Projective Modules for the Group of Twenty-Seven Lines on a Cubic Surface

Dave Benson^{∗} Mathematical Institute 24–29 St. Giles Oxford OX1 3LB England

Contents. 1. Introduction. 2. Notation and preliminary lemmas. 3. Representations of the algebraic group B_{2}(k̄). 4. Modules induced from the Borel subgroup. 5. Calculations using the Jantzen sum formula. 6. Tensor products of Weyl modules. 7. Characters and maximal subgroups of G. 8. Characters and modules for the maximal subgroup 2^{4} :S_{5}. 9. Some permutation modules for G. 10. Tensor products of modules for G and Ĝ. 11. Some Ext^{1} groups for G and Ĝ. 12. More tensor products of Weyl modules. 13. Some Ext^{1} groups for the algebraic group G. 14. The structure of 4^{+}_{H} ↑^{G}. 15. The remaining Ext^{1} groups for G and Ĝ. 16. Some projective kG-modules. 17. Some more induced modules. 18. The remaining projective kG-modules.

Appendix: Characters and decomposition numbers. 3-modular characters of G′, G, Ĝ′ and Ĝ. Decomposition matrix and Cartan matrix for G′ and Ĝ′ mod 3. Decomposition matrix for G and Ĝ mod 3. Cartan matrix for G and Ĝ mod 3. Ordinary characters of H′, H, Ĥ′ and Ĥ. 3-modular characters of H′, H, Ĥ′ and Ĥ. Diagram of highest weights in P_{0}^{+}(Φ).

1 Introduction

Let G be the group of automorphisms of the configuration of 27 lines on a nonsingular cubic surface. Then G is a complete group of order 51,840, whose derived group G′ is a simple group of order 25, 920 = 2^{6}.3^{4}.5. We have |G : G′| = 2, Z(G) = 1, and the Schur multiplier H^{2}(G′, C^{×}) ≅ Z/2Z. We denote by Ĝ′ the universal covering group of G′, of order 51,840.

Other manifestations of this group are the following.

(i) G′ ≅ SU_{4}(2), the group of all 4 × 4 unitary matrices with entries in GF (4), of determinant one. G is obtained by adjoining the graph (twisting) automorphism.

∗Research partially supported by the NSF

(ii) G ≅ SO_{6}^{−}(2), the group of all 6×6 matrices with entries in GF (2), preserving a non-singular quadratic form of Witt index one. All such matrices have determinant one.

(iii) Ĝ′ ≅ Sp_{4}(3), the group of all 4 × 4 symplectic matrices with entries in GF (3). Adjoining a diagonal automorphism, we obtain a group Ĝ ≅ CSp_{4}(3) of order 103,680 consisting of all 4 × 4 symplectic similitudes with entries in GF (3). We have Ĝ/Z(Ĝ) ≅ G.

(iv) G ≅ SO_{5}(3), the group of all 5 × 5 orthogonal matrices with entries in GF (3) and determinant one.

(v) G ≅ W (E_{6}), the Weyl group of a root system of type E_{6}.

For further information on this group, see for example Burnside [8] (note H), Henderson [15], C. Jordan [21], Yu. Manin [23], and the Atlas [12].

The purpose of this paper is to determine the Loewy structure of the projective indecomposable kG-modules, where k is a splitting field of characteristic three. The ordinary and 3-modular character tables, decomposition and Cartan matrices are already known from Frame [14] and Thackray [26]. For convenience, the decomposition matrix, Cartan matrix and 3-modular character table are given in the Appendix. The ordinary character table may be found in the Atlas [12].

Our main result is Theorem 1.1, and our methods come from the theory of modular representations of finite groups and algebraic groups.

Throughout, we shall let R be a complete rank one discrete valuation ring, with the property that the field of fractions K of R, and the residue class field k of R are splitting fields for Ĝ and all its subgroups, of characteristics zero and three respectively.

There are nine simple kĜ′-modules denoted 1, 5, 10, 14, 25, 81, 4, 16, 40 according to their dimensions. These modules fall into three blocks, namely B_{0} = {1, 5, 20, 14, 25}, B_{1} = {81} and B_{2} = {4, 16, 64}. The blocks B_{0} and B_{1} have Z(Ĝ′) in their kernels while B_{2} does not. Thus B_{0} and B_{1} are the blocks of kG′. Each simple kĜ′-module has two extensions to a kĜ-module, and we have chosen to label one with a plus sign and the other with a minus sign. In the Appendix, the character of the extension with the plus sign is given. Thus the simple kGmodules are 1^{+}, 1^{−}, 5^{+}, 5^{−}, 10^{+}, 10^{−}, 14^{+}, 14^{−}, 25^{+}, 25^{−}, 81^{+} and 81^{−}. These fall into three blocks, B_{0}^{±} = {1^{+}, 1^{−}, 5^{+}, 5^{−}, 10^{+}, 10^{−}, 14^{+}, 14^{−}, 25^{+}, 25^{−}}, B_{1}^{+} = {81^{+}} and B_{1}^{−} = {81^{−}}. There is one more block for kĜ, namely B_{2}^{±} = {4^{+}, 4^{−}, 16^{+}, 16^{−}, 40^{+}, 40^{−}}. All the irreducible kG-modules are self dual, while duality changes the signs on the modules in B_{2}^{±}.

The blocks B_{1}^{+} and B_{1}^{−} have defect zero, and are therefore easy to describe. Thus we shall concentrate on B_{0}^{±}. Denote by e_{0} the central idempotent corresponding to this block.

Theorem 1.1 The Loewy structures of the projective indecomposable modules in B_{0}^{±} are the following, together with the diagrams obtained by interchanging the plusses and minuses (i.e., by tensoring with 1^{−}).

\begin{matrix}1^{+} \\ 5^{−} & 14^{−} \\ 1^{+} & 1^{+} & 5^{+} & 10^{+} & 14^{+} \\ 1^{−} & 5^{−} & 5^{−} & 10^{−} & 14^{−} & 25^{−} \\ 1^{+} & 1^{+} & 5^{+} & 10^{+} & 14^{+} \\ 5^{−} & 14^{−} \\ 1^{+} \end{matrix} \begin{aligned}& & & & & 5^{+} \\ 1^{−} & 5^{−} & 10^{−} & 14^{−} & 25^{−} \\ 1^{+} & 5^{+} & 5^{+} & 5^{+} & 5^{+} & 10^{+} & 10^{+} & 14^{+} & 25^{+} \\ 1^{−} & 1^{−} & 5^{−} & 5^{−} & 5^{−} & 10^{−} & 10^{−} & 10^{−} & 14^{−} & 14^{−} & 25^{−} & 25^{−} \\ 1^{+} & 5^{+} & 5^{+} & 5^{+} & 5^{+} & 10^{+} & 10^{+} & 14^{+} & 25^{+} \\ 1^{−} & 5^{−} & 10^{−} & 14^{−} & 25^{−} \\ & & & & & 5^{+} \end{aligned}

\begin{matrix}10 & + \\ 5^{−} & 10^{−} & 14^{−} & 25^{−} \\ 1^{+} & 5^{+} & 5^{+} & 10^{+} & 10^{+} & 10^{+} & 10^{+} & 14^{+} & 25^{+} \\ 1^{−} & 5^{−} & 5^{−} & 5^{−} & 10^{−} & 10^{−} & 10^{−} & 10^{−} & 14^{−} & 14^{−} & 25^{−} & 25^{−} \\ 1^{+} & 5^{+} & 5^{+} & 10^{+} & 10^{+} & 10^{+} & 10^{+} & 14^{+} & 25^{+} & 25^{+} \\ 5^{−} & 10^{−} & 10^{−} & 10^{−} & 14^{−} & 25^{−} \\ 10^{+} & 10^{+} & 25^{+} \\ 10^{−} & 25^{−} \\ 10 & + \end{matrix} \begin{matrix}14 & + \\ 1^{−} & 5^{−} & 10^{−} \\ 1^{+} & 5^{+} & 10^{+} & 14^{+} & 14^{+} & 25^{+} \\ 1^{−} & 5^{−} & 5^{−} & 10^{−} & 10^{−} & 14^{−} \\ 1^{+} & 5^{+} & 10^{+} & 14^{+} & 14^{+} & 25^{+} \\ 1^{−} & 5^{−} & 10^{−} \\ 14 & + \end{matrix}

25^{+}
5^{−} 10^{−}
5^{+} 10^{+} 14^{+} 25^{+}
1^{−} 5^{−} 5^{−} 10^{−} 10^{−} 25^{−}
5^{+} 10^{+} 10^{+} 14^{+} 25^{+} 25^{+}
5^{−} 10^{−} 25^{−}
10^{+}
10^{−}
25^{+}

See the end of Section 2 for an outline of the strategy we shall be using to prove this theorem.

It is worth noting that it follows from Theorem 1.1 that if U and V are simple kG-modules, both of plus type or both of minus type, then dim_{k} Ext_{kG}^{1}(U, V) = 0. Is there a generalisation of this fact, either to a more general finite group context or to a more general Chevalley group context? Note that we also prove later in this paper that this holds for simple kĜ-modules.

2 Notation and preliminary lemmas

If A is a group algebra over k and V is a finitely generated A-module, we write JA for the Jacobson radical of A, and

L_{i}(V) = V.(JA)^{i−1}/V.(JA)^{i}

is the i^{th} Loewy layer of V. The Loewy length of V is the smallest integer m such that V.(JA)^{m} = 0, and the Loewy structure for V is a diagram whose i^{th} layer downwards gives the simple summands of L_{i}(V) with multiplicities (see for example Theorem 1.1). The head of a module is the first Loewy layer.

Let Soc(V) denote the socle of V, namely the sum of all the simple submodules of V. Let S_{1}(V) = Soc(V) and S_{i}(V)/S_{i−1}(V) = Soc(V/S_{i−1}(V)). Then

0 < S_{1}(V) < S_{2}(V) < ⋯ < S_{m−1}(V) < S_{m}(V) = V

is called the socle series of V.

We shall write (U, V)_{A} for dim_{k} Hom_{A}(U, V), V^{∗} for Hom_{k}(V, k) regarded as an A-module, and P_{V} for the projective cover of V. Homomorphisms will usually be written on the right. The exterior n^{th} power of V will be written as Λ^{n}(V), and the symmetric n^{th} power as S^{n}(V).

The lemmas from modular representation theory which we shall need are the following.

Lemma 2.1 (Frobenius reciprocity) Let G_{1} ≤ G_{2}, U a kG_{1}-module and V a kG_{2}-module. Then

(U, V ↓_{G_{1}})_{kG_{1}} = (U ↑^{G_{2}}, V)_{kG_{2}}

and

(V ↓_{G_{1}}, U)_{kG_{1}} = (V, U ↑^{G_{2}})_{kG_{2}}.

Lemma 2.2 (Landrock [22]) Let U and V be simple A-modules. Then the multiplicity of U in L_{i}(P_{V}) is the same as the multiplicity of V^{∗} in L_{i}(P_{U^{∗}}).

Lemma 2.3 (Mackey decomposition) Let G_{1}, G_{2} ≤ G_{3} and V a kG_{1}- module. Then

V ↑^{G_{3}}↓_{G_{2}} ≅ ⨁_{G_{1}xG_{2}} V^{x} ↓_{G_{1}^{x}∩G_{2}}↑^{G_{2}}

where x runs over a set of double coset representatives for G_{1} and G_{2} in G_{3}.

Lemma 2.4 (Scott [25] or [5, Section 2.6]) Any homomorphism between permutation modules over k can be lifted to a homomorphism between the corresponding permutation modules over R. In particular, idempotent endomorphisms lift to idempotent endomorphisms, and hence direct sum decompositions of permutation modules lift, as do homomorphisms between such direct summands.

Lemma 2.5 Let U and V be kG_{1}-modules, and G_{2} ⊴ G_{1} a normal subgroup of index coprime to the characteristic of k. Then the restriction map

res_{G_{1},G_{2}} : Ext_{kG_{1}}^{1}(U, V) → Ext_{kG_{2}}^{1}(U, V)

is a monomorphism whose image is the set of fixed points of G_{1}/G_{2} in its natural action on Ext_{kG_{2}}^{1}(U, V).

Proof : This follows from the collapsing spectral sequence

H^{p}(G_{1}/G_{2}, Ext_{G_{2}}^{q}(U, V)) ⇒ Ext_{G_{1}}^{p+q}(U, V). □

Remark : When we draw diagrams for modules, the meanings of these diagrams are as explained in Benson and Conway [6]. In fact we shall only draw diagrams for modules in case the lattice of submodules is distributive, in which case these diagrams agree with those of Alperin [1].

Strategy : Our strategy will be to play off information from the algebraic group of type B_{2} in characteristic 3 against information from the “non-algebraic” subgroup 2^{4} : S_{5} of small index in the finite group. Thus we first investigate modules for the algebraic group and then concentrate our attention on the finite group.

Our goal when dealing with the algebraic group is to obtain information about the structure of certain modules which when restricted to the finite group are projective. These are obtained by tensoring Weyl modules with the Steinberg module. A theorem of Wang Jian-Pan gives us a filtration of such a tensor product by Weyl modules, and this way we obtain quite strong information about these modules.

The information coming from the subgroup 2^{4} : S_{5} of index 27 is as follows. This subgroup has cyclic Sylow 3-subgroups, and so its projective modules are easy to write down. Inducing these projectives to the group G gives us filtrations of projective G-modules which we may use to obtain information about Loewy structure. The lemmas in Section 8 are relevant here.

In some sense the pieces of information coming from these two different sources complement each other quite well, and we end up switching our concentration between them several times in order to use them to the best effect.

3 Representations of the algebraic group B_{2}(k̄)

Let G denote the simply connected algebraic group of type B_{2} over k̄, the algebraic closure of k. Let F denote the Frobenius morphism G → G, so that G^{F} = {g ϵ G|g = gF } = Ĝ′. We fix a Borel subgroup B and a maximal torus T ≤ B. Let Φ be the associated root system in E = χ(T ) ⊗_{Z} R, where

[figure: α_{2}; ω_{2}; ρ; ω_{1}; α_{1}]

Figure 1: The B_{2} root system

χ(T ) = Hom(T , k̄^{×}) is the character group of T . Denote by Δ the set of simple roots with respect to which the roots of B are the positive roots Φ^{+}, and denote by B^{−} the opposite Borel subgroup corresponding to the set of negative roots Φ^{−}. Let Δ = {α_{1}, α_{2}}, where α_{1} is short and α_{2} is long. Then Φ^{+} = {α_{1}, α_{2}, α_{1} + α_{2}, 2α_{1} + α_{2}}. Let α^{∨} be the dual root 2α/⟨α, α⟩ ϵ E for a root α, and s_{α} the reflection corresponding to α. Let W = ⟨s_{α}|α ϵ Φ⟩ be the Weyl group of Φ, Q(Φ) = ⟨α ϵ Φ⟩ ⊆ E be the root lattice and P (Φ) = χ(T ) ⊆ E be the weight lattice. Let ω_{1} = α_{1} + \frac{1}{2}α_{2} and ω_{2} = α_{1} + α_{2} be the fundamental weights and let ρ = ω_{1} + ω_{2}. If λ, μ ϵ P (Φ), we write λ ⪰ μ if and only if λ − μ is a sum of positive roots.

