Δ). We indicate below some lemmas which allow for a step-by-step reduction of the size of a fixed subcomplex, making verification of 1.4 practical even in these large cases.

Basic results on contractibility can be found in any standard reference on algebraic topology, such as Spanier [Sp] . We will review briefly the stronger notion of collapsibility, which is especially useful in our context of simpicial complexes. We say a simplex σ ∈ Δ of maximal dimension is free over some maximal face τ, if σ is the only simplex of maximal dimension with τ as a face. In that case, the process of removing σ from Δ (by “collapsing” σ down onto its faces other than τ) is a homotopy equivalence. A complex is collapsible if it can be reduced to a point by a sequence of such steps. Evidently this is in particular a contraction.

We wish to study collapsibility locally, considering only suitable subconfigurations of our complex Δ. If v is a vertex of Δ (that is, a simplex of dimension 0), then the star of v, denoted Star(v), is the set of simplexes of Δ containing v as a vertex. And the residue of v, denoted Res(v), is the subcomplex of the next lower dimension, obtained by deleting v from each simplex of Star(v). Because of our flag-transitive hypothesis, Star(v) is in fact just the simplicial cone v∗Res(v). (We use * to denote the usual simplicial join). The following local lemma is surely clear to algebraic topologists; though entirely elementary, it turns out to be exactly what is needed to treat our examples.

2.1 Lemma. Suppose for a vertex v that Res(v) is collapsible. Then collapsing Δ to Δ \ Star(v) is a homotopy. Indeed all such vertices v of a fixed type can be homotopically deleted from Δ.

Proof: Note that if σ is a face free over τ in Res(v), then the geometric product σ ∗ v is free over τ ∗ v in Star(v). So a sequence collapsing Res(v) to a point (vertex) w translates, on adjoining v, to a sequence collapsing Star(v) to the edge v ∗ w; but this edge is now free over v, and so can be collapsed to w. Thus we have removed Star(v) from Δ. Finally since vertices of the same type cannot be incident, their stars have no intersection, and so they can be removed independently. □

The referee points out that more is true in the situation of 2.1: in fact, Res(v) is a strong deformation retract of Star(v), so that Δ\Star(v) is a strong deformation retract of Δ.

Since our geometries admit flag-transitive action by the group G, vertices of a fixed type will be G-conjugate; this will usually allow them to be treated all at once, as the Lemma suggests. To see why the condition on residues is likely to occur, recall that in 1.4 we actually consider fixed subcomplexes Δ^{z} for elements z ∈ G; so 2.1 reduces us to investigating Res_{Δ^{z}}(v) = Res(v)^{z}. Since our geometries are closely related to Tits buildings, the following special case of 2.1 is often applicable:

2.2 Lemma. If Res(v) is a building, on which z acts non-trivially, then Res(v)^{z} is collapsible.

Proof: This is an observation of Quillen [Q] relevant to 1.3; it essentially relies on the argument of Solomon-Tits for the collapsibility of the subcomplex obtained by deleting from the building all the maximal faces opposite to a particular maximal face. □

We mention also point-line subgeometries, which are in effect a special case of 2.2. Loosely speaking, the terms point and line are usually reserved for vertex types corresponding to an end node and a neighboring node in a linear diagram. More precisely, we use this terminology when there are p + 1 points per line, permuted as a projective line by a subgroup SL_{2}(p) in the stabilizer G_{l} of a line l; and furthermore Res(l) is the geometric product Q ∗ R where Q consists of the points of l, and R is the non-point part of Res(l). We get:

2.3 Lemma. If z of order p in a line stabilizer G_{l} does not fix l pointwise, then Res(l)^{z} is a cone (and so contractible).

Proof: A p-element z of SL_{2}(p) fixes a unique point q of the projective line l. Then Res(l)^{z} is the cone q ∗ R^{z}. And contractibility of cones is a standard elementary fact. □

We close the section with some remarks on the general strategy of using these tools. Given explicit z, the structure of the centralizer C_{G}(z) usually distinguishes a particular vertex v_{0} which in some sense lies at the heart of Δ^{z}. By repeated application of 2.1, 2.2, and 2.3, we can attempt to strip off the stars of the more “distant” vertices of the fixed complex, ideally reducing down to Star(v_{0}), which is a cone and collapsible as in 2.3.

This reduction often requires several steps in which we successively restrict to fixed vertices v for which z even acts trivially on the residue Res(v); we shall refer to such vertices as res-fixed. We adopt a shorthand notation useful for these reductions, as follows. Suppose the geometry has vertex types called (say) points, lines, … denoted by P, L, . . .. Then Δ_{P}^{z} will mean the subcomplex of Δ^{z} whose points are only those which are only those res-fixed by z; with Δ_{PL}^{z} and so on defined accordingly.

We note also that the fixed subcomplex Δ^{z} can be studied via the equivalent definition of the geometry in terms of cosets of the subgroups of G stabilizing various simplexes, so that questions of geometric incidence reduce to questions of fusion of elements in these various subgroups. One possible technique for determining z-fixed simplexes is to determine first their number, by computing the value at z of the permuation character on the simplexes; these characters are usually available from the [Atlas] , and are freely used in the analysis of the character of L̃(Δ) in Section 4. However, it is not necessary to rely on these values in Section 3; and to emphasize the more elementary nature of the method, we base our analysis there just on information about fusion (which is of course a prerequisite for the computation of permutation characters). More precisely, for incident vertices v and w, we usually just need to know the distribution in G_{v} of the conjugates of the common stabilizer G_{vw}. Ordinarily we will simply state the relevant fact about conjugacy, which should either be clear or else can be found in the literature on G. We have abstracted below several common situations in which we use such group-theoretic information.

In a vertex stabilizer G_{v}, we let K_{v} denote the kernel of the action on Res(v). Then v res-fixed is equivalent to z ∈ K_{v}. This is particularly useful in case K_{v} is abelian, when z ∈ K_{v} is equivalent to K_{v} ⊆ C_{G}(z):

2.4 Lemma. Suppose K_{v} is abelian. Then res-fixed vertices of the type of v correspond to conjugates of K_{v} in the centralizer C_{G}(z). □

Once we have used 2.4 to determine the res-fixed vertices v, we often want to know that other fixed vertices must be incident to some such v. A group-theoretic condition for this is provided by:

2.5 Lemma. Assume all conjugates of z in a vertex stabilizer G_{w} are contained in subgroups K_{v}, for vertices v in Res(w). Then w is incident to some res-fixed v. □

The process of checking the hypothesis of 2.5 usually begins by investigating conjugates of z first in the quotient G_{w}/K_{w}, for which we adopt the notation Ḡ_{w}; then it remains to determine the distribution of conjugates in a coset zK_{w}.

Finally we indicate a situation in which the same analysis can be applied to two related geometries. When we apply 2.1 to reduce to res-fixed vertices v of some type V, it sometimes happens that the kernel K_{v} is a p′-group (in practice, when G_{v} is simple); thus z ∈ K_{v} is impossible, and there are no res-fixed vertices of that type. Removal of all such vertices results in a subcomplex we denote by Δ_{V}, of dimension one less; in the literature this is called a truncation of Δ. It is natural to use a superscript z to denote the fixed subcomplex in the truncation; since we will only consider the truncation when there are no res-fixed vertices of type V, the resulting notation Δ_{V}^{z} will in fact agree with our earlier definition. In particular, Δ_{V}^{z} is homotopic to Δ^{z}. So we can reduce to considering the smaller geometry, to establish the desired result for both:

2.6 Lemma. Suppose for all vertices v of a fixed type that Res(v)^{z} is collapsible, and the kernel K_{v} is a p′-group. Then contractibility of all fixed subcomplexes Δ_{V}^{z} in the truncation implies projectivity of both L̃(Δ_{V}) and L̃(Δ). □

The reader will have observed that the lemmas in Section 2 do not lie very deep. The applications in the next section should demonstrate that they are nevertheless quite useful.

3. Proofs of projectivity.

We now consider in turn the geometries of Table 1. In each case, we establish 1.4 (contractibility of Δ^{z}), by a sequence of applications of the results in Section 2. The first few proofs will be presented in fuller detail; afterwards, details may be indicated more briefly, as the common aspects of the proofs become evident. As background on the geometries, we indicate for each vertex type the structure of the stabilizer and of the residue; for fuller details, a reference is given at the beginning of each case. 3.1: The C_{3} geometry for A_{7}.

The geometry was discovered by Neumaier [Neu] . Projectivity of L̃(Δ) was verified, and the character computed, by Ronan and Smith with Webb (see [RoSm88] [We88] ). However, a proof via Section 2 is quite short, and is indicated below.

The set P of points is given by the 7 letters permuted by G = A_{7}; a typical point ^{2} p has stabilizer G_{p} ≅ A_{6}, with residue Res(p) the generalized quadrangle for Sp_{4}(2)′ ≅ A_{6}. The lines L are the 35 = \binom{7}{3} triples of points, with typical stabilizer G_{l} ≅ 2^{2}(3 × 3)2 and Res(l) the complete bipartite graph { 3 points of l} ∗ { 3 planes above l}. The planes Π are given by one A_{7}-class of 15 projective planes, each consisting of 7 points and 7 lines; the typical stabilizer G_{π} ≅ L_{3}(2) acts naturally on Res(π) ≅ P G(2, 2).

