The group ring of SL_{2}(2^{f}) over 2-adic integers.

Gabriele Nebe

Dedicated to Prof. K. W. Roggenkamp in occasion of his 60th birthday

Abstract: Let R = ℤ_{2}[ζ_{2^{f} −1}] and G = SL_{2}(2^{f}). The group ring RG is calculated nearly up to Morita equivalence. In particular the irreducible RG-lattices can be described purely combinatorically in terms of subsets of {1, …, f}.

1991 Mathematics Subject Classification: 20C05, 20C11

1 Introduction

The group G = SL_{2}(p^{f}) of all 2×2-matrices over the field k with p^{f} elements is one of the simplest examples of a nonabelian finite group of Lie type. Its representation theory in characteristic 0 was already investigated by I. Schur [13] and its modular representation theory is also well understood ([1], [2]). The next step is to describe the integral group ring RG of G when R is the ring of integers in a finite extension of the field ℚ_{l} of l-adic numbers, to bring together the characteristic 0 and the characteristic l information. If l ≠ p and l ≠ 2 then the defect groups of the ring direct summands of RG are cyclic, so RG is described by the general theory of blocks with cyclic defect groups ([10], [12], [7]). For odd primes p the Sylow 2-subgroups of G are dihedral groups and [10], Chapter VII investigates RG for l = 2. So the only remaining case is l = p, where the Sylow p-subgroups of G are elementary abelian of rank f. If f = 1 one again has the cyclic defect case and for f = 2 the group ring ℤ_{p}G is described up to Morita equivalence in [8]. In the present paper the remaining cases f ≥ 3 are treated for p = 2.

To find kG, one uses methods from the representation theory for groups of Lie type in defining characteristic. However these methods are not directly applicable for calculating RG, when R = ℤ_{2}[ζ_{2^{f} −1}] is the ring of integers in the unramified extension K of degree f of ℚ_{2}. The new idea used in this paper is to start from the explicit presentation of kG given in [6] and lift the generators of kG to generators of RG. The explicit knowledge of kG together with the decomposition numbers calculated in [4] and [5] do not seem to be sufficient to determine RG up to Morita equivalence. But they give enough information to describe the inclusion patterns of the irreducible RG-lattices (Theorem 3.15) as well as the endomorphism rings of the projective indecomposable RG-lattices (Theorem 3.12). In particular it turns out that the endomorphism ring of the projective cover of the trivial RG-module is isomorphic to the group ring of the Sylow 2-subgroup of G. From Theorem 3.15 one gets the following explicit description of the projections of RG onto the simple summands of KG. The simple kG-modules M_{I} are naturally indexed with the subsets I of {1, …, f} such that dim_{k}(M_{I}) = 2^{|I|}. Theorem. Let V be a simple KG-module of dimension n and let M_{I_{1}}, …, M_{I_{r}} be the 2-modular constituents of V of dimension n_{j} := 2^{|I_{j}|} = dim_{k}(M_{I_{j}}), 1 ≤ j ≤ r. Then there is a basis of V such that the corresponding matrix representation Δ_{V} satisfies

Δ_{V} (RG) = {(X_{ij})_{1≤i,j≤r} ∈ R^{n×n} | X_{ij} ∈ 2^{|I_{i}−I_{j}|}R^{n_{i}×n_{j}}}.

This is the first time that such a detailed description of an infinite series of p-adic group rings has been found, where not only the order but also the number of generators of the Sylow p-subgroups grows. It is astonishing that, though the situation gets more and more complicated, the group rings RSL_{2}(2^{f}) can be described in a uniform way for all f ≥ 3. Similar methods can be applied for G = SL_{2}(p^{f}) where p is odd, to get analogous information about the group ring as is given here for p = 2. There are additional technical difficulties for odd primes p that make the basic ideas less transparent, so the case p > 2 will be treated in a separate paper.

This paper is intended to make one part of my habilitation thesis [9] available to a wider audience. I thank Dr. Alexander Zimmermann for pointing out to me reference [6].

2 Generalities.

[figure: Throughout the paper let K be; a finite extension of ℚ_{p},; R its ring of integers with; maximal ideal πR and; residue class field k := R/πR.; K; R; πR; ℚ_{p}; ℤ_{p}; pℤ_{p}; k {]

Let A be a finite-dimensional semisimple K-algebra

A = ⨁_{t=1}^{s} Aϵ_{t} ≅ ⨁_{t=1}^{s} K_{t}^{n_{t}×n_{t}}

where ϵ_{1}, …, ϵ_{s} are the central primitive idempotents of A and K_{t} are K-division algebras.

This section develops a language for describing the ring theoretic structure of group rings RG of finite groups G up to Morita equivalence. Group rings are certain R-orders Λ in a semisimple K-algebra K ⊗_{R} Λ =: A.