If V is a finite dimensional T -module and λ ϵ P (Φ), we write

V^{λ} = {x ϵ V |xt = λ(t)x ∀t ϵ T },

the weight space of V with weight λ. Then V = ⨁_{λ} V^{λ}, and we write ch(V) for the formal character of V, namely the P (Φ)-tuple (dim(V^{λ}))_{λϵP (Φ)} as an element of the group algebra Z[P (Φ)]. We denote by P^{+}(Φ) the set of dominant weights with respect to Δ, namely the weights ω such that ω = aω_{1} + bω_{2} with a, b ϵ N, and by P_{0}^{+}(Φ) the set of restricted dominant weights, namely the weights ω such that ω = aω_{1} + bω_{2} with a, b ϵ {0, 1, 2} (i.e., 0 ≤ a, b ≤ p − 1 with p = 3).

4 Modules induced from the Borel subgroup

The basic definitions and properties of rational induction of modules for algebraic groups are given in Cline, Parshall and Scott [9, 10].

μ →
λ↓ 00 10 01 20 11 30 02 21 40 12 31 50 03 22 41 60 13 32 51 70 04 23 42 dim(W (λ))
00 1  1
10 . 1  4
01 1 . 1  5
20 2 . 1 1  10
11 . 2 . . 1  16
30 . 2 . . 1 1  20
02 2 . 1 1 . . 1  14
21 3 . 3 2 . . 1 1  35
40 3 . 2 2 . . 1 1 1  35
12 . 3 . . 2 1 … 1  40
31 . 4 . . 3 2 … 1 1  64
50 . 3 . . 2 2 … 1 1 1  56
03 2 . 2 1 . . 1 1 …. 1  30
22 5 . 4 4 . . 3 2 1 … 1 1  81
41 5 . 5 4 . . 3 3 2 … 1 1 1  105
60 4 . 3 3 . . 2 2 2 … 1 1 1 1  84
13 . 4 . . 3 2 … 2 1 . . . . . 1  80
32 . 6 . . 5 4 … 3 2 1 …. 1 1  140
51 . 6 . . 5 4 … 3 3 2 …. 1 1 1  160
70 . 4 . . 3 3 … 2 2 2 …. 1 1 1 1  120
04 3 . 2 2 . . 2 1 1 … 1 1 . . . . . . 1  55
23 6 . 6 5 . . 4 4 2 … 3 2 1 . . . . . 1 1  154
42 8 . 7 7 . . 6 5 4 … 3 3 2 1 …. 1 1 1  220

Table 1: dim(W (λ))^{μ} (dots signify μ ⪯̸ λ)

For λ ϵ P (Φ) we let λ̃ be the one dimensional B-module whose restriction to T is λ, and the induced module is

H^{0}(G/B, −λ̃) ≅ \{ \begin{matrix}W (λ)^{∗} & if λ ϵ P^{+}(Φ) \\ 0 & otherwise \end{matrix}

where W (λ)^{∗} is the dual of the Weyl module W (λ) with highest weight λ. The formal character ch(W (λ)) is given by Freudenthal’s recursive formula (see for example Humphreys [16, p. 122]), and extensive tables may be found in Bremner, Moody and Patera [7]. Table 1 gives the values for λ small.

Note : In the row and column headings in the tables, ab signifies λ = aω_{1} + bω_{2}.

μ →
λ↓ 00 10 01 20 11 30 02 21 40 12 31 50 03 22 41 60 13 32 51 70 04 23 42 L(λ)
00 1  1
10 . 1  4
01 1 . 1  5
20 2 . 1 1  10
11 . 2 . . 1  16
30 . 0 . . 0 1  4^{F}
02 2 . 1 1 . . 1  14
21 1 . 2 1 . . 1 1  25
40 0 . 0 1 . . 0 1 1  4 ⊗ 4^{F}
12 . 3 . . 2 1 … 1  40
31 . 0 . . 1 1 … 0 1  5 ⊗ 4^{F}
50 . 1 . . 1 2 … 1 1 1  10 ⊗ 4^{F}
03 1 . 0 0 . . 0 0 …. 1  5^{F}
22 5 . 4 4 . . 3 2 1 … 1 1  81
41 0 . 2 2 . . 2 2 2 … 0 1 1  16 ⊗ 4^{F}
60 2 . 0 0 . . 0 0 0 … 1 0 0 1  10^{F}
13 . 1 . . 0 0 … 1 0 . . . . . 1  4 ⊗ 5^{F}
32 . 1 . . 2 2 … 1 1 1 …. 0 1  14 ⊗ 4^{F}
51 . 3 . . 3 1 … 2 2 1 …. 1 1 1  25 ⊗ 4^{F}
70 . 2 . . 0 0 … 1 0 1 …. 1 0 1 1  4 ⊗ 10^{F}
04 1 . 1 0 . . 1 0 0 … 1 1 . . . . . . 1  5 ⊗ 5^{F}
23 2 . 1 1 . . 1 1 0 … 2 1 0 . . . . . 1 1 10 ⊗ 5^{F}
42 4 . 4 5 . . 4 4 3 … 2 2 2 1 …. 0 1 1 40 ⊗ 4^{F}

Table 2: dim(L(λ))^{μ} (dots signify μ ⪯̸ λ)

The module W (λ)^{∗} has a unique minimal submodule L(λ)^{∗}, where L(λ) is the simple module with highest weight λ. The formal character ch(L(λ)) may be calculated using the Steinberg tensor product theorem in terms of ch(L(λ)) for λ ϵ P_{0}^{+}(Φ), and these L(λ) restrict to give the simple modules for the finite group G^{F} = Ĝ′. The simple modules for Ĝ′ are known from Thackray’s thesis [26], and the character table for Ĝ′ appears in the Appendix. It turns out that there is only one possibility for which L(λ) corresponds to which of the simple modules in Thackray’s list, and this information is displayed in the diagram of highest weights in the Appendix. The formal characters of these modules can be calculated from the formal characters of the W (λ), and so using Steinberg’s tensor product theorem we can calculate the formal characters of modules L(λ) with λ outside the restricted fundamental dominant region. This way we obtain Table 2.

μ →
λ↓ 00 10 01 20 11 30 02 21 40 12 31 50 03 22 41 60 13 32 51 70 04 23 42 61
00 1
10 . 1
01 . . 1
20 0 . . 1
11 …. 1
30 …. 1 1
02 . . 0 … 1
21 0 . . 1 … 1
40 . . 1 … 1 . 1
12 . 0 . . . . . . . 1
31 . 1 . . . . . . . 1 1
50 …. 1 0 . . . . . 1
03 0 . . 0 … 1 …. 1
22 . . . . . . . . . . . . . 1
41 1 . . 1 … 1 …. 1 . 1
60 0 . . 1 … 0 …. 0 . 1 1
13 . 0 . . . . . . . 1 1 . . . . . 1
32 . 1 . . . . . . . 1 1 . . . . . 1 1
51 …. 1 1 . . . . . 1 . . . . . . 1
70 . 1 . . . . . . . 0 1 . . . . . 0 1 . 1
04 . . 0 … 1 . 1 . . . . . . . . . . . 1
23 0 . . 1 … 1 …. 1 . 1 . . . . . . 1
42 . . 1 … 1 . 1 . . . . . . . . . . . 1 . 1
61 . . 1 … 0 . 1 . . . . . . . . . . . 0 . 1 1

Table 3: Multiplicity of L(μ) as composition factor in W (λ) (dots signify μ ⪯̸ λ or μ is not strongly linked to λ)

It is now an easy matter to calculate the multiplicy of L(μ) as a composition factor of W (λ) by comparing formal characters. The result can only be non-zero of μ ⪯ λ and λ is strongly linked to μ in the sense of Andersen [2, Defn. 1]. These multiplicities are given for λ and μ small in Table 3.

5 Calculations using the Jantzen sum formula

Let us first establish some notation. Recall that to each root α ϵ Φ we have a reflection s_{α} ϵ W.

i →
λ↓ 1 2 3 4 5
00 1
10 4
01 5
20 10
11 16
30 4^{F}  16
02 14
21 25  10
40 4 ⊗ 4^{F} 5 ⊕ 14
12 40
31 5 ⊗ 4^{F} 4 ⊕ 40
50 10 ⊗ 4^{F} 16
03 5^{F}  25
22 81
41 16 ⊗ 4^{F} 1 ⊕ 10 ⊕ 5^{F}  25
60 10^{F}  16 ⊗ 4^{F}  10
13 4 ⊗ 5^{F} 5 ⊗ 4^{F}  0  40
32 14 ⊗ 4^{F} 4 ⊗ 5^{F}  4  5 ⊗ 4^{F} 40
51 25 ⊗ 4^{F} 4^{F}  10 ⊗ 4^{F}  16
70 4 ⊗ 10^{F} 5 ⊗ 4^{F} ⊕ 14 ⊗ 4^{F}  4
04 5 ⊗ 5^{F} 4 ⊗ 4^{F}  0  14
23 10 ⊗ 5^{F} 16 ⊗ 4^{F}  5^{F}  10  25
42 40 ⊗ 4^{F} 5 ⊗ 5^{F}  5  4 ⊗ 4^{F} 14
61 5 ⊗ 10^{F} 4 ⊗ 4^{F} ⊕ 40 ⊗ 4^{F}  5
80 10 ⊗ 10^{F} 16 ⊗ 4^{F}  1
05 14 ⊗ 5^{F} 4 ⊗ 4^{F}  0  5
24 25 ⊗ 5^{F} 1 ⊕ 10 ⊗ 5^{F} ⊕ 10^{F}  16 ⊗ 4^{F}  0  10
43 4 ⊗ 16^{F} 5 ⊗ 10^{F} ⊕ 5 ⊗ 5^{F} ⊕ 14 ⊗ 5^{F} 4 ⊗ 4^{F} ⊕ 40 ⊗ 4^{F} 5

Table 4: Layers in the Jantzen filtration for the Weyl modules

Denote by s_{α,m}, m ϵ Z, the affine reflection λ → s_{α}λ + mα = λ − (⟨λ, α^{∨}⟩ − m)α on E. The affine Weyl group W_{aff} is the group generated by the reflections s_{α,mp} (in our situation p = 3) with m ϵ Z, α ϵ Φ. We define the “dot action” of the affine Weyl group on E via w.λ = w(λ + ρ) − ρ, for w ϵ W_{aff} and λ ϵ E. For μ ϵ P (Φ) define

ch(μ) = (−1)^{ℓ(w)}ch(W (w.μ))

where w is the element of the Weyl group such that w.μ ϵ P^{+}(Φ) if such exists, and ℓ(w) is the length of w (i.e., the shortest length of an expression of w as a product of fundamental reflections). If no such w exists, set ch(μ) = 0.

It is shown is Jantzen [18, 19] (see also [20] for a summary of these results) that W (λ) has a filtration

W (λ) = W (λ)_{0} ≥ W (λ)_{1} ≥ ⋯ ≥ W (λ)_{r} = 0

with W (λ)_{1} = Rad W (λ), W (λ)_{i−1}/W (λ)_{i} a self dual module, and

∑_{i≥1} ch(W (λ)_{i}) = ∑_{αϵΦ^{+}} ∑_{0<mp<⟨λ+ρ,α^{∨}⟩} ν_{p}(mp)ch(s_{α,mp}.λ).

In this formula, ν_{p}(p^{n}q) = n if p |̸ q. This formula is known as the “Jantzen sum formula”.

In general, this formula is not enough to determine the exact modules occurring in the filtration. However, in our case it turns out that for small values of λ this formula leaves only one possibility for the self dual quotients. This data is displayed in Table 4.

6 Tensor products of Weyl modules

We shall need the following theorems.

Theorem 6.1 (Wang Jian-Pan [27, Theorem 4.12]) The tensor product of two arbitrary Weyl modules W (λ) ⊗ W (μ) has a filtration in which the quotients are Weyl modules. □

Remark : In [27, Theorem 4.12], this is proved for G of type B_{ℓ}, C_{ℓ} or D_{ℓ} and p + 1 not smaller than the Coxeter number of Φ. In our case, p = 3 and the Coxeter number of B_{2} is 4, so that the hypothesis is fulfilled. Note also that the general case has now been proved by Donkin [13].

We shall also make use of the explicit form of the filtration of Theorem 6.1 given in [27, Theorem 2.7], in the case where for all weights μ′ of W (μ), either λ + μ′ ϵ P^{+}(Φ) or there is an α ϵ Δ such that ⟨α^{∨}, λ + μ′⟩ = −1.

Theorem 6.2 (Andersen [3, Theorem 4.1]) Let λ ϵ P^{+}(Φ), α ϵ Φ^{+}, n ϵ N such that np < ⟨α^{∨}, λ + ρ⟩ < (n + 1)p. Suppose μ = s_{α,np}.λ is maximal among weights strongly linked to λ (see [2, Defn 1]). Then

dim_{k} HomG(H^{0}(G/B, −λ̃), H^{0}(G/B, −μ̃)) = 1,

i.e.,

dim_{k} HomG(W (μ), W (λ)) = 1. □

Remark : In [3, Theorem 4.1] this is proved in the context where G is of type A_{ℓ} or B_{ℓ}.

Proposition 6.3 (Andersen [3, Proposition 4.3]) Suppose λ ϵ P^{+}(Φ) is in general position with respect to α ϵ Δ (see [3] for the definition of general position) such that 0 < np < ⟨α^{∨}, λ + ρ⟩ < (n + 1)p. Then the formal character of the kernel of the homomorphism

H^{0}(G/B, −λ̃) → H^{0}(G/B, −(s_{α}.λ + npα)^{∼})

given by Theorem 6.2 is equal to − ∑_{m=1}^{n} ch(λ − mpα). □

Proposition 6.4 (Cline, Parshall, Scott, van der Kallen) [11, Corollary 3.10] Suppose λ, μ ϵ P^{+}(Φ) with λ ⊀ μ. Let M(λ) denote the unique maximal submodule of W (λ). Then

ExtG^{1}(L(λ), L(μ)) ≅ HomG(M(λ), L(μ)).

In other words, if λ ⊀ μ then dimk̄ ExtG^{1}(L(λ), L(μ)) is equal to the multiplicity of L(μ) in the second Loewy layer of W (λ). □

6.1 The module W (2ω_{1} + 2ω_{2}) ⊗ W (ω_{2})

Since W (2ω_{1} + 2ω_{2}) is projective as a G^{F} -module (it is the Steinberg module), by Brauer characters we have

(W (2ω_{1} + 2ω_{2}) ⊗ W (ω_{2}))↓G^{F} ≅ 81 ⊕ P_{25}.

Now W (ω_{2}) ↓B is uniserial, with quotients (from the top) ω̃_{2}, (2ω_{1} − ω_{2})^{∼}, ˜0, (ω_{2} − 2ω_{1})^{∼}, (−ω_{2})^{∼}. Thus W (2ω_{1} + 2ω_{2}) ⊗ W (ω_{2}) has a filtration with quotients W (2ω_{1} + ω_{2}), W (3ω_{2}), W (2ω_{1} + 2ω_{2}), W (4ω_{1} + ω_{2}), W (2ω_{1} + 3ω_{2}). By the linkage principle, W (2ω_{1} + 2ω_{2}) is a direct summand. If we call the other direct summand P̂_{25}, then we have P̂_{25} ↓G^{F} ≅ P_{25}. Using Table 4, we may draw the following consequences.