In G = A_{7} there is just one conjugacy class of involutions, so it suffices to consider the subcomplex Δ^{z} fixed by one representative z = (12)(34). Evidently z fixes the three points 5,6,7; and the unique line fixed pointwise is l_{0} = 567. We will aim to collapse Δ^{z} down into the cone Star(l_{0}).

We begin by applying 2.6 to the plane residues: Note that for a plane π, Res(π) is the building P G(2, 2), and so by 2.2 Res(π)^{z} is contractible. But also as G_{π} ≅ L_{3}(2) is simple, we have trivial plane kernel K_{π} = 1. Thus by 2.6 it suffices to consider z-fixed vertices in the truncation Δ_{Π}— that is, the bipartite graph Δ_{Π}^{z} on the z-fixed points and lines. But using 2.3 we can remove from this graph any line not fixed pointwise. What remains is the tree l_{0} ∗ {5, 6, 7}, which is collapsible. We conclude using 2.6 that both L̃(Δ_{Π}) and L̃(Δ) are projective.

We remark here that similar arguments apply in certain other geometries; but no new modules arise. For example, we could observe above that the point stabilizer G_{p} ≅ A_{6} is simple, so arguing via 2.6 we obtain L̃(Δ_{P}) projective—but this module is isomorphic to the 56-dimensional module L̃(Δ) for the full geometry. We can also consider A_{7} acting on the building P G(3, 2) for its overgroup L_{4}(2). Its Lefschetz module reproduces the 64dimensional L̃(Δ_{Π}) above. Furthermore for x a point or plane of the building, we have (in A_{7}) simple stabilizer G_{x} ≅ L_{3}(2); the projective module produced by the truncation using 2.6 is the 56-dimensional L̃(Δ) again. 3.2: The 2-local geometry for M_{24}.

The geometry was introduced in Ronan-Smith [RoSm80] . Ronan showed (unpublished) Δ is homotopy-equivalent to the 2-subgroups complex Δ_{2}, so that in this case L̃(Δ) is projective by 1.2; and the character was analyzed by Ronan-Smith (see [RoSm88] ). For comparison, we indicate here a proof of projectivity via 1.4 and Section 2. Since the arguments are typical, we indicate them in some detail.

The vertices are determined by the Steiner triple system S(5, 8, 24); for details see Conway [Con] . The points P are given by the octads of the system (certain 8-sets from the 24 letters permuted by M_{24}), with stabilizer G_{p} ≅ 2^{4}L_{4}(2) and Res(p) given by the subgeometry of a projective space P G(3, 2) obtained by ignoring its planes. The lines L are given by the trios (triples of disjoint octads) with stabilizer G_{l} ≅ 2^{6}[S_{3} × L_{3}(2)] with Res(l) the complete bipartite graph { 3 point of l} ∗ { 7 quads above l}. The quads Q are given by the sextets (partitions of the 24 letters into six “tetrads”, each pair of which gives an octad); a quad Q has stabilizer G_{Q} ≅ 2^{6}3Sp_{4}(2) with Res(Q) the corresponding symplectic generalized quadrangle. A useful representation of the geometry, described in [RoSm80] , arises from the 11-dimensional F_{2}-module determined by the Golay code; in it points, lines, and quads are certain F_{2}-subspaces of dimensions 1,2, and 4.

2It is often convenient to use p to denote a point; in any subsection, the prime previously denoted p has the value indicated in Table 1.

The argument divides into two cases, according to the two conjugacy classes of involutions z.

Case 1: z is not 2-central in G. We observe that C_{G}(z) ⊆ G_{Q_{0}} for some quad Q_{0}. In fact, the abelian kernel K_{0} ≅ 2^{6} is weakly closed (that is, has a unique conjugate) in C_{G}(z); consequently by 2.4, Q_{0} is the unique quad res-fixed by z. We will collapse down into the cone Star(Q_{0}).

We remark first that the point and line residues involve not buildings but “truncations” of them, and so we cannot apply 2.2 to them. But for a quad Q, Res(Q) is a full generalized quadrangle and so a building; using 2.2, we reduce Δ^{z} to the subcomplex Δ_{Q}^{z} with all fixed points and lines, but only those quads which are res-fixed. By uniquess above, Q_{0} is the only such quad.

Next we will check using 2.5 that the fixed points are just those of Q_{0}. Since such arguments will occur frequently in what follows, we indicate fuller details in this first application. Certainly they are fixed, since they lie in Res(Q_{0}) and Q_{0} is res-fixed. Conversely we now let p denoted any z-fixed point. Since the involutions in K_{p} are all of 2-central type, we see z must be non-trivial in Ḡ_{p} = G_{p}/K_{p}. We note for a quad Q incident to p that K_{Q} ∩ K_{p} is a subgroup of order 2^{2} in K_{p} ≅ 2^{4}, and in the quotient determines a subgroup of order 2^{4} which is the unipotent radical in Ḡ_{p} ≅ L_{4}(2) of the parabolic stabilizing Q; and any involution of L_{4}(2) is contained in one of the corresponding 35 unipotent radicals. So we may assume z lies in the coset xK_{p} for some x ∈ K_{Q}; we check that [x, K_{p}] = K_{p} ∩ K_{Q}, and it follows from this free action on K_{p} that all involutions (including z) in xK_{p} in fact lie in x(K_{p} ∩ K_{Q}), hence in K_{Q}. We abbreviate this verification with the phrase “all conjugates of z in G_{p} are covered by K_{Q} for incident quads Q”. So we can apply 2.5 to conclude that any z-fixed p must lie in a res-fixed Q, hence in Q_{0}. Thus Δ_{Q}^{z} has only the 15 points of Q_{0}.

Next we reduce to res-fixed lines. Notice that an involution fixing a line l pointwise, but not res-fixing it (that is, permuting some of the quads above it), would lie in the L_{3}(2)-part of Ḡ_{l} ≅ S_{3} × L_{3}(2); such an involution must lie in the 2-central class. (For example, its centralizer has a 3-element fixing a subgroup of order 2^{4}, whereas only 2^{3} can occur for z not 2-central). It follows that if l is not res-fixed by our chosen z, then z must fix exactly one point p of l. By 2.3 we may remove such lines, reducing to the subcomplex Δ_{LQ}^{z} where now all lines are also res-fixed.

In particular any such line is pointwise fixed, and so as previously noted has all its points in Q_{0}; but a line in this geometry is uniquely determined by any two of its points, hence the line is one of the 15 lines of Q_{0}. In fact all 15 lines of Q_{0} are res-fixed; and we have reduced Δ^{z} to the cone Star(Q_{0}), which is collapsible as required.

Case 2: z is 2-central. As a permutation on 24 letters, this z has cycle type 1^{8}2^{8}, with the 8 fixed letters defining a distinguished octad (point) which we will denote by p_{0}; we try to collapse to Star(p_{0}), eventually in fact to a simpler subcomplex.

As in the previous paragraph, we begin by using 2.2 to reduce Δ^{z} to the subcomplex Δ_{Q}^{z} with only res-fixed quads. This time, there will be more than one such quad, and we study carefully the configuration they determine: We note that for incident line l and quad Q, the (abelian) kernels K_{l} and K_{Q} of order 2^{6} intersect O_{2}(C_{G}(z) ≅ 2^{1+6} in subgroups of order 2^{4}, and so determine the two 4-groups in a dihedral Sylow 2-group of the quotient C_{G}(z)/O_{2}(C_{G}(z)) ≅ L_{3}(2). The conjugates of these 4-groups under L_{3}(2) conversely cover the 7 conjugates each of K_{l} and K_{Q}; furthermore they afford the incidence geometry of a projective plane P G(2, 2). ^{3} From this description, we conclude using 2.4 that there are just 7 res-fixed lines l_{1}, l_{2}, …, l_{7} and 7 res-fixed quads Q_{1}, Q_{2}, …, Q_{7}. The plane incidence shows that each quad Q_{i} contains 3 of the lines l_{j}, which are the 3 lines of the quad on p_{0}; similarly each line l_{j} appears as the intersection (in terms of points or lines) of any pair from a set of 3 “plane-collinear” quads Q_{i}. In particular, these 7 quads are the res-fixed quads defining Δ_{Q}^{z}.

With this information in hand, we proceed to further reductions. Let l be a line which is fixed but not res-fixed. If l is not pointwise fixed, it is removable just as in Case 1, using 2.3. So now assume l is pointwise fixed. In this case, we can extend 2.5 in an obvious way: we check that involutions in the subgroup 2^{6}L_{3}(2) of G_{l} fixing l pointwise are covered by conjugates of incident-quad kernels K_{Q}; it follows that l lies in one of the res-fixed quads Q_{i}. But since the res-fixed l_{j} give the pairwise intersections of the Q_{i}, we conclude that Q_{i} is the only res-fixed quad on l. Thus the residue in Δ_{Q}^{z} of l is the cone Q_{i}∗ { 3 points of l }. So in this case also we can remove l by 2.1. Thus we have reduced to the subcomplex Δ_{LQ}^{z} with only the res-fixed lines l_{j}.