It turns out that it is easier to describe the overorder

Γ := ⨁_{t=1}^{s} Λϵ_{t}

of Λ, the direct sum of the projections of Λ into the simple components of A. If the decomposition numbers of RG are ≤ 1 (and R is big enough), then the order Γ is a so called graduated order (see Definition 2.1). Then one may describe Γ by purely combinatorical data, which also allows one to read off the Λ-lattices in the irreducible A-modules. A language for describing such graduated orders is developed in [10] and will be repeated briefly in section 2.1.

The main tool for computing Λ and Γ is the observation that group rings and their ring direct summands are symmetric orders (see Definition 2.4). This means that Λ = Λ^{#} is a self dual lattice with respect to some associative symmetric bilinear form on A. The knowledge of this form allows one in particular to calculate the index of Λ in a maximal overorder in A.

Let J(Λ) denote the Jacobson radical of Λ, the smallest Λ-ideal I in Λ such that Λ/I is semisimple. Then Λ/J(Λ) is a semisimple k-algebra and one can lift the central primitive idempotents of Λ/J(Λ) to a system e_{1}, …, e_{h} of orthogonal idempotents in Λ with 1 = ∑_{i=1}^{h} e_{i} (see e.g. [11], Theorem 6.19). Then one obtains a direct sum decomposition

Λ = ⨁_{i,j=1}^{h} e_{i}Λe_{j}.

Under certain conditions one can embed the summands e_{i}Λe_{j} simultaneously into a commutative K-algebra E (isomorphic to the center of A) such that the multiplication e_{i}Λe_{j} × e_{j}Λe_{l} → e_{i}Λe_{l} can be performed in E. This is described in Section 2.3.

2.1 Graduated orders.

Definition 2.1 An R-order Γ ⊂ A is called graduated if there are orthogonal idempotents e_{1}, …, e_{h} ∈ Γ satisfying e_{i}e_{j} = δ_{ij}e_{i} and 1 = ∑_{i=1}^{h} e_{i} such that e_{i}Γe_{i} is a maximal order in e_{i}Ae_{i}.

Let Γ be a graduated order in A. Then Γ contains the central primitive idempotents ϵ_{1}, …, ϵ_{s} ∈ Γ and each order Γϵ_{t} is a graduated order in the simple algebra Aϵ_{t}.

Definition 2.2 Let Γ be a graduated order in the simple K-algebra A = D^{n×n} and Ω the maximal R-order in the division algebra D with maximal ideal P. Then there are h, n_{1}, …, n_{h} ∈ ℕ (n = n_{1} + … + n_{h}) and M = (m_{ij}) ∈ ℤ_{≥0}^{h×h} such that Γ is conjugated to

Λ(Ω, n_{1}, …, n_{h}, M) :=
{X = (x_{ij})_{i,j=1,...,h} ∈ D^{n×n} | x_{ij} ∈ (P^{m_{ij}})^{n_{i}×n_{j}} for all 1 ≤ i, j ≤ h}.

If Γ/J(Γ) ≅ ⊕_{i=1}^{h}(Ω/P)^{n_{i}×n_{i}} then m_{ii} = 0, m_{ij} + m_{ji} > 0 and m_{ij} + m_{jl} ≥ m_{il} for all 1 ≤ l, i ≠ j ≤ h. In this case M is called an exponent matrix of Γ.

The Γ-lattices in the simple A-module can be easily described by means of its exponent matrix M, see [10], Remark (II.4). One can always conjugate Γ such that m_{1j} = 0 for all 1 ≤ j ≤ h.

From [10], Proposition (IV.1) one finds

Lemma 2.3 Let M ∈ ℤ^{h×h} be an exponent matrix of a graduated R-order Γ in the simple K-algebra A. Assume that there is an involution ^{∘} : Γ → Γ (i.e. an R-order antiautomorphism of order ≤ 2) fixing the central primitive idempotents of Γ/J(Γ). Then

m_{ij} + m_{jl} + m_{li} = m_{ji} + m_{il} + m_{lj} for all 1 ≤ i, j, l ≤ h.

If M is normalized such that the first row of M consists of 0 only then

m_{ij} + m_{j1} = m_{ji} + m_{i1} for all 1 ≤ i, j ≤ h.

This lemma will be applied to epimorphic images of group rings RG. The natural involution ^{∘} : RG → RG is the R-linear map defined by g ↦ g^{−1} for all g ∈ G. If ϵ is a central primitive idempotent of KG with ϵ^{∘} = ϵ such that Γ := RGϵ is a graduated order and all p-modular constituents of the character belonging to ϵ are self dual, then Γ satisfies the conditions of Lemma 2.3.