(i) W (2ω_{1} + ω_{2})↓G^{F} is a non-trivial extension \begin{matrix}25 \\ | \\ 10 \end{matrix} , since it is a quotient of P_{25}.

(ii) W (2ω_{1} + 3ω_{2}) has a unique minimal submodule, which is isomorphic to the 25-dimensional simple module L(2ω_{1} + ω_{2}).

(iii) By Theorem 6.2 there is a non-zero homomorphism

W (4ω_{1} + ω_{2}) → W (2ω_{1} + 3ω_{2}).

Using the information given in Table 4, we see that there is now only one possibility for the structures of these modules. Their diagrams are as

\begin{aligned}& & 25 \\ 10 ⊗ 5^{F} & \begin{matrix}5^{F} \\ | \\ 16 ⊗ 4^{F} \\ | \\ 1 \\ | \\ 16 ⊗ 4^{F} \\ | \\ 5^{F} \end{matrix} & & 10 ; | ; \begin{matrix}25 \\ / \ \\ 10 5^{F} \\ \ / \\ 25 \end{matrix} ; | ; 10 \\ & & 25 \end{aligned} Figure 2: P̂_{25}

follows.

W (4ω_{1} + ω_{2}) = \begin{matrix}16 ⊗ 4^{F} \\ / | \ \\ 1 10 5^{F} \\ \ / \\ 25 \end{matrix} W (2ω_{1} + 3ω_{2}) = 10 ⊗ 5^{F} ; | ; \begin{matrix}16 ⊗ 4^{F} \\ / \ \\ 10 5^{F} \\ \ / \\ 25 \end{matrix}

(iv) Now by self-duality, there is only one possibility for the structure of P̂_{25}. Namely, the diagram is as in Figure 2.

We come back for a further discussion of tensor products of Weyl modules after obtaining some information about modules for the finite group G^{F} = Ĝ′.

7 Characters and maximal subgroups of G

In the Appendix, we give the decomposition matrix, Cartan matrix and 3modular character tables of G, G′, Ĝ and Ĝ′. These have been taken from Thackray [26]. The ordinary character table may be found in the Atlas [12].

The maximal subgroups of G′ are each of index two in a maximal subgroup of G. The list may be found in the Atlas [12], but is reproduced in Table 5 for convenience.

Structure of Structure of Character of
max’l subgp corr. subgp  KG-permutation Objects
of G′  of G  module  permuted  Index
2^{4} :A_{5}  2^{4} :S_{5}  1^{+} + 6^{+} + 20^{+}  lines on cubic  27
S_{6}  S_{6} × 2  1^{+} + 15_{2}^{+} + 20^{+} double sixers  36
3^{1+2} :2A_{4}  3^{1+2} :2S_{4}  1^{+} + 15_{2}^{+} + 24^{+} ninefold triples 40_{1}
3^{3} :S_{4}  3^{3} :(S_{4} × 2) 1^{+} + 15_{1}^{−} + 24^{+} threefold nines 40_{2}
2(A_{4} × A_{4})2 2(A_{4} × A_{4})2^{2} 1^{+} + 20^{+} + 24^{+} planar triangles 45

Table 5: Maximal subgroups of G and G′

8 Characters and modules for the maximal subgroup 2^{4} :S_{5}

Denote by H the subgroup of G of index 27 and shape 2^{4} : S_{5}. Similarly, there are subgroups H′ of G′ of shape 2^{4} : A_{5}, Ĥ of Ĝ of shape 2^{1+4} : S_{5} and Ĥ′ of Ĝ′ of shape 2^{1+4} : A_{5}, in each case of index 27. The ordinary and 3-modular character tables, decomposition matrices and Cartan matrices for these groups are given in the Appendix. The projective indecomposable kH-modules have the following structures:

\begin{matrix}1^{+} \\ | \\ 4^{−} \\ | \\ 1^{+} \end{matrix}  \begin{matrix}4^{+} \\ | \\ 1^{−} \\ | \\ 4^{+} \end{matrix} 6 \begin{matrix}5^{+} \\ | \\ 5^{−} \\ | \\ 5^{+} \end{matrix}  \begin{matrix}10_{1} & + \\ | \\ 10_{2} & − \\ | \\ 10_{1} & + \end{matrix}  \begin{matrix}10_{2} & + \\ | \\ 10_{1} & − \\ | \\ 10_{2} & + \end{matrix} 15_{1}^{+}

(together with the modules obtained from these by interchanging the plus and minus signs).

Notice that the only simple kH′-modules which do not extend to give kHmodules are the two 3-dimensional modules, which fuse to give a 6-dimensional simple kH-module.

The restrictions of the simple kG-modules to kH are as follows:

\begin{matrix}1^{+} ↓_{H}= 1^{+} & 10^{+} ↓_{H}= 10_{2}^{+} & 25^{+} ↓_{H}= 15^{+} ⊕ 10_{1}^{+} \\ 5^{+} ↓_{H}= 5^{+} & 14^{+} ↓_{H}= 10_{1}^{+} ⊕ 4^{+} & 81^{+} ↓_{H}= P_{10_{1}}^{+} ⊕ P_{10_{2}}^{+} ⊕ 6 ⊕ 15^{+} \end{matrix}

We can identify J(kH)↑^{G} as a subring of kG via

J(kH)↑^{G}= J(kH) ⊗_{kH} kG ≤ kH ⊗_{kH} kG ≅ kG.

Lemma 8.1 We have J(kH)↑^{G} .e_{0} ≤ J(kG).

Proof : This follows as in [4, Lemma 4.1.1] from the fact that for each simple module in B_{0}^{±}, the restriction to H is semisimple. □

As in [4, Section 4.1], we have the following consequences.

Lemma 8.2 We have J(kH)^{n} ↑^{G} .e_{0} ≤ J(kG)^{n} for all n ≥ 0. □

Theorem 8.3 If V is any kH-module, then

(\frac{V ↑^{G}}{V ↑^{G} .J(kG)^{n}} ) .e_{0} ≅ (\frac{W ↑^{G}}{W ↑^{G} .J(kG)^{n}} ) .e_{0}

where W = V/(V.J(kH)^{n}). □

Proposition 8.4 If V is any kH-module, then

L_{1}(V ↑^{G}).e_{0} ≅ L_{1}((L_{1}(V))↑^{G}).e_{0}. □

Proposition 8.5 If V is a kH-module and 0 → V′ → V → V′′ → 0 is a non-split short exact sequence with V′, V′′ simple, then

L_{1}(V ↑^{G}).e_{0} ≅ L_{1}(V′′ ↑^{G}).e_{0}. □

9 Some permutation modules for G

In this section, we establish the structures of the permutation modules for G on maximal subgroups, as well as a couple of other subgroups.

9.1 The 36 double sixers

The composition factors of this kG-module M_{36} are (by Brauer character theory) 1^{+} + 1^{+} + 1^{−} + 5^{−} + 14^{+} + 14^{+}. Frobenius reciprocity shows that L_{1}(M_{36}) ≅ S_{1}(M_{36}) ≅ 1^{+} ⊕ 14^{+}. It follows from self duality, Scott’s Lemma and the fact that

dim_{k} Ext_{kG}^{1}(1^{+}, 1^{+}) = dim_{k} Ext_{kG}^{1}(1^{+}, 1^{−}) = 0

(since G has no subgroup of index 3) that the diagram for this module is

\begin{matrix}1^{+} & 14^{+} \\ \ / \ \\ 5^{−} & 1^{−} \\ / \ / \\ 1^{+} & 14^{+} \end{matrix}

9.2 The 27 lines

The composition factors of this kG-module M_{27} are 1^{+} + 1^{+} + 1^{+} + 5^{−} + 5^{−} + 14^{+}. Frobenius reciprocity shows that L_{1}(M_{27}) ≅ S_{1}(M_{27}) ≅ 1^{+}. Since dim_{k} Ext_{kG}^{1}(1^{+}, 1^{+}) = 0, and by self duality, there are two possible diagrams, namely

\begin{matrix}1^{+} \\ / \ \\ \begin{matrix}5^{−} \\ | \\ 1^{+} \\ | \\ 5^{−} \end{matrix} & 14^{+} \\ \ / \\ 1^{+} \end{matrix} and 1^{+} ; | ; \begin{matrix}5^{−} \\ / \ \\ 1^{+} & 14^{+} \\ \ / \\ 5^{−} \end{matrix} ; | ; 1^{+} .

By Scott’s Lemma, (M_{27}, M_{36})_{kG} = 2, and so the first diagram is impossible. Hence,

M_{27} = 1^{+} ; | ; \begin{matrix}5^{−} \\ / \ \\ 1^{+} & 14^{+} \\ \ / \\ 5^{−} \end{matrix} ; | ; 1^{+} .

9.3 The 40 ninefold triples

The composition factors of this kG-module M_{40_{1}} are 1^{+} +1^{−} +10^{−} +14^{+} +14^{+}. Since 40 is not divisible by 3, it follows from Scott’s Lemma that 1^{+} is a direct summand. Applying Scott’s Lemma again, and self duality, we see that

M_{40_{1}} = 1^{+} ⊕ \begin{matrix}14 & + \\ / \ \\ 1^{−} & 10^{−} \\ \ / \\ 14 & + \end{matrix} .

9.4 The 40 threefold nines

The composition factors of this kG-module M_{40_{2}} are 1^{+} +5^{+} +10^{−} +10^{−} +14^{+}. Applying the same argument as in Section 9.3, we see that

M_{40_{2}} = 1^{+} ⊕ \begin{matrix}10 & − \\ / \ \\ 5^{+} & 14^{+} \\ \ / \\ 10 & − \end{matrix} .

9.5 The 45 planar triangles

The composition factors of this kG-module M_{45} are 1^{+} +1^{+} +5^{−} +10^{−} +14^{+} + 14^{+}. By Scott’s Lemma and the fact that (M_{27}, M_{45})_{kG} = 2, we see that M_{45} has a submodule whose diagram is

1^{+} ; | ; \begin{matrix}5^{−} \\ / \ \\ 1^{+} & 14^{+} \end{matrix} ,

and hence applying self duality the only possibility is that

M_{45} = \begin{matrix}1^{+} & 14^{+} \\ \ / \ \\ 5^{−} & 10^{−} \\ / \ / \\ 1^{+} & 14^{+} \end{matrix} .

9.6 The 72 cosets of A_{6} × 2

There are two conjugacy classes of subgroups of G isomorphic to A_{6} × 2, each having index two in the stabiliser of a double sixer. The ordinary characters of the two permutation modules corresponding to these are 1^{+} +15_{2}^{+} +20^{+} +6^{+} + 30_{1}^{+} and 1^{+} + 15_{2}^{+} + 20^{+} + 6^{−} + 30_{1}^{−}. Denote the corresponding permutation kG-modules by M_{72}^{+} and M_{72}^{−}. Then M_{36} is a direct summand of each of these, and we have M_{72}^{+} = M_{36} ⊕ M′_{36}, M_{72}^{−} = M_{36} ⊕ (M′_{36} ⊗ 1^{−}). The kG-module M′_{36} has composition factors 1^{+} + 5^{−} + 5^{−} + 25^{+}. By Scott’s Lemma and self duality, the only possibility is

M′_{36} = \begin{matrix}5^{−} \\ / \ \\ 1^{+} & 25^{+} \\ \ / \\ 5^{−} \end{matrix} .

9.7 The 90 double cosets of 2(A_{4} × A_{4})2

As in the case of A_{6}×2 (Section 9.6, there are two conjugacy classes of subgroups of G of shape 2(A_{4} × A_{4})2 (and not contained in G′), each having index two in the stabiliser of a planar triangle. The ordinary characters of the corresponding permutation modules are 1^{+} + 20^{+} + 24^{+} + 15_{1}^{+} + 30_{1}^{+} and 1^{+} + 20^{+} + 24^{+} + 15_{1}^{−} + 30_{1}^{−}. Denote the corresponding permutation modules by M_{90}^{+} and M_{90}^{−}. Then M_{45} is a direct sumand of each of these, and we have M_{90}^{+} = M_{45} ⊕ M′_{45}, M_{90}^{−} = M_{45} ⊕ (M′_{45} ⊗ 1^{−}). The kG-module M′_{45} has composition factors 5^{−} + 5^{−} + 10^{+} + 25^{+}. By Scott’s Lemma and self duality, the only possibility is

M′_{45} = \begin{matrix}5^{−} \\ / \ \\ 10^{+} & 25^{+} \\ \ / \\ 5^{−} \end{matrix} .

10 Tensor products of modules for G and Ĝ

By Brauer characters and self duality, as modules for Ĝ,

4^{+} ⊗ 4^{−} = 1^{+} ⊕ 5^{+} ⊕ 10^{+}.

Similarly (by self duality when restricted to Ĝ′), as modules for Ĝ,

4^{+} ⊗ 5^{+} = 4^{+} ⊕ 16^{+}
4^{+} ⊗ 14^{+} = 16^{+} ⊕ 40^{+},

and as modules for G,

5^{+} ⊗ 5^{+} = 1^{+} ⊕ 10^{+} ⊕ 14^{+}.

10.1 The module 4^{+} ⊗ 40^{+} for Ĝ

As a module for Ĝ′, 4 ⊗ 40 has composition factors 5 + 10 + 14 + 25 + 25 + 81. We have

(5, 4 ⊗ 40)Ĝ = (5 ⊗ 4, 40)Ĝ′ = 0
(10, 4 ⊗ 40)Ĝ′ = (10 ⊗ 4, 40)Ĝ′ = 0

(since 40 is not a composition factor of 10 ⊗ 4) and

(14, 4 ⊗ 40)Ĝ′ = (14 ⊗ 4, 40)Ĝ′ = 1.

Hence by self duality

4 ⊗ 40 = 14 ⊕ \begin{matrix}25 \\ / \ \\ 5 10 \\ \ / \\ 25 \end{matrix} ⊕ 81.

Comparing characters for Ĝ, we deduce that

4^{+} ⊗ 40^{+} = 14^{−} ⊕ \begin{matrix}25 & + \\ / \ \\ 5^{−} & 10^{−} \\ \ / \\ 25 & + \end{matrix} ⊕ 81^{−}.

10.2 The module 4^{+} ⊗ 16^{+} for Ĝ

As a module for Ĝ′, 4 ⊗ 16 has composition factors 5 + 10 + 10 + 14 + 25. We have

(5, 4 ⊗ 16)Ĝ′ = (5 ⊗ 4, 16)Ĝ′ = 1
(14, 4 ⊗ 16)Ĝ′ = (14 ⊗ 4, 16)Ĝ′ = 1

and so 5 and 14 occur as direct summands. It follows from Theorem 6.1 and Section 6.1(i) that 4 ⊗ 16 has a filtration in which one of the quotients is a

uniserial module \begin{matrix}25 \\ | \\ 10 \end{matrix} , and hence by self duality

4 ⊗ 16 = 5 ⊕ 14 ⊕ \begin{matrix}10 \\ | \\ 25 \\ | \\ 10 \end{matrix} .

Comparing characters for Ĝ, we deduce that

4^{+} ⊗ 16^{+} = 5^{−} ⊕ 14^{−} ⊕ \begin{matrix}10 & − \\ | \\ 25 & + \\ | \\ 10 & − \end{matrix} .