Now let p by any fixed point. By arguing much as we did in Case 1, we check that all conjugates of 2-central z in a point stabilizer G_{p} are covered by incident-quad kernels K_{Q}; so 2.5 shows that p must lie in one of the res-fixed quads Q_{i}. So assume now p is in some Q_{i} but not in any l_{j}. Then in Δ_{LQ}^{z} the residue of p is just Q_{i} (since these quads intersect just in the lines l_{j}). We remove these points by 2.1, and get the subcomplex whose only points are those of the l_{j}. (In fact, this is just the subcomplex Δ_{PLQ}^{z} with also all points res-fixed). But here the residue of any quad Q_{i} is the tree given by the 7 points on its 3 lines intersecting at p_{0}, contractible. When we remove the quads by 2.1, we have left only the tree given by the 15 points on the 7 res-fixed lines l_{j} intersecting at p_{0}, again contractible. This finally completes the proof for 2-central z.

3The configuration may be easier to visualize inside the abovementioned 11-dimensional module V: where the quads are 4-spaces intersecting pairwise in the lines given by 2-spaces, all containing the 1-space p_{0}. The plane structure is evident in the 3-dimensional quotient [V, z]/p_{0}, intersected by the lines and quads in spaces of dimensions 1 and 2.

3.3: The 2-local geometry for M_{22}.

The geometry is described in Ronan-Smith [RoSm80] . It is in fact a subgeometry of the M_{24} geometry just considered, as follows: Exhibit M_{22} as the stabilizer of 2 letters a, b from the 24-set permuted by the larger group. The hexads H are the 6-sets determined whenever an M_{24} point, regarded as an 8-set, includes both of a, b. The octads O are the 8-sets given by the M_{24} points containing neither of a, b. The quintets Q are the M_{24} quads with both a, b in one of the six tetrads. Incidence in the geometry is induced from M_{24}: hexad with octad when both are points of an M_{24} line; and hexad or octad with quintet when the former as a point in M_{24} lies in the latter viewed as a quad.

We describe also the stabilizers and residues. For a hexad H, we have G_{H} ≅ 2^{4}Sp_{4}(2)′, with Res(H) a “2-split cover” of the symplectic quadrangle: there are 15 quintets as usual, but now 30 octads—two from each M_{24} trio on H. An octad O has stabilizer G_{O} ≅ 2^{3}L_{3}(2), with Res(O) just the standard projective plane P G(2, 2). And a quintet Q has G_{Q} ≅ 2^{4}S_{5}, with Res(Q) a geometry on 5 hexads (one given by each of the 5 tetrads off a, b with the tetrad on a, b) and 10 octads (given by each of the \binom{5}{2} pairs of tetrads off a, b). Incidence occurs when a hexad and octad occupy 3 distinct tetrads off a, b. Since this is not the usual inclusion, it may be more intuitively satisfactory to replace the tetrad pairs by their complementary triples. Then the geometry is given by a 5-set and all triples from it, with incidence given by inclusion.

With this background, we can begin the proof. There is just one class of involutions z in G = M_{22}. We find that C_{G}(z) = G_{H_{0},O_{1},O_{2}} for a particular line of M_{24} consisting of a hexad H_{0} and two octads O_{1}, O_{2}. We aim to collapse Δ^{z} to the star of this triple of vertices.

As in the case of M_{24}, we look first at the quads, which now determine quintets Q. We cannot apply 2.2 since Res(Q) is not a building; but we can directly verify 2.1 instead: If Q is fixed but not res-fixed, then z acts as an involution in Ḡ_{Q} ≅ S_{5}; up to conjugacy there are just two possibilities. If z acts on the 5 tetrads as the transposition (12), then it fixes the tree rooted at 345 with branches 3 ∗ 123, 4 ∗ 124, 5 ∗ 125 — collapsible. If z acts as (12)(34), then it fixes only 125 ∗ 5 ∗ 345, also collapsible. So in fact 2.1 allows us to remove such Q, and we reduce Δ^{z} to Δ_{Q}^{z} with only res-fixed quintets. But now we observe that the 3 Sylow 2-groups of C_{G}(z)/K_{H} ≅ S_{4} are covered by 3 conjugates of K_{Q}; with 2.4 we conclude there are just three res-fixed quintets Q_{1}, Q_{2}, Q_{3} defining Δ_{Q}^{z}. These are in fact just the three quintets lying above our special trio H_{0}, O_{1}, O_{2}. ^{4}

Now consider any z-fixed octad O. Since involutions in G_{O} ≅ 2^{3}.L_{3}(2) are covered by the conjugates of K_{Q} ≅ 2^{4}, we conclude via 2.5 that O must lie in one of the res-fixed quintets Q_{i}. Similarly for a fixed hexad H, the involutions in G_{H} ≅ 2^{4}.A_{6} are covered by incident quintet kernels K_{Q}, so we conclude with 2.5 that H must lie in one of the Q_{i}. Thus the fixed octads and hexads (which remain in Δ_{Q}^{z}) lie in the res-fixed quintets Q_{i}.

We finish by mimicking the final argument in the 2-central case for M_{24} above. First let X be an octad or hexad of Q_{i}, other than H_{0} or O_{1}, O_{2}. As these latter give the pairwise intersections of the res-fixed quintets, we see Q_{i} is the only quintet in the residue of X in Δ_{Q}^{z}. Just as in 2.3, we may remove such X. (We can check this reduces us to the subcomplex Δ_{HOQ}^{z} where all vertices are res-fixed). But this is just the cone H ∗[{O_{1}, O_{2}}∗{Q_{1}, Q_{2}, Q_{3}}], collapsible. 3.4: The 2-local geometry for U_{4}(3).

4Much as in the M_{24} case, it may help to consider the 10-dimensional M_{22}-submodule W of the 11dimensional module V described above for M_{24}, in which these three quintets are 4-spaces, intersecting pairwise in the 2-space whose 1-spaces are H_{0}, O_{1}, O_{2}.

The geometry is described in [RoSm80, Ka81] . Points P, lines L, and quads Q can be interpreted as certain isotropic subspaces of dimensions 1,2,3 in a 6-dimensional unitary space over F_{4}, obtained from the inclusion SU_{4}(3) ⊂ SU(6, 2^{2}). However, our analysis of the geometry will proceed entirely in terms of fusion in the stabilizing subgroups, which we now describe. A point p has stabilizer G_{p} ≅ 2^{4}Sp_{4}(2)′, with Res(p) the corresponding generalized quadrangle. The structures for a quad Q are isomorphic (and in fact points and quads are interchanged by an outer automorphism of G). For a line l we have G_{l} ≅ 2^{1+4}(S_{3} × S_{3}), with Res(l) the complete bipartite graph { 3 points of l} ∗ { 3 quads above l}.

Now to the proof. There is one class of involutions z, with C_{G}(z) = G_{l_{0}} for a particular line l_{0}; our aim will be to contract to Res(l_{0}). However, most of the argument will depend not on the particular structure of this centralizer, but only on the “local” structure of the stabilizers and their intersections; we can then quote the more general argument in some non-contractibility arguments later on.

So we begin the general argument. Since the point residue is a generalized quadrangle, we use 2.2 to reduce Δ^{z} to Δ_{P}^{z} containing only res-fixed points. Next we consider a quad Q which is fixed but not res-fixed. We cannot apply 2.2 since in obtaining Δ_{P}^{z} we removed some points; so we must study more carefully the residue of Q in that subcomplex. Now action of z on the quadrangle Res(Q) is determined by the unique class of involutions in Ḡ_{Q} ≅ Sp_{4}(2)′. We find that the point-line subgraph fixed by z can be described as an H-shaped configuration determined by a distinguished point p and line l: it is just the identification along the edge p ∗ l of the two stars p ∗ {l, m, n} and {p, q, r} ∗ l; where l, m, n are all 3 lines on p, and p, q, r are the 3 points of l. But now we observe that the involutions of G_{Q} are covered by the kernels K_{s} for incident points s; using 2.4 we see that at least one point of Q must be res-fixed. By definition of res-fixed, all lines on this point must be z-fixed; as only p among the 3 fixed points above has this property, p must be that res-fixed point, and it is unique. So the residue of Q in Δ_{P}^{z} is the cone p ∗ {l, m, n}; and by 2.1 we remove such quads, reducing to Δ_{PQ}^{z} where now all the quads are also res-fixed.

Of course we turn next to a line l which is fixed but not res-fixed. We observe that the extension in G_{l} of 2^{1+4} by S_{3} × S_{3} is not split, with involutions occurring only in the cosets for elements of either S_{3} (not in “diagonal” cosets). And so involutions in G_{l} are in fact covered either by conjugates of an incident point kernel K_{p} or a quad kernel K_{Q}. In the former case we conclude with 2.4 that the unique point of l fixed by z is in fact res-fixed, so that the residue in Δ_{PQ}^{z} of l has unique point p. As in 2.3, we can remove such lines. The case z ∈ K_{Q} is entirely similar, leading to a unique res-fixed quad Q in Res(l); so we in fact reduce to the subcomplex Δ_{PLQ}^{z} in which all vertices are res-fixed.

Next we will eliminate the quads entirely: If Q is a res-fixed quad, then z ∈ K_{Q}, and we wish to determine the residue of Q in Δ_{PLQ}^{z}. We regard K_{Q} as a symplectic 4-space for Ḡ_{Q} ≅ Sp_{4}(2)′; then incident point kernels K_{p} intersect K_{Q} in isotropic 2-spaces, and line kernels K_{l} in 3-spaces with radical Z(K_{l}). Consdering z as a vector of K_{Q}, we see z lies in the 3 isotropic 2-spaces in the 3-space z^{⊥}; correspondingly we conclude that there are just 3 res-fixed points p_{1}, p_{2}, p_{3}, forming a res-fixed line l with z ∈ Z(K_{l}). However, z lies in a total of seven 3-subspaces of K_{Q}; we see the remaining 6 res-fixed lines are just the two lines other than l on each of the p_{i}. Thus the residue in Δ_{PLQ}^{z} of Q consists of 3 trees of form p_{i} ∗ { 3 lines on p_{i}}, identified at the common line l— which is easily seen to be a tree and therefore contractible. So by 2.1, we may remove such quads Q. We have reduced now to the bipartite graph on the res-fixed points and lines.