2.2 Symmetric orders.

Definition 2.4 An R-order Λ in A is called symmetric if there is a nondegenerate symmetric associative K-bilinear form Φ : A × A → K such that Λ is self dual with respect to Φ, i.e. Λ = Λ^{#} = {a ∈ A | Φ(Λ, a) ⊂ R}.

One easily shows that the nondegenerate symmetric associative K-bilinear form on A are precisely the forms

T r_{u} : A × A → K, (a, b) ↦ ∑_{t=1}^{s} tr_{red}(auϵ_{t}b)

where u ∈ Z(A)^{∗} and tr_{red} denotes the reduced trace of Aϵ_{t} to K.

Example. Let G be a finite group. Then RG is a symmetric order in A = KG with respect to |G|^{−1} times the regular trace bilinear form. If χ_{t}(1) denotes the dimension of an absolutely irreducible constituent of the simple KGϵ_{t}-module, then this associative symmetric bilinear form equals T r_{u}, where u = |G|^{−1} ∑_{t=1}^{s} χ_{t}(1)ϵ_{t}.

Lemma 2.5 ([14], Proposition (1.6.2)) If Λ is a symmetric R-order with respect to Φ and e, f are idempotents in Λ then Φ_{|(eΛf)×(fΛe)} is a nondegenerate R-bilinear pairing. In particular eΛe is a symmetric order.

2.3 A language for describing certain basic orders.

Let Δ be an R-order in A. In this section it is assumed that k = R/πR is a splitting field for k ⊗_{R} Δ and that the division algebras K_{t} are commutative. Let P_{1}, …, P_{h} represent the isomorphism classes of projective indecomposable Δ right modules.

Then Δ is Morita equivalent to

Λ := End_{Δ}(P_{1} ⊕ … ⊕ P_{h}) = ⨁_{i,j=1}^{h} Hom_{Δ}(P_{i}, P_{j})

and Λ is a basic order in the sense that the simple Λ-modules are one dimensional vector spaces over k.

Since there is an idempotent e ∈ Δ such that Λ ≅ eΔe, Lemma 2.5 shows that Λ is symmetric if Δ is symmetric. Note that the module categories of Δ and Λ are equivalent. In particular the decomposition numbers of Δ and Λ are equal. We assume that for 1 ≤ i ≤ h the endomorphism rings End_{Δ}(P_{i}) are commutative which is equivalent to say that the decomposition numbers of Δ are ≤ 1.

The main new idea for describing the order Λ is to embed the R-lattices Hom_{Δ}(P_{i}, P_{j}) simultaneously for all 1 ≤ i, j ≤ h into a commutative finitedimensional K-algebra E such that the multiplication Hom_{Δ}(P_{i}, P_{j})×Hom_{Δ}(P_{j}, P_{l}) → Hom_{Δ}(P_{i}, P_{l}) can be performed in E.

To this purpose let

V := ⨁_{t=1}^{s} V_{t}

be the sum over a system of representatives of the isomorphism classes of simple A-modules and

E := End_{A}(V) ≅ ⨁_{t=1}^{s} K_{t} ≅ Z(A).

Let 1 ≤ j ≤ h. Since End_{Δ}(P_{j}) is commutative, the A-module V has a unique A-submodule isomorphic to K ⊗_{R}P_{j} and up to isomorphism a unique Δ-sublattice isomorphic to P_{j}. For all 1 ≤ j ≤ h choose an embedding

φ_{j} : P_{j} ↪ V.

Let Q_{j} be the unique A-invariant complement of K ⊗_{R} φ_{j}(P_{j}) in V,

V = (K ⊗_{R} φ_{j}(P_{j})) ⊕ Q_{j}.

Then the Δ-homomorphisms φ ∈ Hom_{Δ}(P_{j}, P_{i}) for 1 ≤ i, j ≤ h are considered as elements of E by letting

φ_{|Q_{j}} = 0.

Definition 2.6 For i = 1, …, h let φ_{i}^{−1} : V → K ⊗_{R} P_{i} be the right inverse of φ_{i} with φ_{i}^{−1}(Q_{i}) = 0. Then for 1 ≤ i, j ≤ h there are embeddings

Hom_{Δ}(P_{i}, P_{j}) ↪ E, φ ↦ φ_{i}^{−1}φφ_{j}.

Via these embeddings Hom_{Δ}(P_{i}, P_{j}) is viewed as a subset

Λ_{ij} := (φ_{i}^{−1})Hom_{Δ}(P_{i}, P_{j})φ_{j} ⊂ E.

Remark 2.7 For 1 ≤ i ≠ j ≤ h the endomorphism ring End_{Δ}(P_{j}) is canonically (i.e. independent of the choice of φ_{j}) embedded into E, whereas the embedding Hom_{Δ}(P_{i}, P_{j}) ↪ E depends on the choice of φ_{i} and φ_{j}.