10.3 The module 4^{+} ⊗ 25^{+} for Ĝ

As a module for Ĝ′, 4 ⊗ 25 has composition factors 4 + 16 + 40 + 40. We have (4, 4 ⊗ 25)Ĝ′ = (4 ⊗ 4, 25)Ĝ′ = 0

(16, 4 ⊗ 25)Ĝ′ = (16 ⊗ 4, 25)Ĝ′ = 0.

Hence by self duality

4 ⊗ 25 = \begin{matrix}40 \\ / \ \\ 4 16 \\ \ / \\ 40 \end{matrix} .

Comparing characters for Ĝ, we deduce that

4^{+} ⊗ 25^{+} = \begin{matrix}40 & − \\ / \ \\ 4^{+} & 16^{+} \\ \ / \\ 40 & − \end{matrix} .

10.4 The module 5^{+} ⊗ 10^{+} for G

As a module for G′, 5 ⊗ 10 has composition factors 5 + 10 + 10 + 25. We have

(5, 5 ⊗ 10)_{G′} = (5 ⊗ 5, 10)_{G′} = 1,

and so 5 is a direct summand. Again it follows from Theorem 6.1 that 5 ⊗ 10 has a filtration in which one of the quotients is a uniserial module \begin{matrix}25 \\ | \\ 10 \end{matrix} , and so by self duality

5 ⊗ 10 = 5 ⊕ \begin{matrix}10 \\ | \\ 25 \\ | \\ 10 \end{matrix} .

Comparing characters for G, we deduce that

5^{+} ⊗ 10^{+} = 5^{+} ⊕ \begin{matrix}10 & + \\ | \\ 25 & − \\ | \\ 10 & + \end{matrix} .

10.5 The module 5^{+} ⊗ 25^{+} for G

As a module for G′, 5 ⊗ 25 has composition factors 1 + 5 + 10 + 14 + 14 + 81. We have

(10, 5 ⊗ 25)_{G′} = (10 ⊗ 5, 25)_{G′} = 0
(5, 5 ⊗ 25)_{G′} = (5 ⊗ 5, 25)_{G′} = 0
(1, 5 ⊗ 25)_{G′} = (5, 25)_{G′} = 0

and so by self duality

5 ⊗ 25 = 81 ⊕ \begin{matrix}14 \\ / | \ \\ 1 5 10 \\ \ | / \\ 14 \end{matrix} .

Comparing characters for G, we deduce that

5^{+} ⊗ 25^{+} = 81^{−} ⊕ \begin{matrix}14 & − \\ / | \ \\ 1^{+} & 5^{+} & 10^{+} \\ \ | / \\ 14 & − \end{matrix} .

10.6 The modules 5^{+} ⊗ 14^{+} and 10^{+} ⊗ 14^{+} for G

As modules for Ĝ, we have

(4^{−} ⊗ 4^{+}) ⊗ 14^{+} = (1^{+} ⊕ 5^{+} ⊕ 10^{+}) ⊗ 14^{+}
= 14^{+} ⊕ (5^{+} ⊗ 14^{+}) ⊕ (10^{+} ⊗ 14^{+}).

On the other hand,

4^{−} ⊗ (4^{+} ⊗ 14^{+}) = 4^{−} ⊗ (16^{+} ⊕ 40^{+})
= (4^{−} ⊗ 16^{+}) ⊕ (4^{−} ⊗ 40^{+})
= 5^{+} ⊕ 14^{+} ⊕ \begin{matrix}10 & + \\ | \\ 25 & − \\ | \\ 10 & + \end{matrix} ⊕ 14^{+} ⊕ \begin{matrix}25 & − \\ / \ \\ 5^{+} & 10^{+} \\ \ / \\ 25 & − \end{matrix} ⊕ 81^{+}.

Comparing dimensions, we have

5^{+} ⊗ 14^{+} = 5^{+} ⊕ \begin{matrix}25 & − \\ / \ \\ 5^{+} & 10^{+} \\ \ / \\ 25 & − \end{matrix}

and

10^{+} ⊗ 14^{+} = 14^{+} ⊕ \begin{matrix}10 & + \\ | \\ 25 & − \\ | \\ 10 & + \end{matrix} ⊕ 81^{+}.

10.7 The module 4^{+} ⊗ 10^{+} for Ĝ

The composition factors of this module are 4^{+}+4^{−}+16^{+}+16^{+}. By Section 10.2 we have

(16^{+}, 4^{+} ⊗ 10^{+})Ĝ = (16^{+} ⊗ 4^{−}, 10^{+})Ĝ = 1,
(4^{+}, 4^{+} ⊗ 10^{+})Ĝ = (4^{+} ⊗ 4^{−}, 10^{+})Ĝ
= (1^{+} ⊕ 5^{+} ⊕ 10^{+}, 10^{+})Ĝ
= 1

and

(4^{−}, 4^{+} ⊗ 10^{+})Ĝ = (4^{−} ⊗ 4^{−}, 10^{+})Ĝ
= (1^{−} ⊕ 5^{−} ⊕ 10^{−}, 10^{+})Ĝ
= 0.

Hence

4^{+} ⊗ 10^{+} = 4^{+} ⊕ \begin{matrix}16 & + \\ | \\ 4^{−} \\ | \\ 16 & + \end{matrix} .

10.8 The module 5^{+} ⊗ 16^{+} for Ĝ

We have

(5^{+} ⊗ 16^{+}) ⊕ 16^{+} ⊕ 4^{+} = 5^{+} ⊗ (16^{+} ⊕ 4^{+})
= 5^{+} ⊗ (5^{+} ⊗ 4^{+})
= (5^{+} ⊗ 5^{+}) ⊗ 4^{+}
= (1^{+} ⊕ 10^{+} ⊕ 14^{+}) ⊗ 4^{+}
= 4^{+} ⊕ 4^{+} ⊕ \begin{matrix}16 & + \\ | \\ 4^{−} \\ | \\ 16 & + \end{matrix} ⊕ 16^{+} ⊕ 40^{+}

by 10.7. Hence

5^{+} ⊗ 16^{+} = 4^{+} ⊕ \begin{matrix}16 & + \\ | \\ 4^{−} \\ | \\ 16 & + \end{matrix} ⊕ 40^{+}.

11 Some Ext^{1} groups for G and Ĝ

Proposition 11.1 For V a simple kG-module,

dim_{k} Ext_{kG}^{1}(1^{+}, V) = \{ \begin{matrix}1 & if V ≅ 5^{−} or 14^{−} \\ 0 & otherwise. \end{matrix}

Proof : Comparing characters, we see that (P_{1^{+}})_{G} = (P_{1^{+}})_{H} ↑^{G}. By Theorem 8.3 with n = 2, we see that L_{2}((P_{1^{+}})_{G}) is the sum of L_{2}(M_{27}) (see Section 9.2) and some summand of L_{1}(4^{−}_{H} ↑^{G}). By Frobenius reciprocity, L_{1}(4^{−}_{H} ↑^{G}) ≅ 14^{−}. By Section 9.3, there is an extension of 1^{−} by 14^{+}, and hence of 1^{+} by 14^{−}, proving the proposition. □

Note that this implies that for V simple,

dim_{k} Ext_{kG}^{1}(1^{−}, V) = \{ \begin{matrix}1 & if V ≅ 5^{+} or 14^{+} \\ 0 & otherwise. \end{matrix}

In future we shall use such deductions without writing them down explicitly.

Lemma 11.2 For G′, there is no module of shape \begin{matrix}1 \\ | \\ 5 \\ | \\ 25 \end{matrix} .

Proof : This module would have to be a quotient of P_{1}. Since L_{1}(4_{H′} ↑^{G′}) ≅ 14, it would have to be a quotient of M_{27}, which it is not. □

Lemma 11.3 We have dim_{k} Ext_{kG′}^{1}(1, \begin{matrix}25 \\ / \ \\ 5 10 \\ \ / \\ 25 \end{matrix} ) = 0.

Proof : This follows from Proposition 11.1 and Lemma 11.2 using the long exact Ext sequence. □

Proposition 11.4 For V a simple kG-module,

dim_{k} Ext_{kG}^{1}(5^{+}, V) = \{ \begin{matrix}1 & if V ≅ 1^{−}, 5^{−}, 10^{−} or 14^{−} \\ 0 & if V ≅ 1^{+}, 5^{+}, 10^{+} or 14^{+} \end{matrix}

(Note that we cannot yet give information for the simple modules 25^{−} or 25^{+}). Proof : For V = 1^{+} and 1^{−}, the result follows from Proposition 11.1. We have

dim_{k} Ext_{kG}^{1}(5^{+}, 5^{−}) = dim_{k} Ext_{kG}^{1}(1^{+}, 5^{+} ⊗ 5^{+})
= dim_{k} Ext_{kG}^{1}(1^{+}, 1^{+} ⊕ 10^{+} ⊕ 14^{+}) = 0

by Section 10 and Proposition 11.1. Similarly

dim_{k} Ext_{kG}^{1}(5^{+}, 5^{−}) = dim_{k} Ext_{kG}^{1}(1^{+}, 1^{−} ⊕ 10^{−} ⊕ 14^{−}) = 1
dim_{k} Ext_{kG}^{1}(5^{+}, 10^{+}) = dim_{k} Ext_{kG}^{1}(1^{+}, 5^{+} ⊗ 10^{+})
= dim_{k} Ext_{kG}^{1}(1^{+}, 5^{+} ⊕ \begin{matrix}10 & + \\ | \\ 25 & − \\ | \\ 10 & + \end{matrix} ) = 0

dim_{k} Ext_{kG}^{1}(5^{+}, 10^{−}) = dim_{k} Ext_{kG}^{1}(1^{+}, 5^{−} ⊕ \begin{matrix}10 & − \\ | \\ 25 & + \\ | \\ 10 & − \end{matrix}) = 1
dim_{k} Ext_{kG}^{1}(5^{+}, 14^{+}) = dim_{k} Ext_{kG}^{1}(1^{+}, 5^{+} ⊗ 14^{+})
= dim_{k} Ext_{kG}^{1}(1^{+}, 5^{+} ⊕ \begin{matrix}25 & − \\ / \ \\ 5^{+} & 10^{+} \\ \ / \\ 25 & − \end{matrix}) = 0

dim_{k} Ext_{kG}^{1}(5^{+}, 14^{−}) = dim_{k} Ext_{kG}^{1}(1^{+}, 5^{−} ⊕ \begin{matrix}25 & + \\ / \ \\ 5^{−} & 10^{−} \\ \ / \\ 25 & + \end{matrix} ) = 1.

In the last two cases, we use Lemma 11.3. □ Proposition 11.5 For V a simple Ĝ-module,

dim_{k} Ext_{kĜ}^{1}(4^{+}, V) = \{ \begin{matrix}2 & if V ≅ 16^{−} \\ 1 & if V ≅ 4^{−} or 40^{−} \\ 0 & otherwise \end{matrix}

Proof : By Section 10 and Proposition 11.1, we have

dim_{k} Ext_{kĜ}^{1}(4^{+}, 4^{+}) = dim_{k} Ext_{kĜ}^{1}(1^{+}, 4^{−} ⊗ 4^{+})
= dim_{k} Ext_{kĜ}^{1}(1^{+}, 1^{+} ⊕ 5^{+} ⊕ 10^{+}) = 0
dim_{k} Ext_{kĜ}^{1}(4^{+}, 4^{−}) = dim_{k} Ext_{kĜ}^{1}(1^{+}, 1^{−} ⊕ 5^{−} ⊕ 10^{−}) = 1
dim_{k} Ext_{kĜ}^{1}(4^{+}, 16^{+}) = dim_{k} Ext_{kĜ}^{1}(1^{+}, 4^{−} ⊗ 16^{+})

= dim_{k} Ext_{kĜ}^{1}(1^{+}, 5^{+} ⊕ 14^{+} ⊕ \begin{matrix}10 & + \\ | \\ 25 & − \\ | \\ 10 & + \end{matrix} ) = 0

dim_{k} Ext_{kĜ}^{1}(4^{+}, 16^{−}) = dim_{k} Ext_{kĜ}^{1}(1^{+}, 5^{−} ⊕ 14^{−} ⊕ \begin{matrix}10 & − \\ | \\ 25 & + \\ | \\ 10 & − \end{matrix}) = 2

dim_{k} Ext_{kĜ}^{1}(4^{+}, 40^{+}) = dim_{k} Ext_{kĜ}^{1}(1^{+}, 4^{−} ⊗ 40^{+})

= dim_{k} Ext_{kĜ}^{1}(1^{+}, 14^{+} ⊕ \begin{matrix}25 & − \\ / \ \\ 5^{+} & 10^{+} \\ \ / \\ 25 & − \end{matrix} ⊕ 81^{+}) = 0

dim_{k} Ext_{kĜ}^{1}(4^{+}, 40^{−}) = dim_{k} Ext_{kĜ}^{1}(1^{+}, 14^{−} ⊕ \begin{matrix}25 & + \\ / \ \\ 5^{−} & 10^{−} \\ \ / \\ 25 & + \end{matrix} ⊕ 81^{−}) = 1

by Lemma 11.3.

We have now dealt with all the simple modules in the same block as 4^{+}, and in all other cases we get zero. □

Proposition 11.6 (i) dim_{k} Ext_{kG}^{1}(10^{+}, 14^{+}) = 0.

(ii) dim_{k} Ext_{kG}^{1}(10^{+}, 14^{−}) = 1.

Proof :

(i) dim_{k} Ext_{kG}^{1}(10^{+}, 14^{+}) = dim_{k} Ext_{kG}^{1}(1^{+}, 10^{+} ⊗ 14^{+})

= dim_{k} Ext_{kG}^{1}(1^{+}, 14^{+} ⊕ \begin{matrix}10 & + \\ | \\ 25 & − \\ | \\ 10 & + \end{matrix} ⊕ 81^{+}) = 0

(ii) dim_{k} Ext_{kG}^{1}(10^{+}, 14^{−}) = dim_{k} Ext_{kG}^{1}(1^{+}, 14^{−} ⊕ \begin{matrix}10 & − \\ | \\ 25 & + \\ | \\ 10 & − \end{matrix} ⊕ 81^{−}) = 1

(by Proposition 11.1). □

12 More tensor products of Weyl modules

We continue here the calculations started in Section 6.

12.1 The module W (2ω_{1} + 2ω_{2}) ⊗ W (2ω_{1})

This module has a filtration with quotients (from the top) W (2ω_{2}), W (2ω_{1}+ω_{2}), W (4ω_{1}), W (3ω_{2}), (W (2ω_{1} + 2ω_{2}) ⊕ W (2ω_{1} + 2ω_{2})), W (4ω_{1} + ω_{2}), W (4ω_{2}), W (2ω_{1} +3ω_{2}), W (4ω_{1} +2ω_{2}). By the linkage principle, this module decomposes as a direct sum

W (2ω_{1} + 2ω_{2}) ⊗ W (2ω_{1}) = W (2ω_{1} + 2ω_{2}) ⊕ W (2ω_{1} + 2ω_{2}) ⊕ P̂_{25} ⊕ P̂_{14},

where P̂_{14} is a module which when restricted to G^{F} gives P_{14} ⊕81, and which as a module for G has a filtration with factors W (2ω_{2}), W (4ω_{1}), W (4ω_{2}), W (4ω_{1} + 2ω_{2}). Using Table 4, we may draw the following consequences.