Only one further reduction is needed. We suppose l is a res-fixed line, with z ∈ K_{l}\Z(K_{l}). For the three points p of l, the intersections K_{p} ∩K_{l} are elementary subgroups 2^{3} intersecting just in Z(K_{l}). Thus the residue of l in the graph is a unique point p; and we may remove such lines by 2.1.

Call the resulting graph Δ_{Pz}^{z}. Its lines l satisfy z ∈ K_{l}; then z ∈ K_{p_{i}} for the 3 points p_{1}, p_{2}, p_{3} of l, so the p_{i} are all res-fixed and hence lie in the graph. If now l′ is a line on p_{i} other than l, then l and l′ determine some common quad Q aove p_{i}; and our argument two paragraphs above shows z ∉ Z(K_{l′}), so this line is not in our graph. It follows that Δ^{z}Pz consists of connected components of form l ∗ {p_{1}, p_{2}, p_{3}}, for each line l with z ∈ Z(K_{l}).

We return at last to the special case of G = U_{4}(3). In this case, there is a unique line l_{0} satisfying z ∈ Z(K_{l_{0}})— since C_{G}(z) = G_{l_{0}}. And so we have single connected component, visibly contractible. 3.5: The G̃_{2}-geometry for G_{2}(3).

The geometry was discovered by Cooperstein (unpublished); it is described in Aschbacher-Smith [AsSm] . No “natural” representation of the geometry is known; its points P, lines L, and planes Π are simply defined as cosets (or conjugates) of the corresponding stabilizers: For a point we have G_{p} ≅ G_{2}(2) with Res(p) the generalized hexagon for that group. A line has stabilizer G_{l} ≅ 2^{1+4}(3 × 3)2, with Res(l) the complete bipartite graph on the { 3 points of l} ∗ { 3 planes over l}. A plane has stabilizer G_{π} ≅ 2^{3}L_{3}(2) with Res(π) the projective plane P G(2, 2).

There is one class of involutions z, and C_{G}(z) is just the stabilizer of a particular line l_{0}; we wish to reduce into the cone Star(l_{0}). But as in the case of U_{4}(3), we will make the bulk of the argument depend only on the structure of the stabilizers, and not on the particular structure of C_{G}(z) in G_{2}(3).

We begin the general argument. For a point p, we notice that Res(p) is a generalized hexagon, so 2.2 applies; and the kernel K_{p} = 1. So by 2.6 it suffices to consider fixed vertices in Δ_{P}, the bipartite graph on lines and planes.

Next if a line l is not res-fixed, then action of z on Res(l) is determined by the unique class of involutions in Ḡ_{l} ≅ (3 × 3)2, so that z must fix a unique plane (and point) on l; but then by 2.1 we may remove such lines, reducing to the subcomplex Δ_{PL}^{z}, just the graph with fixed planes and only res-fixed lines.

So suppose π is a fixed but not res-fixed plane. Action of z on the plane Res(π) is determined by the unique class of involutions in Ḡ_{π} ≅ L_{3}(2); the subgraph fixed by z is isomorphic to the H-shaped configuration of the previous section (which actually arose for Sp_{4}(2)′). Mimicking the argument there, we observe that involutions in G_{π} are covered by incident line kernels, and conclude with 2.5 that π contains a res-fixed line. But only one line of the H-configuration is res-fixed; thus the residue of π in Δ_{PL}^{z} is a single line. By 2.1 we remove such planes, reducing now to the subcomplex Δ_{PLΠ}^{z}, which is the graph on res-fixed lines and planes.

We can now imitate closely the final arguments of the previous section. For a line l, the three incident planes π_{1}, π_{2}, π_{3} determine elementary subgroups K_{π_{i}} of K_{l} of order 2^{3}, intersecting pairwise in Z(K_{l}). Thus if z ∈ K_{l} \ Z(K_{l}), its residue in the graph is a single point. We remove such lines using 2.1. So any remaining line l satisfies z ∈ Z(K_{l}); and its 3 planes π_{i} are also in the graph. But if now l′ is a line other than l on π_{i}, we see z ∉ Z(K_{l′}), so l′ is not in the graph. Thus the graph consists of connected components of form l ∗ {π_{1}, π_{2}, π_{3}}, for each l with z ∈ K_{l}.

Finally we return to the special case G = G_{2}(3). Here there is a unique line l_{0} with z ∈ Z(K_{l_{0}}), since C_{G}(z) = G_{l_{0}}. So the remaining graph consists of just one connected component, visibly contractible. With 2.6 we conclude both L̃(Δ_{P}) and L̃(Δ) are projective. 3.6: The characteristic-2 geometry for McL.

The geometry was discovered by Ronan, and is described in Ronan-Stroth [RoSt]. Again the geometry is described in terms of the stabilizing subgroups. We name the vertex types points P, lines L, planes Π, and spaces S. We remark that points and lines do no behave as indicated in 2.3; rather the notation is suggested by the structure of the residue of a space s, which is in fact a projective space P G(3, 2), with action by G_{s} ≅ 2L_{4}(2). The vertices incident to s afford the corresponding geometry of points, lines, and planes. Away from Res(s), though, the structures are genuinely sporadic: For a line l, we have G_{l} ≅ U_{4}(3) and Res(l) its 2-local geometry from Section 3.4. For p a point, we have G_{p} ≅ 2^{4}.A_{7}, and Res(p) the sporadic C_{3} geometry for A_{7} in Section 3.1. The structures for a plane are isomorphic to those for a point (and in fact points and planes are interchanged by an outer automorphism of G).

The group G = McL has a unique class of involutions z, and C_{G}(z) is just the stabilizer of a particular space s_{0}. We will contract into the cone Star(s_{0}).

We first apply 2.6 to the line residues: since G_{l} ≅ U_{4}(3) is simple, the kernel K_{l} is trivial; and the work of Section 3.4 showed that Res(l)^{z} is contractible. So it suffices to consider fixed vertices in the truncation Δ_{L} from which all ines have been removed.

Now let p be a point which is fixed but not res-fixed. Then the residue in Δ_{L}^{z} of p is a configuration which we saw at the end of Section 3.1—just the z-fixed vertices there called lines and planes, contractible as we then commented. Thus by 2.1 we reduce to considering the sucomplex Δ_{PL}^{z} where now all points are res-fixed.

Next let π be a plane which is fixed, but not res-fixed. Then the residue of π in Δ_{PL}^{z} consists, in the language of Section 3.1, of the unique res-fixed A_{7}-line and its 3 planes, contractible. So by 2.1 we reduce to the subcomplex Δ_{PLΠ}^{z} where now all planes are resfixed.

Finally let s be a space which is fixed, but not res-fixed. We notice that the extension of the group K_{s} of order 2 by Ḡ_{s} ≅ L_{4}(2), and involutions occur only in cosets corresponding to transvections in the quotient. And on the projective space P G(3, 2) such an involution resfixes a unique point in a unique res-fixed plane. So the residue of s in Δ_{PLΠ}^{z} is contractible; we reduce by 2.1 to the complex Δ_{SPLΠ}^{z} with all vertices res-fixed.

But now we observe that for the res-fixed space s_{0} above, K_{s_{0}} is just ⟨z⟩, which is central in G_{s_{0}}. Arguing as in 2.4, we see for a res-fixed space s that G_{s} would like in C_{G}(z) = G_{s_{0}}. We conclude that s_{0} is the unique res-fixed space. And we note also that involutions of a point or plane kernel are covered by conjugates of incident space kernels; so that by 2.5 the res-fixed points and lines must be just those of the unique res-fixed space s_{0}. Consequently we have reduced to the subcone of Star(s_{0}) given by the product of s_{0} with the graph on its 15 points and 15 planes. This is contractible, and with 2.6 we conclude that L̃(Δ_{L}) and L̃(Δ) are both projective. 3.7: The 2-local geometry for J_{3}.

Apparently only rank-2 truncations of this geometry had been previously studied [Wei, RoSt] [Bu, Sec. 11, no. 54] ; as a rank-3 geometry, it was first described by Yoshiara, and fuller details will appear in [Yo] . (We remark that the geometry does not have a characteristic in the rigorous sense; but the module it determines is interesting in characteristic 2). We define vertex types as vectors V, points P, and lines L. The names are suggested by the form of a line stabilizer G_{l} ≅ 2^{4}.GL_{2}(4); regarding K_{l} as a 2-space over F_{4}, we get 5 projective points p of this projective line l. But each point p can in turn be regarded as a 2-space over F_{2}, with 3 non-zero vectors. Thus Res(l) consists of 5 connected components of form p ∗ { 3 vectors of p}. A point p has stabilizer G_{p} ≅ 2^{2+4}(3 × S_{3}), with Res(p) the complete bipartite graph { 3 vectors of p}∗{ 2 lines above p}. Finally a vector has stabilizer G_{v} ≅ 2^{1+4}A_{5}, with Res(v) consisting of 5 connected components, one for each point p above v, of form p ∗ { 2 lines above p}.