For simplicity we now assume that K is a splitting field for A. Then K_{t} = K for all 1 ≤ t ≤ s and the central primitive idempotents ϵ_{1}, …, ϵ_{s} form a canonical K-basis of E.

Definition 2.8 Let φ = ∑_{t=1}^{s} a_{t}ϵ_{t} ∈ E with a_{t} ∈ K. The fractional R-ideal ∑_{t=1}^{s} a_{t}R is called the norm of φ,

n(φ) := ∑_{i=t}^{s} a_{t}R ⊆ K.

The norm has a certain multiplicative property.

Remark 2.9 If φ, ψ ∈ E − {0} then

n(φ)n(ψ) divides n(φψ)

and for i ∈ ℕ

n(φ^{i}) = n(φ)^{i}.

One may characterize the unit groups of the local rings Λ_{ii} (cf. Definition 2.6) with help of the norm.

Lemma 2.10 For 1 ≤ i ≤ h the unit group Λ_{ii}^{∗} of the local ring Λ_{ii} is

Λ_{ii}^{∗} = {φ ∈ Λ_{ii} | n(φ) = R}.

\underline{Proof.} Let x ∈ Λ_{ii} with n(x) = R. Then n(x^{j}) = R and hence x^{j} ∉ πΛ_{ii} for all j ∈ ℕ. Therefore x does not lie in the unique maximal ideal of Λ_{ii} and is a unit. The other inclusion is trivial since n(id_{|P_{i}}) = R. □

Since for i ≠ j the modules P_{i} and P_{j} are not isomorphic one gets the following remark from Lemma 2.10.

Remark 2.11 Let 1 ≤ i ≠ l ≤ h, 0 ≠ φ ∈ Λ_{il} ⊂ E, and 0 ≠ ψ ∈ Λ_{li} ⊂ E. Then φψ = ψφ ∈ Λ_{ii} ∩ Λ_{ll} with

πR ⊇ n(φψ).

Assume now that Δ (and hence Λ) is a symmetric order with respect to the associative bilinear form T r_{u}, u = ∑_{t=1}^{s} u_{t}ϵ_{t} ∈ E, u_{t} ∈ K.

Some additional notation is needed: If L ⊂ M are two R-lattices with M/L ≅ ⊕_{i=1}^{t}R/π^{x_{t}}R the index of L in M is the ideal [M : L] := π^{x_{1}+...+x_{t}}R.

For 1 ≤ i ≤ h let

c_{i} := {1 ≤ t ≤ s | ϵ_{t}P_{i} ≠ 0}

denote the constituents of the KG-module K ⊗_{R} P_{i}.

Lemma 2.12 If (ψ_{1}, …, ψ_{l}) is an R-basis of Λ_{jj}, then

(∏_{i=1}^{l} n(ψ_{i}))^{2} divides ∏_{t∈c_{j}} u_{t}^{−1}R.

\underline{Proof.} Let M := ⊕_{t∈c_{j}}Rϵ_{t} be the maximal R-order in K ⊗_{R} Λ_{jj}. Then the dual of M with respect to T r_{u} is M^{#} = ⊕_{t∈c_{j}}u_{t}^{−1}Rϵ_{t}. Since M^{#} ⊆ Λ_{jj} = Λ_{jj}^{#} ⊆ M with [Λ_{jj} : M^{#}] = [M : Λ_{jj}] one has

[M : Λ_{jj}]^{2} = [M : M^{#}] = ∏_{t∈c_{j}} u_{t}^{−1}R.