(i) W (4ω_{1} + 2ω_{2}) has a unique minimal submodule, which is isomorphic to the 14 dimensional simple module L(2ω_{2}).

(ii) By Theorem 6.2, there is a non-zero homomorphism W (4ω_{2}) → W (4ω_{1} + 2ω_{2}). Thus W (4ω_{1} + 2ω_{2}) has a submodule whose composition factors are either 14 and 5 ⊗ 5^{F} or 14, 4 ⊗ 4^{F} and 5 ⊗ 5^{F}. In the former case, this would imply that there was a subquotient of W (4ω_{1} + 2ω_{2}) isomorphic to 4 ⊗ 4^{F} ⊕ 5 ⊗ 5^{F}. Restricting this to G^{F} using the results of Section 10, we would have dimk̄ ExtG^{F}^{1} (1, 14) ≥ 2, contradicting Proposition 11.1. Hence the second possibility holds, and we have

W (4ω_{2}) = \begin{matrix}5 ⊗ 5^{F} \\ | \\ 4 ⊗ 4^{F} \\ | \\ 14 \end{matrix} .

(iii) Since P̂_{14} is self dual, there is now only one possibility for its structure. Namely, the diagram is given in Figure 3.

\begin{aligned}& & 14 \\ & & | \\ & & 4 ⊗ 4^{F} \\ & & | \\ 14 & & 5 & & 5 ⊗ 5^{F} \\ & 4 ⊗ 4^{F} & & 40 ⊗ 4^{F} \\ 14 & & 5 & & 5 ⊗ 5^{F} \\ & & | \\ & & 4 ⊗ 4^{F} \\ & & | \\ & & 14 \end{aligned} Figure 3: P̂_{14}

(iv) Hence

W (4ω_{1} + 2ω_{2}) = \begin{matrix}40 ⊗ 4^{F} \\ / \ \\ 5 5 ⊗ 5^{F} \\ \ / \\ 4 ⊗ 4^{F} \end{matrix} ; | ; 14 ,

and

( \begin{matrix}4 ⊗ 4^{F} \\ | \\ 14 \end{matrix} )↓G^{F} = \begin{matrix}1 5 10 \\ \ | / \\ 14 \end{matrix} .

12.2 The module W (2ω_{1} + 2ω_{2}) ⊗ W (2ω_{2})

Repeating the above process for this module, we obtain a module P̂_{10} which when restricted to G^{F} gives P_{10} ⊕ 81, and which has a filtration whose factors (from the top) are W (2ω_{1}), W (2ω_{1}+ω_{2}), W (4ω_{1}+ω_{2}), (W (6ω_{1})⊕W (2ω_{1}+3ω_{2})), W (2ω_{1} + 4ω_{2}). Using Table 4 again, we may draw the following consequences.

(i) By Theorem 6.2, there is a non-zero homomorphism W (6ω_{1}) → W (2ω_{1} + 4ω_{2}), and by Proposition 6.3 the image has formal character equal to that of W (6ω_{1}). Hence this homomorphism is injective.

(ii) W (2ω_{1} + 4ω_{2}) has a unique minimal submodule, which is isomorphic to the 10-dimensional simple module L(2ω_{1}). This implies that the diagrams for W (6ω_{1}) and W (2ω_{1} + 4ω_{2}) are as follows.

W (6ω_{1}) = \begin{matrix}10^{F} \\ | \\ 16 ⊗ 4^{F} \\ | \\ 10 \end{matrix} W (2ω_{1} + 4ω_{2}) = \begin{matrix}25 ⊗ 5^{F} \\ / | \ \\ 1 10 ⊗ 5^{F} 10^{F} \\ \ | / \\ 16 ⊗ 4^{F} \end{matrix} ; | ; 10

(iii) By Proposition 6.4 and Table 4, we have dimk̄ ExtG^{1}(L(2ω_{1}), V) = 0 for V isomorphic to 1, L(2ω_{1}) or L(3ω_{2}). This implies (using the self duality of P̂_{10}), that the quotient of P̂_{10} with filtration whose factors are W (2ω_{1}), W (2ω_{1} + ω_{2}), W (4ω_{1} + ω_{2}), has Loewy series

\begin{matrix}10 \\ 25 16 ⊗ 4^{F} \\ 1 10 10 5^{F} \\ 25 \end{matrix} .

We do not try to write down the exact diagram for this module.

(iv) We now have enough information to write down the Loewy series for P̂_{10}. It is given in Figure 4.

(v) By Section 10.2 and the fact that Soc (( \begin{matrix}16 ⊗ 4^{F} \\ | \\ 10 \end{matrix} )↓G^{F} ) = 10,

we have ( \begin{matrix}16 ⊗ 4^{F} \\ | \\ 10 \end{matrix} )↓G^{F} = \begin{matrix}5 14 \begin{matrix}10 \\ | \\ 25 \\ | \\ 10 \end{matrix} \\ \ | / \\ 10 \end{matrix} .

(vi) It now follows from Section 6.1(iv) and the fact that Soc(P̂_{10} ↓G^{F}) = 10⊕81, that the second Loewy layer of P_{25} for G^{F} is 5 ⊕ 10.

Proposition 12.1 For V a simple kG-module,

dim_{k} Ext_{kG}^{1}(25^{+}, V) = \{ \begin{matrix}1 & if V ≅ 5^{−} or 10^{−} \\ 0 & otherwise. \end{matrix}

Proof : This follows from Sections 12.2(vi) and 10.1. □

25 16 ⊗ 4^{F} 1 10 10 5^{F} 10^{F} 10 ⊗ 5^{F} 25 16 ⊗ 4^{F} 16 ⊗ 4^{F} 25 ⊗ 5^{F} 1 10 10 5^{F} 10^{F} 10 ⊗ 5^{F} 25 16 ⊗ 4^{F} 10

Figure 4: Loewy series of P̂_{10}

12.3 The module W (2ω_{1} + 2ω_{2}) ⊗ W (ω_{1})

The tensor product W (2ω_{1}+2ω_{2})⊗W (ω_{1}) is a module P̂_{40} which when restricted to G^{F} gives P_{40}, and which has a filtration whose factors (from the top) are W (ω_{1} + 2ω_{2}), W (3ω_{1} + ω_{2}), W (ω_{1} + 3ω_{2}), W (3ω_{1} + 2ω_{2}). Since we do not need the conclusions of this section for the rest of the paper, we simply state the results without proof. The methods are the same as before.

(i) W (3ω_{1} + ω_{2}) = \begin{matrix}5 ⊗ 4^{F} \\ / \ \\ 4 40 \end{matrix} . (ii) W (ω_{1} + 3ω_{2}) = \begin{matrix}4 ⊗ 5^{F} \\ | \\ 5 ⊗ 4^{F} \\ | \\ 40 \end{matrix} .

(iii) W (3ω_{1} + 2ω_{2}) = \begin{matrix}14 ⊗ 4^{F} \\ / \ \\ 4 4 ⊗ 5^{F} \\ \ / \\ 5 ⊗ 4^{F} \end{matrix} ; | ; 40 .

(iv) The Loewy structure of P̂_{40} is as given in Figure 5.

13 Some Ext^{1} groups for the algebraic group G

By Proposition 6.4, we have now amassed enough information to check Table 6, which gives dimk̄ ExtG^{1}(L(λ), L(μ)) for small values of λ and μ. In this table, dots signify that μ is not strongly linked to λ; question marks indicate that we have not obtained this information, and it is not needed in the proof of the main theorem.

5 ⊗ 4^{F} 4 40 4 ⊗ 5^{F} 5 ⊗ 4^{F} 14 ⊗ 4^{F} 4 40 4 ⊗ 5^{F} 5 ⊗ 4^{F} 40

Figure 5: Loewy series of P̂_{40}

μ →
λ↓ 00 10 01 20 11 30 02 21 40 12 31 50 03 22 41 60 13 32 51 70 04 23 42 61
00 0
10 . 0
01 . . 0
20 0 . . 0
11 …. 0
30 …. 1 0
02 . . 0 … 0
21 0 . . 1 … 0
40 . . 1 … 1 . 0
12 . 0 . . . . . . . 0
31 . 1 . . . . . . . 1 0
50 …. 1 0 . . . . . 0
03 0 . . 0 … 1 …. 0
22 . . . . . . . . . . . . . 0
41 1 . . 1 … 0 …. 1 . 0
60 0 . . 0 … 0 …. 0 . 1 0
13 . 0 . . . . . . . 0 1 . . . . . 0
32 . 1 . . . . . . . 0 0 . . . . . 1 0
51 …. ? 1 . . . . . ? . . . . . . 0
70 . ? . . . . . . . 0 1 . . . . . 0 1 . 0
04 . . 0 … 0 . 1 . . . . . . . . . . . 0
23 0 . . 0 … 0 …. 0 . 1 . . . . . . 0
42 . . 1 … 0 . 0 . . . . . . . . . . . 1 . 0
61 . . ? … 0 . 1 . . . . . . . . . . . 0 . 1 0

Table 6: dimk̄ ExtG^{1}(L(λ), L(μ)) for λ ⪰ μ

14^{+}
5^{−} 10^{−}
1^{+} 10^{+} 14^{+} 25^{+}
5^{−} 10^{−}
14^{+}

Figure 6: Loewy series for 4^{+}_{H} ↑^{G}

14 The structure of 4^{+}_{H} ↑^{G}

By Brauer characters, the composition factors of the induced module 4^{+}_{H} ↑^{G} are 1^{+} + 5^{−} + 5^{−} + 10^{+} + 10^{−} + 10^{−} + 14^{+} + 14^{+} + 14^{+} + 25^{+}. By Frobenius reciprocity and the results of Section 8, S_{1}(4^{+}_{H} ↑^{G}) ≅ L_{1}(4^{+}_{H} ↑^{G}) ≅ 14^{+}. Now

by Section 9.5 there is a module whose diagram is \begin{matrix}14 & + \\ / \ \\ 5^{−} & 10^{−} \\ \ / \\ 14 & + \end{matrix} , which must be a

quotient of P_{14}^{+} = (P_{4^{+}})_{H} ↑^{G}= ( \begin{matrix}4^{+} \\ | \\ 1^{−} \\ | \\ 4^{+} \end{matrix} )_{H} ↑^{G}. Hence by Theorem 8.3 and the

fact that L_{1}(1^{−}_{H} ↑^{G}) = 1^{−} (see Section 9.2), the above module is a quotient of 4^{+}_{H} ↑^{G}. By self duality, it is also a submodule. Now by Propositions 11.1, 11.6 and 12.1, we have dim_{k} Ext_{kG}^{1}(14^{+}, V) = 0 for V = 1^{+}, 10^{+} or 25^{+}. Hence L_{2}(4^{+}_{H} ↑^{G}) ≅ 5^{−}⊕10^{−}, and the Loewy series for 4^{+}_{H} ↑^{G} is as given in Figure 6.

15 The remaining Ext^{1} groups for G and Ĝ

In this section we complete the calculation started in Section 11 of the dimensions of Ext1 ; Ĝ(V, W) for V and W simple kĜ-modules.

Proposition 15.1 For V a simple kG-module,

dim_{k} Ext_{kG}^{1}(14^{+}, V) = \{ \begin{matrix}1 & if V ≅ 1^{−}, 5^{−} or 10^{−} \\ 0 & otherwise. \end{matrix}

Proof : From the calculations of Section 14, we see that

L_{2}(P_{14}^{+}) ≅ 1^{−} ⊕ L_{2}(4^{+}_{H} ↑^{G}) ≅ 1^{−} ⊕ 5^{−} ⊕ 10^{−}. □

Proposition 15.2 We have dim_{k} Ext_{kG}^{1}(14^{+}, \begin{matrix}10 & − \\ | \\ 25 & + \\ | \\ 10 & − \end{matrix}) = 1.

Proof : Since there is a module \begin{matrix}_{14^{+}} & \begin{matrix}10 & − \\ | \\ 25 & + \end{matrix} \\ \ / \\ 10 & − \end{matrix} , obtained by forming an amalga-

mated sum of \begin{matrix}14 & + \\ | \\ 10 & − \end{matrix} with \begin{matrix}10 & − \\ | \\ 25 & + \\ | \\ 10 & − \end{matrix}, it suffices, by Proposition 15.1, to show that there

is no uniserial module with composition factors (from the top) 14^{+}, 10^{−}, 25^{+}, 10^{−}. If there were such a module, it would have to be a quotient of P_{14}^{+}, and hence of 4^{+}_{H} ↑^{G}. This is impossible, by the information given in Section 14. □

Proposition 15.3 For V a simple kĜ-module,

dim_{k} Ext_{kĜ}^{1}(16^{+}, V) = \{ \begin{matrix}2 & if V ≅ 4^{−} \\ 1 & if V ≅ 16^{−} or 40^{−} \\ 0 & otherwise. \end{matrix}

Proof : Using the results of Section 10, we have

dim_{k} Ext_{kĜ}^{1}(16^{+}, 16^{+} ⊕ 40^{+}) = dim_{k} Ext_{kĜ}^{1}(16^{+}, 4^{+} ⊗ 14^{+})
= dim_{k} Ext_{kĜ}^{1}(16^{+} ⊗ 4^{−}, 14^{+})

= dim_{k} Ext_{kĜ}^{1}(5^{+} ⊕ 14^{+} ⊕ \begin{matrix}10 & + \\ | \\ 25 & − \\ | \\ 10 & + \end{matrix} , 14^{+})

= 0

by Proposition 15.1.

Similarly

dim_{k} Ext_{kĜ}^{1}(16^{+}, 16^{−} ⊕ 40^{−}) = dim_{k} Ext_{kĜ}^{1}(5^{−} ⊕ 14^{−} ⊕ \begin{matrix}10 & − \\ | \\ 25 & + \\ | \\ 10 & − \end{matrix} , 14^{+})

= 2

by Propositions 15.1 and 15.2. Also

dim_{k} Ext_{kĜ}^{1}(16^{+}, 16^{−}) = dim_{k} Ext_{kĜ}^{1}(16^{+}, 4^{+} ⊗ 5^{−}) − dim_{k} Ext_{kĜ}^{1}(16^{+}, 4^{−})
= dim_{k} Ext_{kĜ}^{1}(16^{+} ⊗ 4^{−}, 5^{−}) − 2

= dim_{k} Ext_{kĜ}^{1}(5^{+} ⊕ 14^{+} ⊕ \begin{matrix}10 & + \\ | \\ 25 & − \\ | \\ 10 & + \end{matrix} , 5^{−}) − 2

≥ 1

by Propositions 11.5 and 11.4, and Section 10.2. Since we also have

dim_{k} Ext_{kĜ}^{1}(16^{+}, 40^{−}) ≥ 1

by Section 10.3, we have equality in both cases.

Combining this information with Proposition 11.5 completes the proof. □

Proposition 15.4 We have

dim_{k} Ext_{kĜ}^{1}(40^{+}, 40^{+}) = dim_{k} Ext_{kĜ}^{1}(40^{+}, 40^{−}) = 0.