There is just one conjugacy class of involutions z, with C_{G}(z) ≅ 2^{1+4}.A_{5} the stabilizer of a particular vector v_{0}. As usual, we try to collapse down to Star(v_{0}).

Consider first a line l which is fixed but not res-fixed. As there is just one involution class in Ḡ_{l} ≅ 3 × A_{5}, z must permute the 5 components of Res(l) with cycle type 1^{1}2^{1}, in particular fixing only one point p and possibly some of its vectors. So the residue of l in Δ^{z} is a cone with vertex p, collapsible. By 2.1 we remove such lines, reducing to the subcomplex Δ_{L}^{z} with all lines res-fixed. But now we observe that there are just 10 conjugates of a line kernel K_{l} in C_{G}(z), so we conclude using 2.4 that the res-fixed lines are just the 10 lines l_{i}, m_{i} which in pairs cover the 5 points p_{i}(i = 1, …, 5) in Res(l_{0}).

We turn to a point p which we assume to be fixed but not res-fixed. Action of z on Res(p) is determined by the unique class of involutions in Ḡ_{p} ≅ 3×S_{3}, so z must fix a unique vector v of p. As Res(v) is a complete bipartite graph, it follows as in 2.3 that the residue in Δ_{P}^{z} of p is a cone on v and therefore collapsible. Thus we reduce to the subcomplex Δ_{PL}^{z} where now also the points are res-fixed. We observe that involutions of a point kernel K_{p} are covered by the two incident line kernels K_{l}, so we conclude as in 2.5 that any res-fixed point lies in one of the 10 res-fixed lines determined above. Conversely if p is a point on one of these lines l, then z fixes the three vectors of p since l is res-fixed; and as there are just two lines above p, z must fix also fix the second line (other than l) above p—that is, z res-fixes p. Thus the res-fixed points are just those of the 10 res-fixed lines; these include the 5 points p_{i} over v_{0}, where l_{i} and m_{i} intersect, and the 4 remaining points of each of the 10 lines.

So let v denote a fixed but not res-fixed vector. It is convenient here to identify the vector v_{0} with the involution z, and v with an involution in Ḡ_{z} ≅ A_{5}. Now the 10 line kernels in G_{z} intersect at involutions only in the corresponding point 4-groups Z(K_{p_{i}}) which lie entirely in K_{z}. Since the line kernels (in pairs) cover the 5 trivially-intersecting Sylow 2-groups of A_{5}, we conclude v lies in just one res-fixed line, say l_{i}. But also the only involutions in the extra-special group K_{z} fixed by z form a 4-group giving Z(K_{p_{i}}), so there is also a unique res-fixed point centralized by v. We conclude finally that the residue of v in Δ_{PL}^{z} is just p_{i}∗l_{i}, contractible. So we reduce using 2.1 to the subcomplex Δ_{PLV}^{z} with all vertices res-fixed. Here we observe that involutions of a vector kernel K_{v} ≅ 2^{1+4} are covered by the kernels K_{p} for incident points p; we conclude as in 2.5 that a res-fixed vector is on one of the 5 res-fixed points p_{i}. Conversely, such vectors are on res-fixed lines, and our earlier argument using the fact that only two lines lie above a vector shows that these vectors are indeed res-fixed.

The rest is easy. First let v be a vector of some p_{i}, other than v_{0}. Then as the points p_{i} intersect just at v_{0}, the residue of v in Δ_{PLV}^{z} is p_{i} ∗ {l_{i}, m_{i}}, collapsible. So by 2.1 we reduce to the subcomplex Δ_{PLv_{0}}^{z} which now has unique vector v_{0}. Then if p is a point of l_{i} other than p_{i}, since the res-fixed lines intersect in pair just at the p_{i}, the residue of p in Δ_{PLv_{0}}^{z} is l_{i}. We use 2.1 to remove such lines, and we have finally reduced to the cone Star(v_{0}), collapsible. 3.8: The pairs-geometry for M_{11} in characteristic 3.

The geometry is described by Buekenhout [Bu, Sec. 6, no. 4] . It is just the flag graph from the complete graph on 11 vertices. Thus the points are the 11 letters on which M_{11} acts, and the lines are the 55 = \binom{11}{2} pairs of points. There is just one class of elements z of order 3, with cycle type 1^{2}3^{3}. Evidently z fixes just one line and its two points, visibly collapsible. 3.9: The characteristic-3 geometry for L_{3}(4).

The geometry is described in Buekenhout [Bu, Sec. 6, no. 16]. It is just the flag graph for the graph on 56 vertices corresponding to a class of subgroups A_{6}, joined when they intersect in a Sylow 3-normalizer. Thus the vertices give the points P, with stabilizer G_{p} ≅ A_{6} and 10 lines on a point. And the edges give the lines L, with stabilizer G_{l} ≅ 3^{2}Q_{8} and 2 points on a line.

There is a single class z of elements of order 3, with centralizer elementary of order 3^{2}. But this group is just O_{3}(K_{l_{0}}) for a particular line l_{0}, and contains all 3-elements of N_{G}(⟨z⟩). So l_{0} is the unique z-fixed line, and is in fact res-fixed. As there are 10 lines on a point, any z-fixed point must lie on some z-fixed line, which can only be l_{0}. Thus Δ^{z} consists of the tree Star(l_{0}), collapsible. 3.10: The Sylow-3 overgroup geometry for U_{5}(2).

The geometry is described in Buekenhout [Bu, Sec. 7, no. 20, case n = 5] and Aschbacher [As, p. 40]. The points P have stabilizer G_{p} ≅ 3 × Sp_{4}(3), with Res(p) the corresponding generalized quadrangle. The lines L have stabilizer G_{l} ≅ S_{3} × 3^{1+2}SL_{2}(3), with Res(l) the complete bipartite graph {2 points of l} ∗ {4 quintets above l}. The quintets Q have stabilizer G_{Q} ≅ 3^{4}S_{5} with residue given by the flag graph of the complete bipartite graph on 5 points—that is, 5 points and 10 lines given by pairs of points.

Now the geometry is preserved by the outer automorphsim group of U_{5}(2), so we may consider just 4 fused classes of elements z of order 3, rather than the 6 classes in the simple group. We distinguish these by the notation 3AB, 3CD, 3E, 3F in the [Atlas].

Case 1: z of type 3F. Suppose Q is a quintet which is fixed but not res-fixed. Then z must act as a 3-element of Ḡ_{Q} ≅ S_{5}, and so has cycle type 1^{2}3^{1}. Thus z fixes in Res(Q) just a single line (pair) and its two points, a tree. By 2.1, we can reduce to the subgeometry Δ_{Q}^{z} with only res-fixed quintets. However, C_{G}(z) is of form K_{Q_{0}}2^{2} for some particular quintet Q_{0}. Since K_{Q_{0}} then contains all the 3-elements of N_{G}(⟨z⟩), it is certainly the only conjugate of K_{Q} in that group. We conclude Q_{0} is the unique z-fixed quintet. We finish just as in the L_{3}(4) case above: any z-fixed point or line must lie on some z-fixed quintet, hence on Q_{0}. Thus Δ_{Q}^{z} consists just of the cone Star(Q_{0}), collapsible.

Case 2: z of type 3CD or 3E. We begin just as in Case 1, by reducing to Δ_{Q}^{z} with resfixed quintets. Since conjugates of z in a line stabilizer G_{l} are covered by incident-quintet kernels K_{Q}, we see from 2.5 that any z-fixed line l lies on at least one res-fixed quintet Q. If now l is not res-fixed, then Q is the only one of the four quintets above l fixed by z, so that the residue in Δ_{Q}^{z} of l is just Q ∗ {2 points of l}, collapsible. So we reduce to Δ_{LQ}^{z} with also all lines res-fixed. Now we observe (for either conjugacy class type) that C_{G}(z) is contained in G_{l_{0}} for a particular line l_{0}. The elementary group K_{Q} ≅ 3^{4} is weakly closed in a Sylow 3-group; so each of the 4 Sylow 3-groups of G_{l_{0}} determines a single conjugate of K_{Q} in N_{G}(⟨z⟩). Using 2.4 we see the 4 quintets of Res(l_{0}) are the only res-fixed quintets. And now it is easy to finish: Since any two of these quintets intersect exactly in l_{0}, we see any res-fixed line other than l_{0} would lie in just one of the 4 quintets, and so can be removed as in 2.3. And then any point not in l_{0} would determine just a single quintet in its residue, and be removable by 2.1. What then remains is just the cone Star(l_{0}), collapsible.

Case 3: z of type 3A. We begin as in Case 2 by reducing to Δ_{LQ}^{z} with lines and quintets res-fixed. Also we note that incident line and quintet kernels cover conjugates of z in G_{p}, so that any fixed point p must lie in some res-fixed line and some res-fixed quintet by 2.5. Now C_{G}(z) is just the stabilizer G_{p_{0}} for some particular point p_{0}. The quintet kernel K_{Q} determines an elementary group 3^{3} in the quotient Ḡ_{p_{0}} ≅ Sp_{4}(3), and the 40 conjugates of this group determine by 2.4 that the res-fixed quintets are just the 40 quintets of Res(p_{0}). These quintets intersect pairwise only in lines above p_{0}; so any res-fixed line not above p_{0} would have in its residue just one of the quintets, and so be removable as in 2.3. Thus we have further reduced to the 40 res-fixed lines of Res(p_{0}). Next, a point not on one of these lines but in one of the quintets Q determines just Q as its residue, and so may be removed by 2.1. And finally, a point on one of these lines l but distinct from p_{0} determines as its residue the cone l ∗ { 4 lines above l}, removable by 2.1. We have reduced at last to the cone Star(p_{0}), collapsible.