The Lemma follows, because ∏_{i=1}^{l} n(ψ_{i}) divides [M : Λ_{jj}]. □

3 The group ring ℤ_{2}[ζ_{2^{f} −1}]SL_{2}(2^{f}).

Let G be a finite group and R and k be as in Section 2. This section presents a method for obtaining the ring theoretic structure of the integral group ring RG from the group algebra kG. Assume that k is a splitting field for kG. Then one usually describes the finite-dimensional k-algebra kG by giving a presentation of the Morita equivalent basic algebra \overline{Λ} := End_{kG}(\overline{P_{1}} ⊕ … ⊕ \overline{P_{h}}), where \overline{P_{1}}, …, \overline{P_{h}} are the projective indecomposable kG-modules. \overline{Λ} is generated by id\overline{P_{i}} ∈ End_{kG}(\overline{P_{i}}) (1 ≤ i ≤ h) and preimages in Hom_{kG}(\overline{P_{j}}, \overline{P_{i}}) of a k-basis of Hom_{kG}(\overline{P_{j}}, J(kG)\overline{P_{i}})/Hom_{kG}(\overline{P_{j}}, J(kG)^{2}\overline{P_{i}}) (1 ≤ i, j ≤ h) (see [3], Proposition 4.1.7) usually encoded as vertices and arrows in the Ext-quiver. This generating set can be lifted to obtain a generating set of Λ := End_{RG}(P_{1} ⊕ … ⊕ P_{h}), where P_{i} is the projective RG-module with P_{i}/πP_{i} = \overline{P_{i}} (i = 1, …, h) and the lifts of the generators in Hom_{kG}(\overline{P_{i}}, \overline{P_{j}}) lie in Hom_{RG}(P_{i}, P_{j}) (1 ≤ i, j ≤ h). Now Remark 2.11 gives upper bounds on the norm of the basis elements of End_{RG}(P_{i}) obtained as product of the generators. The fact that End_{RG}(P_{i}) is a symmetric order yields lower bounds on these norms.

In the particular situation of this section upper and lower bounds coincide. So let 3 ≤ f ∈ ℕ, R be the ring of integers in the unramified extension K of

degree f of ℚ_{2} and k := R/2R ≅ 𝔽_{2^{f}} the residue class field. Let G := SL_{2}(2^{f}) denote the group of 2 × 2-matrices over k of determinant 1. Then (K, R, k) is a 2-modular splitting system for G. Since the decomposition numbers of RG are ≤ 1 (cf. [5], Corollary 2.8), the order ⊕_{i=1}^{s}ϵ_{i}RG, where ϵ_{1}, …, ϵ_{s} are the central primitive idempotents of KG, is a graduated order in KG, the graduated hull of RG. Therefore the methods of the previous section can be applied to describe RG. In particular the graduated hull of RG has a very nice description given in Theorem 3.15.

Since an explicit presentation of kG is used to obtain information on the group ring RG, the description of kG given in [6] is repeated in the first paragraph.

3.1 The group algebra in characteristic 2

Steinberg’s tensor product theorem establishes a bijection between the simple kGmodules and the subsets of N := {1, …, f}: let M_{1} be the natural kG-module k^{2} and let F be the Frobenius automorphism F : k → k; x ↦ x^{2} of k. For i = 0, …, f − 1 one defines M_{i+1} to be the set M_{1} with scalar multiplication am := F^{i}(a)m for a ∈ k. Then the simple kG-modules are the tensor products

M_{I} := ⊗_{i∈I}M_{i}

where I runs through the subsets of N. Note that dim_{k}(M_{I}) = 2^{|I|}. In particular M_{∅} is the trivial kG-module and M_{N} is the projective simple kG-module, and hence lies in a block of defect zero.

The projective kG-modules are described in [1]. The projective cover of the simple kG-module M_{I} is denoted by \overline{P_{I}} (I ⊆ N). Then by Theorem 1 in [1]

\overline{P_{I}} = M_{N} ⊗ M_{N−I}, if ∅ ≠ I ⊆ N

and

\overline{P_{∅}} ⊕ M_{N} = M_{N} ⊗ M_{N}.

Alperin also calculates the Ext-groups between the simple kG-modules. Koshita [6] extends this result to give a presentation of the basic algebra that belongs to kG by describing the homomorphism spaces between the projective indecomposable kG-modules explicitly.

If I, I′ ⊂ N then I + I′ := I ∪ I′ − I ∩ I′ denotes the symmetric difference and for elements i, j ∈ N let i + j ∈ N (resp. i − j ∈ N) be the element of N, that is congruent to i + j ∈ ℤ (bzw. i − j ∈ ℤ) modulo f. Then one has the following theorem.

Theorem 3.1 ([6]) Let f ≥ 3 and

\overline{Λ} := End_{kG}(⊕_{N≠I⊂N} \overline{P_{I}})

be the basic algebra belonging to the principal block of kG. Let Q be the quiver with vertices corresponding to the proper subsets of N and arrows

α_{i,I} : I + {i} → I for all I ⊆ N, i ∈ N, i − 1 ∉ I.

Let kQ be the path algebra of Q. Then the paths from I to J = I +{i_{1}}+…+{i_{a}} are written as

(I|i_{1}, …, i_{a}|J) := α_{i_{1},I+{i_{1}}}α_{i_{2},I+{i_{1}}+{i_{2}}} … α_{i_{a},J}.

(I||I) is the idempotent in kQ corresponding to the vertex I. With this notation let X be the ideal of kQ generated by

(I + {i} + {j}|i, j|I) − (I + {i} + {j}|j, i|I) (i − 1, j − 1 ∉ I, j ≠ i − 1, i, i + 1)

(I|i, i|I) (i − 1 ∉ I, i ∈ I)
(I + {i + 1}|i + 1, i, i|I) − (I + {i + 1}|i, i, i + 1|I) (i − 1, i ∉ I)
(I + {i + 1}|i, i + 1, i|I) (i − 1 ∉ I, i ∈ I).