Proof : It suffices to prove that dim_{k} Ext_{kĜ′}^{1}(40, 40) = 0. Now by Theorem 8.3 and Section 14, we see that there is only one copy of 25 in L_{3}(P_{14}), and hence by Landrock’s lemma, only one copy of 14 in L_{3}(P_{25}). Hence there is no module

with Loewy series 25 5 10 14 25 14 . Hence by Propositions 11.4, 11.6 and 12.1, and the

long exact Ext sequence, we have

dim_{k} Ext_{kĜ′}^{1}(14, \begin{matrix}25 \\ / \ \\ 5 10 \\ \ / \\ 25 \end{matrix} ) ≤ 1.

Now by Section 10 and Propositions 15.3 and 15.1 we obtain

dim_{k} Ext_{kĜ′}^{1}(40, 40) = dim_{k} Ext_{kĜ′}^{1}(4 ⊗ 14, 40) − dim_{k} Ext_{kĜ′}^{1}(16, 40)
= dim_{k} Ext_{kĜ′}^{1}(14, 4 ⊗ 40) − 1

= dim_{k} Ext_{kĜ′}^{1}(14, 14 ⊕ \begin{matrix}25 \\ / \ \\ 5 10 \\ \ / \\ 25 \end{matrix} ⊕ 81) − 1

≤ 0. □

Proposition 15.5 (i) dim_{k} Ext_{kG}^{1}(10^{+}, 10^{+}) = 0

(ii) dim_{k} Ext_{kG}^{1}(10^{+}, 10^{−}) = 1.

Proof : (i) By Propositions 11.1, 11.4, 11.5 and 11.6, and Section 10 we have

dim_{k} Ext_{kG}^{1}(10^{+}, 10^{+}) = dim_{k} Ext_{kG}^{1}(1^{+} ⊕ 5^{+} ⊕ 10^{+}, 1^{+} ⊕ 10^{+} ⊕ 14^{+})
= dim_{k} Ext_{kĜ}^{1}(4^{+} ⊗ 4^{−}, 5^{+} ⊗ 5^{+})
= dim_{k} Ext_{kĜ}^{1}(4^{+} ⊗ 5^{+}, 4^{+} ⊗ 5^{+})
= dim_{k} Ext_{kĜ}^{1}(4^{+} ⊕ 16^{+}, 4^{+} ⊕ 16^{+})
= 0.

(ii) By Propositions 11.1, 11.4, 11.5 and 11.6, and Section 10 we have

dim_{k} Ext_{kG}^{1}(10^{+}, 10^{−}) = dim_{k} Ext_{kG}^{1}(1^{+} ⊕ 5^{+} ⊕ 10^{+}, 1^{−} ⊕ 10^{−} ⊕ 14^{−}) − 5
= dim_{k} Ext_{kĜ}^{1}(4^{+} ⊗ 4^{−}, 5^{+} ⊗ 5^{−}) − 5
= dim_{k} Ext_{kĜ}^{1}(4^{+} ⊗ 5^{+}, 4^{+} ⊗ 5^{−}) − 5
= dim_{k} Ext_{kĜ}^{1}(4^{+} ⊕ 16^{+}, 4^{−} ⊕ 16^{−}) − 5
= 1. □

This completes the calculation of dim_{k} Ext_{kĜ}^{1}(V, W) for V and W simple kĜmodules. The results of Sections 11 and 15 and Proposition 12.1 are summarised in Table 7.

16 Some projective kG-modules

16.1 The structure of P_{14}^{+}

Since P_{14}^{+} = (P_{4^{+}})_{H} ↑^{G}, we have a filtration of P_{14}^{+} whose quotients are (by Section 8) 4^{+}_{H} ↑^{G}, 1^{−}_{H} ↑^{G}, 4^{+}_{H} ↑^{G}. The Loewy series for these quotients are given in Sections 9.2 and 14. Combining this with the information given by Section 12.1(iii) and using Theorem 8.3, it is easy to see that the Loewy series for P_{14}^{+} is as given in Theorem 1.1, and that our filtration is as in Figure 7.

16.2 The structure of P_{1^{+}}

Since P_{1^{+}} = (P_{1^{+}})_{H} ↑^{G}, we have a filtration of P_{1^{+}} whose quotients are 1^{+}_{H} ↑^{G}, 4^{−}_{H} ↑^{G}, 1^{+}_{H} ↑^{G}. Using the information given in Section 16.1 and Table 7, we deduce that P_{1^{+}} has Loewy series as given in Theorem 1.1, and that our filtration is as in Figure 8.

1^{+} 5^{+} 10^{+} 14^{+} 25^{+} 1^{−} 5^{−} 10^{−} 14^{−} 25^{−} 81^{+} 81^{−} 4^{+} 16^{+} 40^{+} 4^{−} 16^{−} 40^{−}
1^{+} 1 1
5^{+}  1 1 1 1 1
10^{+}  1 1 1 1
14^{+}  1 1 1
25^{+}  1 1
1^{−} 1 1
5^{−} 1 1 1 1 1
10^{−}  1 1 1 1
14^{−} 1 1 1
25^{−}  1 1

81^{+}

81^{−}

4^{+}  1 2 1
16^{+}  2 1 1
40^{+}  1 1
4^{−}  1 2 1
16^{−}  2 1 1
40^{−}  1 1

Table 7: dim_{k} Ext_{kĜ}^{1}(V, W) for V and W simple (Only non-zero entries are shown)

14^{+}

5^{−} 10^{−} 1^{−}
1^{+} 10^{+} 14^{+} 25^{+} 5^{+}  14^{+}
5^{−} 10^{−}  1^{−} 14^{−}  5^{−} 10^{−}
14^{+}  5^{+} 1^{+} 10^{+} 14^{+} 25^{+}
1^{−}  5^{−} 10^{−}
14^{+}

Figure 7: The filtration of P_{14}^{+} 1^{+}

5^{−}  14^{−}
1^{+} 14^{+}  5^{+} 10^{+}  1^{+}
5^{−} 1^{−} 10^{−} 14^{−} 25^{−} 5^{−}
1^{+} 5^{+} 10^{+} 1^{+} 14^{+}
14^{−}  5^{−}

1^{+}

Figure 8: The filtration of P_{1^{+}}

5^{+}
1^{−} 10^{−} 14^{−} 25^{−}
5^{+} 5^{+} 5^{+} 10^{+}
1^{−} 10^{−} 14^{−} 25^{−}
5^{+}

Figure 9: Loewy series for 5^{+}_{H} ↑^{G}

17 Some more induced modules

17.1 The module 5^{+}_{H} ↑^{G}

By Brauer characters, the module 5^{+}_{H} ↑^{G} has composition factors 1^{−}+1^{−}+5^{+}+ 5^{+} + 5^{+} + 5^{+} + 5^{+} + 10^{+} + 10^{−} + 10^{−} + 14^{−} + 14^{−} + 25^{−} + 25^{−}. By Frobenius reciprocity, S_{1}(5^{+}_{H} ↑^{G}) ≅ L_{1}(5^{+}_{H} ↑^{G}) ≅ 5^{+}. Also by Brauer characters we have P_{5^{+}} = (P_{5^{+}})_{H} ↑^{G}, and so P_{5^{+}} has a filtration whose quotients are 5^{+}_{H} ↑^{G}, 5^{−}_{H} ↑^{G}, 5^{+}_{H} ↑^{G}. Since L_{2}(P_{5^{+}}) ≅ 1^{−} ⊕ 5^{−} ⊕ 10^{−} ⊕ 14^{−} ⊕ 25^{−} by Table 7, we have L_{2}(5^{+}_{H} ↑^{G}) ≅ 1^{−} ⊕ 10^{−} ⊕ 14^{−} ⊕ 25^{−}. Now by Sections 9.2, 9.6 and 9.7, P_{5^{+}}, and hence 5^{+}_{H} ↑^{G}, has quotients whose diagrams are

\begin{matrix}5^{+} \\ / \ \\ 1^{−} & 14^{−} \\ \ / \\ 5^{+} \end{matrix} , \begin{matrix}5^{+} \\ / \ \\ 1^{−} & 25^{−} \\ \ / \\ 5^{+} \end{matrix} , and \begin{matrix}5^{+} \\ / \ \\ 10^{−} & 25^{−} \\ \ / \\ 5^{+} \end{matrix} .

Hence there are at least three copies of 5^{+} in L_{3}(5^{+}_{H} ↑^{G}). This forces 5^{+}_{H} ↑^{G} to have Loewy length five, and the Loewy series given in Figure 9.

17.2 The modules 10_{1}^{+}_{H} ↑^{G} and 10_{2}^{−}_{H} ↑^{G}

By Brauer characters, the module (10_{1}^{+}_{H} ↑^{G}).e_{0} has composition factors 1^{+} + 1^{−} + 1^{−} + 5^{+} + 5^{−} + 5^{−} + 5^{−} + 5^{−} + 10^{+} + 10^{−} + 10^{−} + 14^{+} + 14^{+} + 14^{+} + 14^{+} + 25^{+} + 25^{+} + 25^{+}. By Frobenius reciprocity,

S_{1}(10_{1}^{+}_{H} ↑^{G}).e_{0} ≅ L_{1}(10_{1}^{+}_{H} ↑^{G}).e_{0} ≅ 14^{+} ⊕ 25^{+}.

Also, (P_{10_{2}}^{+})_{H} ↑^{G} .e_{0} ≅ P_{14}^{+} ⊕ P_{25}^{+} has a filtration whose quotients are

(10_{1}^{+}_{H} ↑^{G}).e_{0}, (10_{2}^{−}_{H} ↑^{G}).e_{0}, (10_{1}^{+}_{H} ↑^{G}).e_{0}.

By Brauer characters, the module (10_{2}^{−}_{H} ↑^{G}).e_{0} has composition factors 5^{+} + 5^{+} + 10^{+} + 10^{+} + 10^{+} + 10^{+} + 10^{−} + 10^{−} + 10^{−} + 10^{−} + 10^{−} + 14^{−} + 25^{+} + 25^{−} + 25^{−}. By Frobenius reciprocity,

S_{1}(10_{2}^{−}_{H} ↑^{G}).e_{0} ≅ L_{1}(10_{2}^{−}_{H} ↑^{G}).e_{0} ≅ 10^{−}.

Also, (P_{10_{2}}^{−})_{H} ↑^{G} .e_{0} ≅ P_{10}^{−} has a filtration whose quotients are

(10_{2}^{−}_{H} ↑^{G}).e_{0}, (10_{1}^{+}_{H} ↑^{G}).e_{0}, (10_{2}^{−}_{H} ↑^{G}).e_{0}.

Lemma 17.1 We have L_{2}(10_{1}^{+}_{H} ↑^{G}).e_{0} ≅ 1^{−} ⊕ 5^{−} ⊕ 5^{−} ⊕ 10^{−}.

Proof : Applying Landrock’s lemma to the Loewy structure of P_{1^{+}} given in Section 16.2, we deduce that there is only one copy of 1^{−} in L_{2}(P_{14}^{+} ⊕ P_{25}^{+}) and two copies in L_{4}(P_{14}^{+} ⊕ P_{25}^{+}). Since only one of the latter pair can come from the first (10_{1}^{+}_{H} ↑^{G}).e_{0} in our filtration, the other must come from the second copy. This forces at least one of 14^{+} and 25^{+} from L_{1} of this second copy to occur in L_{3}(P_{14}^{+} ⊕P_{25}^{+}), and hence the 10^{−} in L_{1}(10_{2}^{−}_{H} ↑^{G}).e_{0} occurs in L_{2}(P_{14}^{+} ⊕ P_{25}^{+}) in the filtration. By Table 7, this completes the proof of the lemma. □

Lemma 17.2 We have ( \begin{matrix}10 & − \\ | \\ 25 & + \end{matrix} )_{G} ↓_{H} ≅ 15^{+} ⊕ \begin{matrix}10_{2} & − \\ | \\ 10_{1} & + \end{matrix} .

Proof : By Section 10.6,

(5^{−} ⊕ \begin{matrix}25 & + \\ / \ \\ 5^{−} & 10^{−} \\ \ / \\ 25 & + \end{matrix} )_{G} ↓_{H} ≅ (5^{+} ⊗ 14^{−})↓_{H}

≅ 5^{+} ⊗ (10_{1}^{−} ⊕ 4^{−})
≅ (5^{+} ⊗ 10_{1}^{−}) ⊕ (5^{+} ⊗ 4^{−})
≅ (5^{−} ⊕ P_{10_{1}}^{+} ⊕ 15^{+}) ⊕ (5^{−} ⊕ 15^{+}).

Now compare composition factors. □

\begin{matrix}14^{+} & 25^{+} \\ 1^{−} & 5^{−} & 5^{−} & 10^{−} \\ 1^{+} & 5^{+} & 10^{+} & 14^{+} & 14^{+} & 25^{+} \\ 1^{−} & 5^{−} & 5^{−} & 10^{−} \\ 14^{+} & 25^{+} \end{matrix} .

Figure 10: Loewy series of (10_{1}^{+}_{H} ↑^{G}).e_{0}

Lemma 17.3 We have L_{2}(10_{2}^{−}_{H} ↑^{G}).e_{0} ≅ 5^{+} ⊕ 10^{+}.

Proof : From Table 7, we see that L_{2}(P_{10}^{−}) ≅ 5^{+} ⊕10^{+} ⊕14^{+} ⊕25^{+}. Thus we must show that the whole of L_{1}(10_{1}^{+}_{H} ↑^{G}).e_{0} ≅ 14^{+} ⊕25^{+} appears in L_{2}(P_{10}^{−}) in our filtration. Since 14^{+} is not a composition factor of (10_{2}^{−}_{H} ↑^{G}).e_{0}, that part is easy. The other follows from Lemma 17.2 and Frobenius reciprocity as follows.

(10_{2}^{−}_{H} ↑^{G}, \begin{matrix}10 & − \\ | \\ 25 & + \end{matrix} )_{G} = (10_{2}^{−}, 15^{+} ⊕ \begin{matrix}10_{2} & − \\ | \\ 10_{1} & + \end{matrix} )_{H} = 0. □

Proposition 17.4 The module (10_{1}^{+}_{H} ↑^{G}).e_{0} has the Loewy series given in Figure 10.

Proof : From Section 6.1 and the tensor product information given in Section 10, we see that L_{3}(P_{25}^{+}) has exactly one copy of 25^{+} in it. Hence from the structure of P_{14}^{+} given in Section 16.1, L_{3}((P_{10_{1}}^{+})_{H} ↑^{G}) = L_{3}(P_{14}^{+} ⊕ P_{25}^{+}) has

two copies of 25^{+} in it. Since the copy in P_{25}^{+} occurs in a quotient \begin{matrix}25 & + \\ / \ \\ 5^{−} & 10^{−} \\ \ / \\ 25 & + \end{matrix}

it follows from Lemma 17.2 and Frobenius reciprocity that it does not occur in (10_{1}^{+}_{H} ↑^{G}).e_{0}. Hence there is exactly one copy of 25^{+} in L_{3}(10_{1}^{+}_{H} ↑^{G}).e_{0}.

Applying Landrock’s lemma to P_{14}^{+}, we see that there are three copies of 14^{+} in L_{3}(10_{1}^{+}_{H} ↑^{G}).e_{0}. By Sections 12.1, 10.2 and 12.2(v), there is a 14^{+} in L_{3}(P_{25}^{+}) which is glued below a 10^{−} in L_{2}(P_{25}^{+}). Hence this comes from L_{1} of the second copy of (10_{1}^{+}_{H} ↑^{G}).e_{0} in our filtration, and the other two must come from L_{3} of the first copy. Hence there are exactly two copies of 14^{+} in L_{3}(10_{1}^{+}_{H} ↑^{G}).e_{0}.