3.11: The rank-3 geometry for McL in characteristic 3.

This geometry is described in Buekenhout [Bu, Sec. 7, no. 22] . Points P are given by 3-central elements of order 3, with stabilizer G_{p} ≅ 3^{1+4}SL_{2}(5); and lines L by subgroups G_{l} ≅ 3^{4}M_{10}, where M′_{10} acts as Ω_{4}^{−}(3) on K_{l} = 3^{4}. Quads Q are given by subgroups G_{Q} ≅ Ω_{6}^{−}(3), with Res(Q) a configuration of points and lines giving the corresponding generalized quadrangle.

We begin by applying 2.6 to the quads: the fixed subcomplex in the residue is collapsible by 2.2, and G_{Q} ≅ Ω_{6}^{−}(3) simple gives trivial kernel K_{Q} = 1. So it suffices to consider the truncation Δ_{Q}, just the bipartite graph on points and lines. (The truncated geometry is described in Buekenhout [Bu, Sec. 6, no. 9] and Ronan-Stroth [RoSt, Sec. 3]).

Case 1: z is 3-central. Here C_{G}(z) is just the full stabilizer G_{p_{0}} of a particular point p_{0}. Now for a line l, a 3-element of Ḡ_{l} ≅ M_{10} fixes just one of the 10 points of l; so by 2.1 we may reduce to the subgraph Δ_{LQ}^{z} with only res-fixed lines. But there are just 10 conjugates of K_{l} in C_{G}(z), so we conclude using 2.4 that the res-fixed lines are just the 10 lines on p_{0}. Furthermore, as the conjugates of z in C_{G}(z) are covered by the kernels K_{l} for these lines, we conclude by 2.5 that the z-fixed points are precisely the points of these 10 lines. Since any pair of these lines intersects precisely in p_{0}, we see that for p any other point, the residue in Δ_{LQ}^{z} of p is just a single line on p_{0}. We can remove such lines by 2.1, and what remains is the cone Star(p_{0}), collapsible.

Case 2: z is not 3-central. We can reduce just as in the previous case to Δ_{LQ}^{z} with only res-fixed lines. But now C_{G}(z) ≅ 3^{4}(2 × S_{3}) is contained in the stabilizer G_{l_{0}} for a particular line l_{0}. In fact K_{l_{0}} is the only line kernel in C_{G}(z), so we obtain using 2.4 that l_{0} is the only res-fixed line. Now conjugates of our new z in a point stabilizer G_{p} are covered by incident line kernels K_{l}, so we conclude using 2.5 that fixed points must lie in the unique res-fixed line l_{0}. Thus Δ_{LQ}^{z} is in fact just the cone Star(l_{0}), collapsible.

With 2.6 we conclude both L̃(Δ_{Q}) and L̃(Δ) are projective. 3.12: The G̃_{2}-geometry for Ly.

This geometry was discovered by Kantor and Tits; it is described in Kantor [Ka81] , again in terms of subgroups. The vertices are points P, with stabilizer G_{p} ≅ G_{2}(5) and Res(p) the corresponding generalized hexagon. The lines L have stabilizer G_{l} ≅ 5^{1+4}2A_{6}.4 and Res(l) given by the complete bipartite graph { 6 points of l} ∗ { 6 planes above l}. The planes Π have stabilizer G_{π} ≅ 5^{3}SL_{3}(5), with Res(π) the projective plane P G(2, 5).

We begin by applying 2.6 to the points: by 2.2 we may reduce to res-fixed points; but as G_{p} ≅ G_{2}(5) is simple, the kernel K_{p} = 1 and there are no such points. Thus it suffices to work in the truncation Δ_{P}, which is just the bipartite graph on lines and planes.

Case 1: z is not 5-central. Suppose l is a line which is fixed but not res-fixed. Then action of z on Res(l) is given by the unique class of order 5 in Ḡ_{l}, which determines a unique fixed plane over l. So using 2.1, we may reduce to the subcomplex Δ_{PL}^{z} with only res-fixed lines. However, we find C_{G}(z) ⊂ G_{l_{0}} for a particular line l_{0}; and the conjugates of z in K_{l_{0}} form a T.I. set, so we conclude by 2.4 that l_{0} is the unique res-fixed line. the 6 points of l_{0} are all the fixed points. Next let π be a fixed plane. Since elements of order 5 in G_{π} are covered by conjugates of incident line kernels, we conclude by 2.5 that π must be one of the 6 planes above the unique res-fixed line l_{0}. So we have reduced to the product of l_{0} with its 6 planes, collapsible.

Case 2: z is 5-central. This time, N_{G}(⟨z⟩) is the full stabilizer of some line, which we denote by l_{0}. The local structure here is closely analogous (for the prime 5) to that found in the G_{2}(3) case (for the prime 2) considered in Section 3.5, so we indicate more briefly how the argument given there can be imitated here. We have already reduced by 2.6 to the graph Δ_{P} on lines and planes. Then as in the previous paragraph, a line which is fixed but not res-fixed has a unique fixed plane, so we reduce to Δ_{PL}^{z} with lines also res-fixed. Again with 2.5 we see any fixed plane π contains at least one res-fixed line; if the plane is not res-fixed, action of z on the projective plane Res(π) shows this line is unique. So using 2.1 we reduce to the complex Δ_{PLΠ}^{z} with also all planes res-fixed. Use of 2.4 shows the res-fixed planes are just the 6 planes above l_{0}. Via 2.5 we check any res-fixed line must lie in some res-fixed plane. But these planes intersect pairwise in l_{0}, so any line l other than l_{0} is contained in just one of the planes. We use 2.1 to remove these lines, and have reduced to the cone Star(l_{0}), collapsible.

With 2.6 we conclude both L̃(Δ_{P}) and L̃(Δ) are projective. 3.13: The graph for O′N in characteristic 7.

The geometry is described in Aschbacher [As] and in the [Atlas]. It is a bipartite graph on points P and lines L, each with stabilizer L_{3}(7).2, with valency 456. There are two classes of elements of order 7, denoted 7A and 7B in the [Atlas].

Case 1: z of type 7A. Here N_{G}(⟨z⟩) is just the stabilizer G_{pl} of a particular edge pl. Now G_{pl} is just a Borel subgroup in the point stabilizer G_{p}, and we see z is conjugate in G_{p} to the root elements of each of two subgroups 7^{2} in the unipotent group 7^{1+2}. Each has 8 subgroups of order 7, with ⟨z⟩ in common; each of the 14 new edges is determined by a line l′ on p other than l. A similar analysis applies to G_{l} (the stabilizers are switched by the outer automorphism group. Furthermore this analysis accounts for all of Δ^{z}, since it covers one of the two orbits (each of length 4) of G_{pl} on subgroups of order 7 in the quotient 7^{1+2}/⟨z⟩, and the other orbit consists of elements of type 7B. We see now that Δ^{z} is contractible, since it is just the identification of two trees p ∗ { 15 lines } and l ∗ { 15 points } along their intersection, the edge pl.

Case 2: z of type 7B. Here there is a unique subgroup of type 7A in N_{G}(⟨z⟩); we conclude z fixes just a single edge pl and its two vertices, contractible.

4. Analysis into projective indecomposables.

In this section our results are character-theoretic rather than geometric and it is convenient to make a few standard definitions which help us to move from geometric to algebraic information. We choose a maximal flag in Δ and for each subset F ⊂ Δ we define G_{F} to be the stabilizer of the subflag of type F in our chosen flag. If F ⊂ F′ ⊂ Δ then G_{F′} ⊂ G_{F} and given any character, ψ_{F′}, of G_{F′} we can form the induced character ψ_{F′}^{F} of G_{F}. From the description of L̃(Δ) as a sum of chain groups we see that its character is given by:

L̃(Δ) = ∑_{F ⊂Δ}(−1)^{(|Δ|−|F |)}1_{F}^{G}.

In order to apply this formula we always need the permuation characters on the vertex stabilizers; these can often be found in the [Atlas] . To obtain the permutation character for the stabilizer of a longer flag, F say, we use the following inductive method. We first choose a vertex v ∈ F. Now Ḡ_{v} = G_{v}/K_{v} acts on Res(v). The group Ḡ_{F} = G_{F} /K_{v} is the Ḡ_{v}-stabilizer of the flag F \ v. Working with the smaller geometry Res(v), we construct the permutation character \overline{1_{F}^{v}}, of Ḡ_{v} on the cosets of Ḡ_{F}. The character 1_{F}^{v} is found by lifting \overline{1_{F}^{v}} to a G_{v}-character in which K_{v} acts trivially. We induce again to end up with 1_{F}^{G} = (1_{F}^{v})^{G}.

The following tables summarize the computation of the characters of the Lefschetz modules for the groups in Table 1. In all but the largest cases we use the known p-modular character table of G to split L̃(Δ) into a sum of projective indecomposable characters. Whenever the characters of a group are already named in the [Atlas] we use the same names without further comment. In each table we describe characters (of G) by their inner products with the irreducible characters of G as ordered in the [Atlas]. In order to represent each inner product by a single entry, we use a,b,..z,A,..Z to denote the integers 10,11,..35,36..61. A few entries exceed 61, these are written as ∗ and their exact values can be deduced from other information which we give. The computations described here were carried out with MOC3 computer programs of Parker, Lux and Hiss (see [Par, Hi] ).