Then there is an epimorphism

Ψ : kQ → \overline{Λ}

with kernel X i.e. \overline{Λ} ≅ kQ/X.

Proposition 3 in [6] determines a k-basis for End_{kG}(\overline{P_{I}}).

Definition 3.2 For i ∉ I ⊆ N let

ω_{I,i} := Ψ((I|j, j + 1, …, i − 1, i, i, i − 1, …, j + 1, j|I))

where j := j(I, i) is the unique element of N such that j − 1 ∉ I and J := J(I, i) := {j, j + 1, …, i − 1} ⊆ I. If i − 1 ∉ I, then j := i, J = ∅ and

ω_{I,i} := Ψ((I|i, i|I)).

The length of ω_{I,i} is the length of the corresponding path in Q,

l(I, i) := 2(|J| + 1).

The elements ω_{I,i} are endomorphisms of \overline{P_{I}}. Since the decomposition numbers of RG are 0 or 1 ([5], Corollary 2.8), the endomorphism rings of the projective indecomposable kG-modules are commutative. Therefore

ω_{I,T} := ∏_{i∈T} ω_{I,i}

for subsets T ⊆ N − I is well defined.

Proposition 3.3 ([6], Proposition 3) Let I ⊂ N be a proper subset of N. The elements ω_{I,T} where T runs through the subsets of N −I form a k-basis of End_{kG}(\overline{P_{I}}).

To describe the vector spaces Hom_{kG}(\overline{P_{H}}, \overline{P_{I}}) for I, H ⊂ N one needs further elements of kQ.

Definition 3.4 Let H ⊂ I ⊂ N, such that there is a path in Q from H to I of length |I − H|. Then let ω_{H⊂I} be the image of such a path under Ψ and ω_{I⊃H} be the image of the corresponding path from I to H of the same length.

The next lemma follows from [6], Proposition 3.

Lemma 3.5 Let H, I ⊂ N with Hom_{kG}(\overline{P_{H}}, \overline{P_{I}}) ≠ 0. Then

Hom_{kG}(\overline{P_{H}}, \overline{P_{I}}) = ω_{H⊃H∩I}End_{kG}(\overline{P_{H∩I}})ω_{H∩I⊂I}.

3.2 The integral group ring RG.

Now this characteristic 2 information is lifted to the characteristic 0 situation to obtain the R-order RG nearly up to Morita equivalence. So let

Λ := End_{RG}(⊕_{N≠I⊂N}P_{I}),

where P_{I} is the projective indecomposable RG-module with head M_{I} (I ⊂ N), be the basic order that is Morita equivalent to the principal block of RG.

As in Section 2.3 let V be the sum of all irreducible KG-modules in the principal block and E := End_{KG}(V). Let ϵ_{1}, …, ϵ_{s} be the central primitive idempotents of KG that belong to the principal block of RG. These idempotents are identified with the primitive idempotents of E. If I ⊂ N then

c_{I} := {1 ≤ t ≤ s | ϵ_{t}P_{I} ≠ 0}

denotes the indices of the irreducibles KG-modules that occur in K ⊗_{R} P_{I}. The sets c_{I} are explicitly described in [5]. In particular c_{∅} = {1, …, s} and hence V ≅ K ⊗_{R} P_{∅}.

Using the fact that Λ is a symmetric order, one gets:

Lemma 3.6 Let I ⊂ N with |N − I| =: n. Then |c_{I}| = 2^{n} = dim_{R}(End_{RG}(P_{I})) and End_{RG}(P_{I}) is a sublattice of ⊕_{t∈c_{I}}Rϵ_{t} of index 2^{2^{n−1}f}R.

\underline{Proof.} Let M := ⊕_{t∈c_{I}}Rϵ_{t} be the maximal R-order in K ⊗_{R} End_{RG}(P_{I}). Now End_{RG}(P_{I}) is a symmetric order with respect to the form T r_{u} where

u := |G|^{−1} ∑_{t∈c_{I}} χ_{t}(1)ϵ_{t},

and χ_{t}(1) is the degree of the irreducible character of G belonging to ϵ_{t}. Since χ_{t}(1) ∈ R^{∗} is odd, the dual of M with respect to T r_{u} is of index (2^{f})^{(2^{n})}R in M. Hence [M : End_{RG}(P_{I})] = \sqrt{2^{f2^{n}}}R = 2^{2^{n−1}f}R. □

Now suitable lifts β′_{I,T} ∈ Λ of the elements ω_{I,T} from Definition 3.2 are constructed.