Now by self duality, Lemma 17.1, and an examination of the set of composition factors, it follows that the Loewy length of (10_{1}^{+}_{H} ↑^{G}).e_{0} is five, and hence the Loewy structure is as given. □

17.3 Another permutation module

Denote by M_{80} the permutation module on the 80 cosets of 3^{3} : (A_{4} × 2) in G, and by M′_{80} the permutation module on the 80 cosets of the subgroup 3^{3} : S_{4} of G not contained in G′. Then M_{80} and M′_{80} have ordinary characters 1^{+}+15_{1}^{−}+24^{+}+(5_{1}+5_{2})+15_{1}^{+}+15_{1}^{+} and 1^{+}+15_{1}^{−}+24^{+}+(5_{1}+5_{2})+15_{1}^{−}+ 15_{1}^{−}. We do not need to work out which is which. Each has M_{40_{2}} as a direct summand, and the other summand of one is a module M_{40_{3}} and of the other, M_{40_{3}} ⊗ 1^{−}. By a theorem of Sawada [24, Proposition 2.8 and Theorem 3.10], the permutation module for G′ on the cosets of a Sylow 3-subgroup is a direct sum of 5 indecomposable modules, each self dual with simple socle and simple head. The simple socles account for the simple G′-modules once each. Since we have already used up 1, 10 and 14 in the socles of summands of M_{40_{1}} ↓_{G′} and M_{40_{2}} ↓_{G′}, the socle of M_{40_{3}} ↓_{G′} must be either 5 or 25. Looking at composition factors, it is 5. We choose the signs so that M_{40_{3}} is the module with socle and head isomorphic to 5^{+}. Since it is a self dual module with composition factors 5^{+} + 5^{+} + 5^{+} + 5^{−} + 10^{−} + 10^{−}, using Table 7 we see that there is only one possibility for its structure. Namely

M_{40_{3}} = \begin{matrix}5^{+} \\ / \ \\ \begin{matrix}10 & − \\ | \\ 5^{+} \\ | \\ 10 & − \end{matrix} & 5^{−} \\ \ / \\ 5^{+} \end{matrix} .

17.4 The structure of (10_{2}^{−}_{H} ↑^{G}).e_{0}

We may now complete the determination of the structure of (10_{2}^{−}_{H} ↑^{G}).e_{0} started in Section 17.2. By Section 17.3, P_{10}^{−}, and hence (10_{2}^{−}_{H} ↑^{G}).e_{0} has a uniserial quotient with composition factors (from the top) 10^{−}, 5^{+}, 10^{−}, 5^{+}. Since this exhausts the set of composition factors isomorphic to 5^{+}, we see that Rad((10_{2}^{−}_{H} ↑^{G}).e_{0})/Soc((10_{2}^{−}_{H} ↑^{G}).e_{0}) decomposes as a direct sum of

\begin{matrix}5^{+} \\ | \\ 10 & − \\ | \\ 5^{+} \end{matrix} and a self dual module X with L_{1}(X) ≅ S_{1}(X) ≅ 10^{+} and composition

factors 10^{+} + 10^{+} + 10^{+} + 10^{+} + 10^{−} + 10^{−} + 14^{−} + 25^{+} + 25^{−} + 25^{−}. Now by Landrock’s lemma, there is a copy of 14^{−} in L_{3}(P_{10}^{−}). Since there is no copy of 14^{−} as a composition factor of (10_{1}^{+}_{H} ↑^{G}).e_{0}, there must be a copy of 14^{−} in L_{3}(10_{2}^{−}_{H} ↑^{G}).e_{0}. Since this is the only copy of 14^{−} in (10_{2}^{−}_{H} ↑^{G}).e_{0},

10^{−}
5^{+} 10^{+}
10^{−} 10^{−} 14^{−} 25^{−}
5^{+} 10^{+} 10^{+} 25^{+}
10^{−} 25^{−}
10^{+}
10^{−}

Figure 11: Loewy series of (10_{2}^{−}_{H} ↑^{G}).e_{0}

we see that Rad(X)/Soc(X) decomposes as a direct sum of 14^{−} and a self dual module Y with composition factors 10^{+} + 10^{+} + 10^{−} + 10^{−} + 25^{+} + 25^{−} + 25^{−}. Now looking at Table 7, we see that there are three possibilities for Soc(Y), namely 10^{−}, 25^{−} or 10^{−} ⊕ 25^{−}. In the first case, Table 7 shows that there is no possible arrangement of the remaining composition factors of Y. In the second case, we see that Y would have to be uniserial, the order of the composition factors being 25^{−}, 10^{+}, 10^{−}, 25^{+}, 10^{−}, 10^{+}, 25^{−}. This in turn would imply that X had a uniserial quotient with composition factors (in order from the top) 10^{+}, 25^{−}, 10^{+}, 10^{−}, 25^{+}, 10^{−}, 10^{+}, 25^{−}. By looking at the projective modules over H (see Section 8), we see that the restriction of this module to H has at least two copies of 10_{2}^{+} in its head. By Frobenius reciprocity, this implies that there are two dimensions of homomorphisms from (10_{2}^{+}_{H} ↑^{G}).e_{0} to this module, contradicting what we know of the structure of (10_{2}^{+}_{H} ↑^{G}).e_{0}. This contradiction shows that the third possibility holds. Thus the Loewy structure of Y is

\begin{matrix}10^{−} & 25^{−} \\ 10^{+} & 10^{+} & 25^{+} \\ 10^{−} & 25^{−} \end{matrix} .

Fitting all this information together, we find that the Loewy series of the module (10_{2}^{−}_{H} ↑^{G}).e_{0} is as given in Figure 11.

18 The remaining projective kG-modules

18.1 The module P_{5^{+}}

Since (P_{5^{+}})_{G} = (P_{5^{+}})_{H} ↑^{G}= ( \begin{matrix}5^{+} \\ | \\ 5^{−} \\ | \\ 5^{+} \end{matrix} )_{H} ↑^{G}, we need to know the Loewy structure

of ( \begin{matrix}5^{+} \\ | \\ 5^{−} \end{matrix} )_{H} ↑^{G}. By Section 6.1, there is a copy of 5 in L_{3}(P_{25}) for G′. Hence

there is a copy of 5^{+} in L_{3}(P_{25}^{+}) for G, and hence by Landrock’s lemma, there is a copy of 25^{+} in L_{3}(P_{5^{+}}). Also applying Landrock’s lemma to the Loewy structure of P_{14}^{+} and P_{1^{+}} given in Sections 16.1 and 16.2, we see that there is a copy of each of 1^{+} and 14^{+} in L_{3}(P_{5^{+}}).

Now by Frobenius reciprocity and Section 17.1, we have ( \begin{matrix}5^{+} \\ | \\ 5^{−} \end{matrix} )_{G} ↓_{H}=

( \begin{matrix}5^{+} \\ | \\ 5^{−} \end{matrix} )_{H}. Hence ( \begin{matrix}5^{+} \\ | \\ 5^{−} \end{matrix} )_{H} ↑^{G}= ( \begin{matrix}5^{+} \\ | \\ 5^{−} \end{matrix} )_{G} ↓_{H}↑^{G}= ( \begin{matrix}5^{+} \\ | \\ 5^{−} \end{matrix} )_{G} ⊗ 1^{+}_{H} ↑^{G}

= \begin{matrix}5^{+} \\ | \\ 5^{−} \end{matrix} ⊗ 1^{+} ; | ; \begin{matrix}5^{−} \\ / \ \\ 1^{+} & 14^{+} \\ \ / \\ 5^{−} \end{matrix} ; | ; 1^{+} .

by Section 9.2. Hence this module has a filtration as follows.

5^{+} ⊗ 1^{+}

(5^{+} ⊗ 5^{−}) ⊕ (5^{−} ⊗ 1^{+})
(5^{+} ⊗ (1^{+} ⊕ 14^{+})) ⊕ (5^{−} ⊗ 5^{−})
(5^{+} ⊗ 5^{−}) ⊕ (5^{−} ⊗ (1^{+} ⊕ 14^{+}))
(5^{+} ⊗ 1^{+}) ⊕ (5^{−} ⊗ 5^{−})

5^{−} ⊗ 1^{+}

5^{+}

1^{−} 10^{−} 14^{−} 25^{−} 5^{−}
5^{+} 5^{+} 5^{+} 10^{+} 1^{+} 10^{+} 14^{+} 25^{+}  5^{+}
1^{−} 10^{−} 14^{−} 25^{−} 5^{−} 5^{−} 5^{−} 10^{−} 1^{−} 10^{−} 14^{−} 25^{−}
5^{+}  1^{+} 10^{+} 14^{+} 25^{+} 5^{+} 5^{+} 5^{+} 10^{+}
5^{−}  1^{−} 10^{−} 14^{−} 25^{−}

5^{+}

Figure 12: The filtration of P_{5^{+}}

This implies that there are at least two copies of 10^{+} in L_{3}(( \begin{matrix}5^{+} \\ | \\ 5^{−} \end{matrix} )_{H} ↑^{G}),

using the results of Section 10. Combining all the above information with the

information in Section 17.1, we see that the Loewy series for ( \begin{matrix}5^{+} \\ | \\ 5^{−} \end{matrix} )_{H} ↑^{G} is

given in terms of our filtration as shown in the first two columns of Figure 12.

Again applying Landrock’s lemma to P_{14}^{−}, we see that there are two copies of 14^{−} in L_{4}(P_{5^{+}}). This implies that the head of the second 5^{+}_{H} ↑^{G} in the filtration of P_{5^{+}} must be in L_{3}(P_{5^{+}}), and so the Loewy series for P_{5^{+}} is as given in Theorem 1.1, and our filtration is as shown in Figure 12.

18.2 The modules P_{10}^{−} and P_{25}^{+}

We now continue the analysis started in Section 17.2. Recall that

( \begin{matrix}10_{1} & + \\ | \\ 10_{2} & − \\ | \\ 10_{1} & + \end{matrix} )_{H} ↑^{G} .e_{0} ≅ P_{14}^{+} ⊕ P_{25}^{+} and ( \begin{matrix}10_{2} & − \\ | \\ 10_{1} & + \\ | \\ 10_{2} & − \end{matrix} )_{H} ↑^{G} .e_{0} ≅ P_{10}^{−},

and that the Loewy structures of (10_{1}^{+}_{H} ↑^{G}).e_{0} and (10_{2}^{−}_{H} ↑^{G}).e_{0} are given in Sections 17.2 and 17.4. Throughout this section, it will help to refer to Figures 13 and 14 at the end of the section.

Lemma 18.1 We have L_{3}(P_{25}^{+}) ≅ 5^{+} ⊕ 10^{+} ⊕ 14^{+} ⊕ 25^{+}.

Proof : By Section 16.1, it suffices to show that L_{3}(P_{14}^{+} ⊕ P_{25}^{+}) ≅ 1^{+} ⊕ 5^{+} ⊕ 5^{+} ⊕10^{+} ⊕10^{+} ⊕14^{+} ⊕14^{+} ⊕14^{+} ⊕25^{+} ⊕25^{+}. Since we already know L_{2}(P_{14}^{+} ⊕ P_{25}^{+}) by Table 7, Theorem 8.3 shows that L_{3}(P_{14}^{+} ⊕P_{25}^{+}) is isomorphic to some summand of L_{3}(10_{1}^{+}_{H} ↑^{G}).e_{0} ⊕ L_{2}(10_{2}^{−}_{H} ↑^{G}).e_{0} ⊕ L_{1}(10_{1}^{+}_{H} ↑^{G}).e_{0}; we need to show that it is the whole of this module. Thus it suffices to show that each summand is present at least the number of times shown. For the modules 1^{+}, 5^{+} and 14^{+}, this follows by applying Landrock’s lemma and Sections 16.2, 18.1 and 16.1. For the module 10^{+}, it follows by applying Landrock’s lemma to the quotient of P_{10}^{+} given by (10_{2}^{+}_{H} ↑^{G}).e_{0}, see Section 17.4. For the module 25^{+}, we argue as follows. There is one copy of 25^{+} in L_{3}(10_{1}^{+}_{H} ↑^{G}).e_{0}. Moreover,

by Lemma 17.2 the module \begin{matrix}25 & + \\ / \ \\ 5^{−} & 10^{−} \\ \ / \\ 25 & + \end{matrix} appearing as a summand of 5^{+} ⊗ 14^{−}

(Section 10.6) is not a quotient of (10_{1}^{+}_{H} ↑^{G}).e_{0}, and so gives us another copy of 25^{+} in L_{3}(P_{14}^{+} ⊕ P_{25}^{+}). □

Lemma 18.2 L_{3}(P_{10}^{−}) ≅ 1^{−} ⊕ 5^{−} ⊕ 5^{−} ⊕ 10^{−} ⊕ 10^{−} ⊕ 10^{−} ⊕ 10^{−} ⊕ 14^{−} ⊕ 25^{−}.

Proof : By Landrock’s lemma applied to Sections 16.1, 16.2 and 18.1 , and Lemma 18.1, we know all of this except the copies of 10^{−}. From Sections 9.4, 10.2 and 17.4, there are the following quotients of P_{10}^{−}.

\begin{matrix}10 & − \\ / \ \\ 5^{+} & 14^{+} \\ \ / \\ 10 & − \end{matrix} , \begin{matrix}10 & − \\ | \\ 25 & + \\ | \\ 10 & − \end{matrix} , \begin{matrix}10 & − \\ / \ \\ \begin{matrix}5^{+} \\ | \\ 10 & − \end{matrix} & \begin{matrix}10 & + \\ | \\ 10 & − \end{matrix} \end{matrix} .

This is sufficient to guarantee at least four copies of 10^{−} in L_{3}(P_{10}^{−}). The fact that there are at most four follows from Theorem 8.3 and the structures of (10_{1}^{+}_{H} ↑^{G}).e_{0} and (10_{2}^{−}_{H} ↑^{G}).e_{0} given in Sections 17.2 and 17.4. □

Lemma 18.3 L_{4}(P_{25}^{+}) ≅ 1^{−} ⊕ 5^{−} ⊕ 5^{−} ⊕ 10^{−} ⊕ 10^{−} ⊕ 25^{−}.

Proof : We now have enough information accumulated in this section to see that Theorem 8.3 gives

L_{4}(P_{14}^{+} ⊕ P_{25}^{+}) ≅ L_{2}(10_{1}^{+}_{H} ↑^{G}).e_{0} ⊕ L_{3}(10_{2}^{−}_{H} ↑^{G}).e_{0} ⊕ L_{4}(10_{1}^{+}_{H} ↑^{G}).e_{0}.

Now use Section 16.1. □

Lemma 18.4 The Loewy series of ( \begin{matrix}10_{1} & + \\ | \\ 10_{2} & − \end{matrix} )_{H} ↑^{G} .e_{0} and ( \begin{matrix}10_{2} & − \\ | \\ 10_{1} & + \end{matrix} )_{H} ↑^{G} .e_{0}

are given by the first two columns and the last two columns respectively of Figure 13.