Table 4.1: The C_{3} geometry for A_{7}.
F = Flag  G_{F}  1_{F}^{G}
A_{7}  1
p  A_{6}  1 1
l  (3 × A_{4}) : 2 1 1  1 1
π  L_{3}(2) 1  1
pl  S_{4}  1 2  2 1 1 1
pπ, lπ  S_{4}  1 1  1 2 1 1
plπ  D_{8}  1 2 0 0 3 3 1 3 4
L̃(Δ)  1 1
L̃(Δ_{Π})  1 1 1

Notes on Table 4.1: G_{π} ≅ L_{3}(2) and from the [Atlas] we obtain the L_{3}(2)-characters 1_{pπ}^{π} = 1_{lπ}^{π} = 1a + 6a, 1_{plπ}^{π} = 1a + 6aa + 8a. Also G_{p} ≅ Sp_{4}(2)′ so that 1_{pl}^{p} and 1_{pπ}^{p} are the A_{6}-characters 1a + 5a + 9a and 1a + 5b + 9a. L̃(Δ) = 21a + 35a is the projective cover of an irreducible 20-dimensional 2-modular representation of A_{7}; and L̃(Δ_{Π}) = 14a + 15a + 35a of a 14-dimensional modular irreducible (see Benson [Ben, p.212] ).

Table 4.2: The 2-local geometry for M_{24}.

F = Flag  G_{F}  1_{F}^{G}
M_{24}  1
p  2^{4} : A_{8}  1 1  1 1
l  2^{6}.(S_{3} × L_{3}(2)) 1  1 1  1  1
Q  2^{6} : 3.S_{6}  1  1 1  1
pl  2^{5}.2^{3}.L_{3}(2) 1 1  2 2  1  1 1 1
pQ  2^{6} : 3.(S_{4} × 2) 1 1  3 3  2  1 1 2 2 1
lQ  2^{6} : 3.(S_{4} × 2) 1  2 2  3  2 1 1 1 1
plQ 2^{6} : 3.(D_{8} × 2) 1 1 0 0 0 0 4 0 4 0 0 0 0 4 0 0 1 0 3 0 4 4 3 0 2 1
L̃(Δ)  1 1 1

Notes on Table 4.2: Ḡ_{p} ≅ L_{4}(2) hence 1_{pl}^{p}, 1_{pQ}^{p} are obtained by lifting the A_{8}-characters 1a + 14a, 1a + 14a + 20a to G_{p}. Ḡ_{Q} ≅ 3.Sp_{4}(2) and so we find 1_{pQ}^{Q}, 1_{lQ}^{Q}, 1_{P lQ}^{Q} by lifting the S_{6}-characters 1a + 5a + 9a, 1a + 5b + 9a and 1a + 5ab + 9aa + 8ab to G_{Q}. L̃(Δ) = 5313a + 5796a + 10395a is the projective cover of the irreducible 2-modular character of degree 1792 (see James[Jam]).

Table 4.3: The 2-local geometry for M_{22}.

F = Flag G_{F} 1_{F}^{G}
M_{22} 1
H  2^{4} : A_{6} 1 1 1
Q  2^{4} : S_{5} 1 1 1 1
O  2^{3} : L_{3}(2) 1 1 1 1 1
HQ 2^{4} : S_{4} 1 2 2 1 2 1 1
HO, QO 2^{3} : S_{4} 1 2 3 2 3 1 2  2
HQO 2^{3} : D_{8} 1 3 5 4 6 3 6 1 1 7
L̃(Δ)  2 1 1 2

Notes on Table 4.3: \overline{G}_{O} ≅ L_{3}(2) hence 1_{HO}^{O} = 1_{QO}^{O} and 1_{HQO}^{O} are obtained by lifting the L_{3}(2)-characters 1a + 6a and 1a + 6aa + 8a to G_{O}. \overline{G}_{H} ≅ Sp_{4}(2)′ so we obtain 1_{HQ}^{H} by lifting the A_{6}-character 1a + 5a + 9a to G_{H}. L̃(Δ) = 231aa + 280ab + 385aa is the sum of the projective covers of the two irreducible 2-modular characters of degree 70 (see [Jam]).

Table 4.4: The 2-local geometry for U_{4}(3).

F = Flag  G_{F}  1_{F}^{G}
U_{4}(3)  1
p  2^{4} : A_{6}  1 1  1 1  1
l  2^{1+4} : (S_{3} × S_{3}) 1 1 1 2 1 1  1 1  1 1
Q  2^{4} : A_{6}  1 1  1 1  1
pl  2^{4} : S_{4}  1 1 1 1 3 2 1  1 4 2 1 1 3 2
pQ  2^{4} : S_{4}  1 2  2 4 1 1  2 2 1 2 2 3 1
Ql  2^{4} : S_{4}  1 1 1 1 3 2 1  4 1 2 1 1 3 2
plQ  2^{4} : D_{8}  1 2 1 1 4 5 2 1 0 0 0 0 5 5 5 0 5 5 8 7
L̃(Δ)  1 1 3

Notes on Table 4.4: Ḡ_{p} ≅ Sp_{4}(2)′ hence 1_{pQ}^{p}, 1_{pl}^{p}, 1_{plQ}^{p} are obtained by lifting to G_{p} the A_{6}-characters 1a + 5a + 9a, 1a + 5b + 9a and 1a + 5ab + 9aa + 8ab. U_{4}(3) has an outer automorphism which interchanges points and quads; this outer automorphism transforms 1_{pl}^{p} into 1_{Ql}^{Q}. L̃(Δ) = 640a + 640b + 896aaa and is a sum of ordinary characters which belong to 2-modular blocks of defect 0.

Table 4.5: The G̃_{2}-geometry for G_{2}(3).
F = Flag  G_{F}  1_{F}^{G}
G_{2}(3)  1
p  G_{2}(2)  1  1 1
l  2^{1+4} : (3 × 3) : 2 1  1 1 2 2 2 2  1 1 1 1  2 2 1
π  2^{3}.L_{3}(2)  1  1  2 1 1  1 1 1
pl  5.S_{4}.2  1  1 2 3 3 4 2 1 3 3 3 4 3 1 5 5 3
pπ, lπ  2^{3}.S_{4}  1  1 1 1 2 4 3 3  2 2 3 3 2 2 5 5 5
plπ  2^{3}.D_{8}  1 0 1 1 1 0 3 3 5 6 6 6 1 1 7 7 a a 8 8 e e d
L̃(Δ)  1 1  1  1 1 1 1 2 1 1 3 2 2 2
L̃(Δ_{P})  1 1 2 2 2 2 2 2 3

Notes on Table 4.5: Only 1_{P}^{G} is given in the [Atlas]. 1_{pl}^{G}, 1_{pπ}^{G}, 1_{plπ}^{G} are obtained by inducing from the G_{2}(2)-characters 1a + 14a + 21a + 27a,1a + 7bc + 21a + 27a, 1a + 7bc + 144 + 21aa + 27aa + 32ab. Now 1_{pπ}^{π} = 1_{lπ}^{π}, so 1_{lπ}^{G} = 1_{pπ}^{G}. We use the permutation characters 1_{pl}^{G}, 1_{pπ}^{G}, 1_{lπ}^{G} to calculate the fusions of conjugacy classes of G_{pπ}, G_{pl} and G_{lπ} into classes of G. This fusion information makes calculation 1_{l}^{G} and 1_{π}^{G} fairly easy. L̃(Δ) = 64ab + 91b + 182b + 273b + 448ab + 546aab + 728abbb + 729aa + 819aa + 832aa. L̃(Δ) is a sum of characters in blocks of defect 0 together with the projective covers 91b + 182b + 273b + 546a + 728bb + 729a + 819a and 546ab + 728ab + 729a + 819a of irreducible 2-modular characters of degrees 90 and 378 (see Ryba [Ry] ). And L̃(Δ_{P}) = 448ab + 546aabb + 728aabb + 729aa + 819aa + 832aaa is the sum of characters of degree 448 and 832 with defect 0, and two copies of a 4096-dimensional projective cover of a single irreducible, as yet undetermined.

Table 4.6: The characteristic-2 geometry for McL.