Definition 3.7 (i) For i − 1 ∉ I ⊂ N let φ_{I,i} ∈ Hom_{RG}(P_{I+{i}}, P_{I}) be a preimage of Ψ(α_{i,I}).

(ii) For i ∉ I ⊆ N let β′_{I,i} ∈ End_{RG}(P_{I}) be defined similarly as ω_{I,i} using the φ_{I,i} instead of Ψ(α_{i,I}): If i − 1 ∈ I, then let j := j(I, i) and J := J(I, i). Then

β′_{I,i} := φ_{I+{j},j}φ_{I−{j,j+1},j+1} … φ_{I−J+{i},i}φ_{I−J,i}φ_{I−J+{i−1},i−1} … φ_{I−{j},j+1}φ_{I,j}.

If i − 1 ∉ I then

β′_{I,i} := φ_{I+{i},i}φ_{I,i}.

For any subset T ⊆ N − I define

β′_{I,T} := ∏_{i∈T} β′_{I,i} ∈ End_{RG}(P_{I}).

Since End_{RG}(P_{I}) is commutative, the definition does not depend on the ordering of the factors.

(iii) If H ⊂ I ⊂ N, such that there is a path ω = α_{i_{1},I_{1}} … α_{i_{l},I_{l}} ∈ Q from H to I and a path ω′ = α_{j_{1},J_{1}} … α_{j_{l},J_{l}} ∈ Q from I to H of length l := |I − H|. Then let β′_{H⊂I} := ∏_{m=1}^{l} φ_{I_{m},i_{m}} and β′_{I⊃H} := ∏_{m=1}^{l} φ_{J_{m},j_{m}}.

From Proposition 3.3 one now gets immediately

Corollary 3.8 (β′_{I,T} , (T ⊆ N − I)) is an R-basis of the R-lattice End_{RG}(P_{I}).

The following lemma is the crucial point in the investigation of Λ.

Lemma 3.9 For I ⊂ N and T ⊂ N − I

n(β′_{I,T}) = 2^{l(I,T)}

where l(I, T) := ∑_{i∈T} l(I, i)/2.

\underline{Proof.} Let i ∈ N − I and l := l(I, i)/2. Then

β′_{I,i} = f_{1} … f_{l}g_{l} … g_{1}

is a product of f_{j} ∈ Hom_{RG}(P_{I_{j}}, P_{I_{j+1}}) and g_{j} ∈ Hom_{RG}(P_{I_{j+1}}, P_{I_{j}}) for certain pairwise distinct subsets I_{1}, …, I_{l+1} of N. In the commutative ring E this product can be evaluated as

β′_{I,i} = (f_{1}g_{1})(f_{2}g_{2}) … (f_{l}g_{l}).

According to Remark 2.11 the norm n(f_{j}g_{j}) is divisible by 2. Then Remark 2.9 says that 2^{l} divides n(β′_{I,i}) and therefore 2^{l(I,T)} divides n(β′_{I,T}).

On the other hand let n := |N − I|. Then

∑_{T ⊆N−I} l(I, T) = ∑_{T ⊆N−I} ∑_{i∈T} l(I, i)/2 = 1/2 ∑_{i∈N−I} l(I, i)2^{n−1} = f2^{n−1}.

Hence n(β′_{I,T}) divides 2^{l(I,T)} by Lemma 3.6 and Lemma 2.12. □

In particular if T = {i} and i, i − 1 ∉ I ⊂ N this crucial lemma yields n(β′_{I,i}) = 2. But

β′_{I,i} = φ_{I+{i},i}φ_{I,i} = φ_{I,i}φ_{I+{i},i} ∈ 2End_{RG}(P_{I+{i}}),

by Theorem 3.1 since (I + {i}|i, i|I + {i}) ∈ X. So Lemma 2.10 implies that β′_{I,i}/2 ∈ End_{RG}(P_{I+{i}})^{∗} is a unit. Hence one gets the following lemma.

Lemma 3.10 If i − 1, i ∉ I, then φ_{I,i} is injective and β′_{I,{i}} ∈ 2(End_{RG}(P_{I+{i}})^{∗}). Therefore there is a unit u_{I,i} ∈ End_{RG}(P_{I+{i}})^{∗}, such that

u_{I,i}β′_{I,{i}} = 2id_{P_{I+{i}}} ∈ End_{RG}(P_{I}).

Definition 3.11 For I ⊂ N let pr_{I} := ∑_{j∈c_{I}} ϵ_{j} ∈ E be the identity on P_{I}. Let i ∉ I ⊂ N. If i − 1 ∉ I, define

β_{I,i} = 2pr_{I+{i}}.