14^{+} 25^{+}
1^{−} 5^{−} 5^{−} 10^{−}  10^{−}
1^{+} 5^{+} 10^{+} 14^{+} 14^{+} 25^{+}  5^{+} 10^{+}  14^{+} 25^{+}
1^{−} 5^{−} 5^{−} 10^{−}  10^{−} 10^{−} 14^{−} 25^{−}  1^{−} 5^{−} 5^{−} 10^{−}
14^{+} 25^{+}  5^{+} 10^{+} 10^{+} 25^{+} 1^{+} 5^{+} 10^{+} 14^{+} 14^{+} 25^{+}
10^{−} 25^{−}  1^{−} 5^{−} 5^{−} 10^{−}
10^{+}  14^{+}

10^{−}

25^{+}
Figure 13: The filtration of P_{14}^{+} ⊕ P_{25}^{+}

Proof : For ( \begin{matrix}10_{1} & + \\ | \\ 10_{2} & − \end{matrix} )_{H} ↑^{G} .e_{0} , this follows rather easily from the information we have so far. For ( \begin{matrix}10_{2} & − \\ | \\ 10_{1} & + \end{matrix} )_{H} ↑^{G} .e_{0}, we need to argue a little further. Everything follows from Landrock’s lemma apart from the position of the 10^{+} from L_{3}(10_{1}^{+}_{H} ↑^{G}).e_{0}, the 10^{−} from L_{4} and the 25^{+} from L_{5}. In particular, the

14^{+} in L_{5}(10_{1}^{+}_{H} ↑^{G}).e_{0} lands in L_{6}(( \begin{matrix}10_{2} & − \\ | \\ 10_{1} & + \end{matrix} )_{H} ↑^{G}).e_{0}. Since (10_{1}^{+}_{H} ↑^{G}).e_{0}

does not have a submodule of shape \begin{matrix}10 & − \\ | \\ 25 & + \end{matrix} (i.e., the 10^{−} is glued above the 14^{+}, see Lemma 17.2) this forces the 10^{−} to be in L_{5}(( \begin{matrix}10_{2} & − \\ | \\ 10_{1} & + \end{matrix} )_{H} ↑^{G}).e_{0}. This in turn forces the 10^{+} to be in L_{4}(( \begin{matrix}10_{2} & − \\ | \\ 10_{1} & + \end{matrix} )_{H} ↑^{G}).e_{0}. The 25^{+} is in L_{8} since the module does not have 10^{−} as a submodule.

Note that the diagram for ( \begin{matrix}10_{2} & − \\ | \\ 10_{1} & + \end{matrix} )_{H} ↑^{G} .e_{0} is slightly misleading in the following sense. It looks as though the 14^{+} in L_{5}(10_{1}^{+}_{H} ↑^{G}).e_{0} has “slipped past” the 10^{−} in L_{7}(10_{2}^{−}_{H} ↑^{G}).e_{0}, which we know it cannot do. What has in fact happened is that it has “slipped past” the diagonal of this 10^{−} and the 10^{−} in L_{4}(10_{1}^{+}_{H} ↑^{G}).e_{0}. □

10^{−}

5^{+} 10^{+}  14^{+} 25^{+}
10^{−} 10^{−} 14^{−} 25^{−} 1^{−} 5^{−} 5^{−} 10^{−} 10^{−}
5^{+} 10^{+} 10^{+} 25^{+} 1^{+} 5^{+} 10^{+} 14^{+} 14^{+} 25^{+}  5^{+} 10^{+}
10^{−} 25^{−}  1^{−} 5^{−} 5^{−} 10^{−}  10^{−} 10^{−} 14^{−} 25^{−}
10^{+}  14^{+}  5^{+} 10^{+} 10^{+} 25^{+}
10^{−}  10^{−} 25^{−}
25^{+}  10^{+}
10^{−}

Figure 14: The filtration of P_{10}^{−}

This is now enough to determine the Loewy series for P_{14}^{+} ⊕ P_{25}^{+}, which is given in terms of our filtration in Figure 13. This completes the proof of the Loewy series for P_{25}^{+} given in Theorem 1.1, by subtracting off the Loewy series for P_{14}^{+} given in Section 16.1. Now using Landrock’s lemma, we can place all but the copies of 10^{+} and 10^{−} coming from the second copy of (10_{2}^{−}_{H} ↑^{G}).e_{0} in our filtration of P_{10}^{−}. The copies in the first three Loewy layers of (10_{2}^{−}_{H} ↑^{G}).e_{0} are forced to be placed as shown in Figure 14, by the structure of (10_{2}^{−}_{H} ↑^{G} ).e_{0} described in Section 17.4. The rest are as shown simply because there is nothing for them to get stuck under! Thus the Loewy series for P_{10}^{+} is as given in Theorem 1.1, by changing the signs on the Loewy series for P_{10}^{−} given in Figure 14.

Appendix: Characters and decomposition numbers

In this Appendix, we give the ordinary and 3-modular characters, decomposition numbers and Cartan matrices used in this paper, with the exception of those which can be found in the Atlas [12] (i.e., the ordinary characters of G′, G, Ĝ′ and Ĝ). Note that we follow ‘Atlas’ notation and conventions where appropriate. In particular, in stands for i\sqrt{n} (so that for example i3 stands for i\sqrt{3}), b5 stands for (−1+i\sqrt{5})/2, and ∗ stands for the algebraic conjugate of the adjacent quadratic irrationality.

The information in the next few tables comes from Thackray’s thesis [26].

3-modular characters of G′, G, Ĝ′ and Ĝ

25920 576 96 48 8 5  720 48 48 16 4 5
p power A A A B A  A A B B A AC
p′ part A A A A A  A A A A A AC
ind 1A 2A 2B 4A 4B 5A fus ind 2C 2D 4C 4D 8A 10A
+ 1 1 1 1 1 1 : ++ 1 1 1 1 1 1
+ 5 −3 1 1 −1 0 : ++ −3 1 3 −1 1 2
+ 10 2 −2 2 0 0 : ++ 2 −2 4 0 0 2
+ 14 6 2 −2 0 −1 : ++ 6 2 4 0 0 1
+ 25 −7 1 −3 1 0 : ++ 7 −1 −1 −1 1 2
+ 81 9 −3 −3 −1 1 : ++ 9 −3 3 −1 1 −1
ind 1 1 4 4 8 5 fus ind 4 2 8 8 8 20
2  4  10  8  8 20
− 4 0 0 2 0 −1 : oo 0 0 2i2 0 −i2 i5
− 16 0 0 0 0 1 : oo 0 0 4i2 0 0 i5
− 40 0 0 −4 0 0 : oo 0 0 4i2 0 0 0

Decomposition matrix and Cartan matrix for G′ and Ĝ′ mod 3

1 5 10 14 25 81 4 16 40 1 1 5_{1} 1 5_{2} 1 6 1 1 10_{1}  1 10_{2}  1 15_{1} 1 1 15_{2} 1  1 20_{1} 1 1 1 24  1 1 30_{1} 1  1 30_{2} 1 1 1 1 30_{3} 1 1 1 1 40_{1} 1 1  1 40_{2} 1 1  1 45_{1}  2  1 45_{2}  2  1 60_{1} 1 2 1 1 1 64_{1} 1 2 1 1 81  1 4_{1}  1 4_{2}  1 20_{2}  1 1 20_{3}  1 1 20_{4}  1 1 20_{5}  1 1 20_{6}  1 1 36_{1}  1 2 36_{2}  1 2 60_{2}  1 1 1 60_{3}  1 1 1 60_{4}  1 1 1 64_{2}  2 1 1 80  2 2 1  1 5 10 14 25 81 4 16 40 1 7 6 3 5 1 5 6 15 9 6 6 10 3 9 21 6 9 14 5 6 6 7 2 25 1 6 9 2 7 81  1 4  20 18 7 16  18 21 6 40  7 6 5

Decomposition matrix for G and Ĝ mod 3

1^{+}1^{−}5^{+}5^{−}10^{+}10^{−}14^{+}14^{−}25^{+}25^{−}81^{+}81^{−}4^{+}4^{−}16^{+}16^{−}40^{+}40^{−}
1^{+}  1
1^{−}  1
(5_{1} + 5_{2})  1 1
6^{+} 1 1
6^{−}  1 1
(10_{1} + 10_{2})  1 1
15_{1}^{+}  1 1
15_{1}^{−}  1  1
15_{2}^{+}  1  1
15_{2}^{−} 1  1
20^{+} 1  1  1
20^{−}  1 1  1
24^{+}  1 1
24^{−}  1  1
30_{1}^{+}  1  1
30_{1}^{−}  1  1
(30_{2} + 30_{3}) 1 1 1 1 1 1 1 1
(40_{1} + 40_{2})  1 1 1 1  1 1
(45_{1} + 45_{2})  2 2  1 1
60_{1}^{+}  1 1 1  1 1  1
60_{1}^{−} 1 1 1 1  1  1
64_{1}^{+}  1 1 1 1  1
64_{1}^{−}  1  1 1  1  1
81^{+}  1
81^{−}  1
(4_{1} + 4_{2})  1 1
20_{2}^{+}  1 1
20_{2}^{−} 1 1
(20_{3} + 20_{4})  1 1 1 1
(20_{5} + 20_{6})  1 1 1 1
(36_{1} + 36_{2})  1 1 2 2
60_{2}^{+}  1  1 1
60_{2}^{−}  1 1  1
(60_{3} + 60_{4})  1 1 1 1 1 1
64_{2}^{+}  1 1 1  1
64_{2}^{−}  1 1  1 1
80^{+}  1 1 1 1 1
80^{−}  1 1 1 1  1

Cartan matrix for G and Ĝ mod 3

1^{+} 1^{−} 5^{+} 5^{−} 10^{+} 10^{−} 14^{+} 14^{−} 25^{+} 25^{−} 81^{+} 81^{−} 4^{+} 4^{−} 16^{+} 16^{−} 40^{+} 40^{−}
1^{+} 6 1 2 4 2 1 2 3 0 1
1^{−} 1 6 4 2 1 2 3 2 1 0
5^{+} 2 4 10 5 4 5 2 4 2 4
5^{−} 4 2 5 10 5 4 4 2 4 2
10^{+} 2 1 4 5 12 9 2 4 4 5
10^{−} 1 2 5 4 9 12 4 2 5 4
14^{+} 2 3 2 4 2 4 6 1 2 0
14^{−} 3 2 4 2 4 2 1 6 0 2
25^{+} 0 1 2 4 4 5 2 0 5 2
25^{−} 1 0 4 2 5 4 0 2 2 5
81^{+}  1
81^{−}  1
4^{+}  11 9 8 10 4 3
4^{−}  9 11 10 8 3 4
16^{+}  8 10 12 9 2 4
16^{−}  10 8 9 12 4 2
40^{+}  4 3 2 4 4 1
40^{−}  3 4 4 2 1 4

Ordinary characters of H′, H, Ĥ′ and Ĥ

960 192 96 16 12 16 8 5 5 12 12 12  48 16 48 16 4 6 4 6
p power A A A A A B A AAB AA AA  A A B B CAD A AC
p′ part A A A A A A A AAB AA AA  A A A A AAD A AC
ind 1A 2A2B2C3A4A4B 5A B∗ 6A 6BC∗∗fusind2D2E4C4D4E 6D 8A12A
+ 1 1 1 1 1 1 1 1 1 1 1 1 : ++ 1 1 1 1 1 1 1 1
+ 3 3 3−1 0−1−1−b5 ∗ 0 0 0 · + 0 0 0 0 0 0 0 0
|
+ 3 3 3−1 0−1−1 ∗−b5 0 0 0 ·
+ 4 4 4 0 1 0 0 −1 −1 1 1 1 : ++ 2 2 2 2 0 −1 0 −1
+ 5 5 5 1−1 1 1 0 0 −1 −1 −1 : ++−1−1−1−1 1 −1 1 −1
+ 5 −3 1 1 2 1−1 0 0 −2 0 0 : ++−3 1 3−1−1 0 1 0
o 5 −3 1 1−1 1−1 0 0 1 i3 −i3 · + 0 0 0 0 0 0 0 0
|
o 5 −3 1 1−1 1−1 0 0 1−i3 i3 ·
+ 10 2−2 2 1−2 0 0 0 1 −1 −1 : ++ 4 0 2−2 0 1 0 −1
+ 10 2−2−2 1 2 0 0 0 1 −1 −1 : ++ 2−2 0 1 0 −1
+ 15 −9 3−1 0−1 1 0 0 0 0 0 : ++ 3−1−3 1−1 0 1 0
+ 20 4−4 0−1 0 0 0 0 −1 1 1 : ++ 2 2−2−2 0 −1 0 1
ind 1 2 4 4 3 4 8 5 5 12 6 6fusind 4 2 8 8 8 12 8 24
2  6 4  10 10  8  8 24
− 4 0 0 0−2 2 0 −1 −1 0 0 0 : oo 0 02i2 0 0 0−i2−i2
− 12 0 0 0 0−2 0 b5 ∗ 0 0 0 · − 0 0 0 0 0 0 0 0
|
− 12 0 0 0 0−2 0 ∗ b5 0 0 0 ·
− 16 0 0 0−2 0 0 1 1 0 0 0 : oo 0 04i2 0 0 0 0 i2
− 20 0 0 0 2 2 0 0 0 0 0 0 : oo 0 02i2 0 0 0 i2−i2

3-modular characters of H′, H, Ĥ′ and Ĥ

960 192 96 16 16 8 5 5  48 16 48 16 4 4
p power A A A A B A A  A A B B C A
p′ part A A A A A A A  A A A A A A
ind 1A 2A 2B 2C 4A 4B 5A B∗ fus ind 2D 2E 4C 4D 4E 8A
+ 1 1 1 1 1 1 1 1 : ++ 1 1 1 1 1 1
+ 3 3 3 −1 −1 −1 −b5 ∗ · + 0 0 0 0 0 0
|
+ 3 3 3 −1 −1 −1 ∗ −b5 ·
+ 4 4 4 0 0 0 −1 −1 : ++ 2 2 2 2 0 0
+ 5 −3 1 1 1 −1 0 0 : ++ −3 1 3 −1 −1 1
+ 10 2 −2 2 −2 0 0 0 : ++ 4 0 2 −2 0 0
+ 10 2 −2 −2 2 0 0 0 : ++ 2 −2 4 0 0 0
+ 15 −9 3 −1 −1 1 0 0 : ++ 3 −1 −3 1 −1 1
ind 1 2 4 4 4 8 5 5 fus ind 4 2 8 8 8 8
2 10 10 8 8
− 4 0 0 0 2 0 −1 −1 : oo 0 0 2i2 0 0 −i2
− 12 0 0 0 −2 0 b5 ∗ · − 0 0 0 0 0 0
|
− 12 0 0 0 −2 0 ∗ b5 ·
− 16 0 0 0 0 0 1 1 : oo 0 0 4i2 0 0 0

Brauer trees: 1^{+}—5_{1}^{+}—4^{−} 5_{2}^{+}—(5_{3} + 5_{4})—5_{2}^{−}

10_{1}^{+}—20_{1}^{+}—10_{2}^{−} 4^{+}—20_{2}^{−}—16^{−} (and their negations)

Diagram of highest weights in P_{0}^{+}(Φ)

[figure: −ρ; 1; 5; 14; 4; 16; 40; 10; 25; 81; 0; ω_{2}; ω_{1}]

Figure 15: Diagram of highest weights in P_{0}^{+}(Φ)

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