F = Flag  G_{F}  1_{F}^{G}
McL  1
l  U_{4}(3)  1 1 1
p, π  2^{4} : A_{7}  1 1 2  2 1  1  1
s  2.A_{8}  1  1  1  1 1  1
lp, lπ  2^{4} : A_{6}  1 2 1 4  5 4 1 3  2 2 4 1 1 1 1
sl  2^{1+4} : (S_{3} × S_{3}) 1 1 4  1 1 7 6 6 2 9 3 5 5 7 7 f 7 7 8 8
pπ  2^{4} : (3 × A_{4}) : 2 1 2 1 6  b a 5 b 2 4 4 8 8 g 7 7 7 7
ps, πs  2^{4} : L_{3}(2) 1 1 3  5 4 2 5 1 2 2 3 3 7 3 3 3 3
spl, sπl, pπs  2^{4} : S_{4}  1 2 1 7  2 2 g i 2 f 6 n 8 g g m m C p p o o
plπ  2^{4} : S_{4}  1 3 3 9 1 1 1 1 j l 4 d 6 n a g g o o A n n o o
spπl  2^{4} : D_{8}  1 3 3 c 1 1 7 7 y J a H o X s Q Q ∗ ∗ ∗ ∗ ∗ ∗ ∗
L̃(Δ)  1 1  2 2  1 1 1 1 2 1 1 2 2
L̃(Δ_{L})  2 2 2 2 6 6 5 5 8 8 8 8 b c c b b

Notes on Table 4.6: 1_{s}^{G} is given in the [Atlas] and from this we find the fusion of conjugacy classes of U_{4}(3) into McL. We use the fusion pattern to induce up from the characters of Table 4.3 – this gives the permutation characters for all flags containing s. The characters 1_{sp}^{G},1_{sπ}^{G}, and 1_{sl}^{G} lead us to fusion patterns for the other vertex stabilizers. Finally we lift and induce the A_{7} characters of Table 4.1 to obtain the permutation characters for all other flags. L̃(Δ) = 896ab + 4500aa + 4752aa + 8019ab + 8250ab + 9265aa + 9856ab + 10395aabb. This is a sum of characters of defect zero together with the projective covers 4500a + 4752a + 8019a + 8250a + 9625a + 10395ab and 4500a + 4752a + 8019b + 8250b + 9625a + 10395ab of a dual pair of irreducible 2-modular characters of degree 2124 (see Thackray [Th] ). Also L̃(Δ_{L}) = 2.896ab+2.3520ab+6.4500a+5.5103a+5.5544a+8.8019ab+8.8250ab+11.9625a+ 12.9856ab + 11.10395ab; those of degree 896 and 9856 are of defect 0, and the others yield (again by reference to Thackray) 3 copies of the projective covers of the 2124-modules plus 5 times the cover of an irreducible of degree 3584 and twice the cover of one of degree 3520.

Table 4.7: The 2-local geometry for J_{3}.

F = F lag  G_{F}  1_{F}^{G}
J_{3}  1
l  2^{4} : (3 × A_{5}) 1  1 1 1  2 2 1 1 1 1 1 1  1
p  2^{2+4} : (3 × S_{3}) : 2 1  1 1 2  2 3 2 2 2 2 2 2 1 1 1 3 1
v  2^{1+4} : A_{5}  1  1 1 1  2 2 2 2 1 1 1 1 1 1 1
lp  2^{4} : (3 × A_{4}) 1  2 2 3 1 1 4 6 3 3 3 4 4 4 2 2 2 6 3
pv  2^{1+4} : A_{4}  1  2 2 3 1 1 2 5 5 5 6 5 5 5 5 5 5 7 6
lv, lpv  2^{4} : (3 × 2^{2}) 1 0 0 4 4 5 3 3 4 a 9 9 b a a a a a a e d
L̃(Δ)  1 1 2 2 2 2 4 3 3 4 5 5 5 5 5 5 8 8

Notes on Table 4.7: We first find the fusion of the conjugacy classes of G_{l} into J_{3}. From this we calculate 1_{lp}^{l}, 1_{lv}^{l} = 1_{lpv}^{ℓ} by lifting the A_{5}-characters 1a + 4a, 1a + 4a + 5aa. As in earlier cases, these characters can be used to obtain the permutation characters of the remaining vertex stabilizers.

Table 4.8: The pairs-geometry for M_{11}.

F = F lag G_{F} 1_{F}^{G}
M_{11} 1
p  M_{10} 1 1
l M_{9} : 2 1 1 1
pl  M_{9} 1 2 0 0 0 0 0 1 1 0
L̃(Δ)  1

Table 4.9: The characteristic-3 geometry for L_{3}(4).

F = F lag G_{F} 1_{F}^{G}
L_{3}(4) 1
p  A_{6} 1 1 1
l  3^{2}Q_{8} 1 1 1 1 1 1 1 0 0 1
pl  3^{2}Z_{4} 1 2 3 1 1 1 1 1 1 2
L̃(Δ)  1  1 1 1

Notes on Table 4.9: The two characters of degree 63 are of defect 0, and the remaining part 35a + 64a is the projective cover of the modular irreducible of degree 15 (the modular character table is widely distributed, for example in Parker’s ”modular Atlas” tables).

Non-Table 4.10: The Sylow-3 overgroup geometry for U_{5}(2).

In this case, it was easier to compute the decomposition of Table 1 directly, using the “modular Atlas”: for the character involves just the two defect-0 modules of degree 1215, and the projective cover of the modular irreducible of degree 891–in a block of defect 1. Thus in a table in the format of this section, most constituents would lie in the principal block, and simply cancel out.

Table 4.11: The characteristic-3 geometry for McL.

F = F lag  G_{F}  1_{F}^{G}
McL 1
Q  U_{4}(3) 1 1 1
p  3^{1+4} : 2.S_{5} 1  1  1 1 1
l  3^{4} : M_{10} 1 1 2  1 1 1 1
Ql  3^{4} : A_{6} 1 2 1 3  2 2 1 1  1
Qp  3^{1+4} : 2S_{4} 1 1 2  1 1 2 2 1 1 1  1  1 1
pl  3^{4} : M_{9} 1 1 3  2 2 3 3 1 2 2 1 1 2 1 1 1 1
Qpl  3^{4} : 3^{2}.4 1 2 1 4 0 0 0 0 4 4 0 4 0 5 2 3 3 2 2 5 2 2 3 3
L̃(Δ) 1 1 1 1 1 1 1 1 1
L̃(Δ_{Q})  1 1 1 1 2 2 1 1 2 1 1 1 1

Notes on Table 4.11: In the table we calculate the Lefschetz characters for both characteristic-3 geometries for McL. The fusion of conjugacy classes of U_{4}(3) into McL is known from 5.6. We use this fusion pattern to induce up from the U_{4}(3)-characters 1_{Ql}^{Q} = 1a + 21a + 90a and 1_{Qp}^{Q} = 1a + 90a + 189a. As in earlier cases it is now easy to obtain the fusion patterns for the other vertex stabilizers.

Table 4.12: The G̃_{2}-geometry for Ly.

F = F lag G_{F}  1_{F}^{G}
Ly  1
p  G_{2}(5) 1 1  1 1 1
l  5^{1+4} : 4.S_{6} 1 2  3 4 3 3 1 1 1 3 2 4 4 2 2 2 1 1 1 6 6 5 5 4 5 2 8 4 3 5 5 5 5 5 5 5 1 7 7 4 5 5 5 7
π  5^{3}.L_{3}(5) 1 1  1 2 1 2  1 1 1 1 1 1 1 1 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 1 1 1  1 1 1
pl  5^{1+4}.4.S_{5} 1 4  1 6 a 1 8 9 5 6 6 f a h h d d d b b b o o m m j q j x q l t t t t t u u k z z x z x B K
pπ, lπ 5^{3}.5^{2}.4.S_{5} 1 0 0 3 0 0 0 0 1 0 6 7 0 8 6 2 8 7 c f f f f d d f f f o o n n m l g u k p t t t t t P r q x x u w B D N

Notes on Table 4.12: The permutation character 1_{plπ}^{G} and Lefschetz characters are not included in the table; they can be deduced from the following list of inner products of L̃(Δ) and L̃(Δ_{P}) with the irreducible characters of Ly:

L̃(Δ) = (0 0 0 0 0 0 1 1 1 1 3 6 5 9 9 9 25 24 35 43 38 38 42 44 44 50 50 50 58 58 60 60 62 60 63 71 74 84 92 92 92 92 92 94 94 101 97 97 111 113 121 139 155).

L̃(Δ_{P}) = (0 0 0 1 0 0 0 0 0 1 2 5 1 4 6 4 5 12 8 13 13 11 11 11 10 10 10 18 18 12 12 15 21 17 25 22 18 14 14 14 14 14 15 15 19 28 28 29 30 28 32 39).

The geometry for Ly has the same diagram as the geometry for G_{2}(3), and this is reflected in the following arguments which are virtually the same as those used for Table 4.4. 1_{p}^{G} is given in the [Atlas] and 1_{pl}^{G}, 1_{pπ}^{G} = 1_{lπ}^{G}, 1_{plπ}^{G} are easily obtained by inducing from the G_{2}(5)- characters 1a + 930a + 1085a + 1890a, 1a + 930a + 1085b + 1890a, 1a + 930aa + 1085ab + 1890aa + 15625a. (The first two of these are given in the [Atlas] and to obtain the third we note that 15625a is the Steinberg character of G_{2}(5) and thus 15625a = 1_{plπ}^{p} −1_{pl}^{p} −1_{pπ}^{p} +1_{p}^{p}). Finally as in Table 4.4 we use 1_{pl}^{G}, 1_{pπ}^{G}, 1_{lπ}^{G} to calculate the permutation characters for the other vertex stabilizers.

Table 4.13: The graph for O′N in characteristic 7.

F = F lag  G_{F}  1_{F}^{G}
O′N  1
p  L_{3}(7) : 2 1 1  1 1  1
l  L_{3}(7) : 2 1 1  1 1 1
pl  7^{1+2}(3 × D_{8}) 1 6 1 1 2 2 a 7 7 4 c 4 8 8 7 8 8 8 f i m m m m p p p p o o
L̃(Δ)  4 1 1 2 2 8 6 6 4 a 4 8 8 7 8 8 8 f i m m m m p p p p o o

Notes on Table 4.13: The defect-0 part of the Lefschetz character is 7.58653a+8.85064a+ 15.116963a + 18.143374a + 22.175616ab.

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