Otherwise let j := j(I, i) and J := J(I, i). Then

β_{I,i} := 2^{|J|+1}pr_{I−J+{i}}pr_{I}.

For T ⊂ N − I let

β_{I,T} := ∏_{i∈T} β_{I,i}

where the empty product

β_{I,∅} := pr_{I}

is the unit element in End_{RG}(P_{I}).

Theorem 3.12 Let H, I ⊂ N.

(a) The β_{I,T} with T ⊂ N − I form an R-basis of the lattice End_{RG}(P_{I}).

(b) If Hom_{RG}(P_{H}, P_{I}) ≠ {0} then

Hom_{RG}(P_{H}, P_{I}) ≅ (pr_{H}(End_{RG}(P_{H∩I}))pr_{I})

as End_{RG}(P_{H}) − End_{RG}(P_{I})-bimodule.

\underline{Proof.} (a) Let I ⊂ N and i ∈ N − I. If i − 1 ∉ I, then β_{I,i} = β′_{I,i}u_{I,i} ∈ End_{RG}(P_{I}) by Lemma 3.10. If i − 1 ∈ I, then let j := j(I, i) and J := J(I, i) be as in Definition 3.2. For l = j, …, i − 1 define I_{l} := I − {j, …, l} and let u_{l} := u_{I_{l},l+1} be the unit of Lemma 3.10. It easily follows from [5] that

c_{I} ⊂ c_{I−{j}} ⊂ … ⊂ c_{I−J}.

Therefore

β′_{I,i} ∏_{l=j}^{i−1} u_{l} = (φ_{I−{j},j}φ_{I,j}u_{j})(φ_{I−{j,j+1},j+1}φ_{I−{j},j+1}u_{j+1}) …

… (φ_{I−J,i−1}φ_{I−J+{i−1},i−1}u_{i−1})(φ_{I−J+{i},i}φ_{I−J,i}u_{I−J,i}) =

(2pr_{I})(2pr_{I−{j}}) … (2pr_{I−J})(2pr_{I−J+{i}}) = 2^{|J|+1}pr_{I}pr_{I−J+{i}}.

Hence the elements β_{I,T} with T ⊂ N − I lie in the ring End_{RG}(P_{I}).

For T ⊂ N − I let a(I, T)′_{j}, a(I, T)_{j} ∈ R (j = 1, …, s) be such that β′_{I,T} = ∑_{j=1}^{s} a(I, T)′_{j}ϵ_{j} and β_{I,T} = ∑_{j=1}^{s} a(I, T)_{j}ϵ_{j}. Since β_{I,T} is obtained from β′_{I,T} by multiplication with units in local rings, there is k_{I,T} ∈ R^{∗} with

k_{I,T} a(I, T)_{j} ≡ a(I, T)′_{j} (mod 2n(β′_{I,T}))

for all 1 ≤ j ≤ s. Replacing β_{I,T} by k_{I,T} β_{I,T} we assume that k_{I,T} = 1. Lemma 3.9 yields that n(β_{I,T}) = n(β′_{I,T}) = 2^{l(I,T)} and the product of the norms is

∏_{T ⊂N−I} n(β_{I,T}) = 2^{2^{n−1}f}

the index of End_{RG}(P_{I}) in ⊕_{j∈c_{I}}Rϵ_{j}. Here n := |N − I|. Let B′ := (a(I, T)′_{j})_{T,j}, B := (a(I, T)_{j})_{T,j} and D := diag(n(β′_{I,T}) | T ⊆ N − I). Then B′ = DU for some U ∈ GL_{2^{n}}(R) and B = DV with V ≡ U (mod 2R). Therefore the determinant of V is also a unit in R and (β_{I,T} | T ⊆ N − I) is an R-basis of End_{RG}(P_{I}).

(b) By Lemma 3.5

Hom_{RG}(P_{H}, P_{I}) = β′_{H⊃H∩I}End_{RG}(P_{H∩I})β′_{H∩I⊂I}.

Multiplication with β′_{H⊃H∩I}β′_{H∩I⊂I} ∈ E induces the desired bimodule isomorphism. □

Remark 3.13 This theorem shows that the endomorphism rings of the projective indecomposable RSL_{2}(2^{f})-modules have a rational structure: Let I ⊂ N and c_{I} = {t_{1}, …, t_{n}}. Then there is a (symmetric) ℤ_{2}-order

O_{I} = ⨁_{T ⊂N−I} ℤ_{2}β_{I,T} ⊂ ⊕_{i=1}^{n}ℤ_{2}ϵ_{t_{i}}

with R ⊗_{ℤ_{2}} O_{I} = End_{RG}(P_{I}).

In particular the order O_{∅} is isomorphic to the group ring ℤ_{2}(C_{2}^{f}) of the Sylow 2-subgroup of G:

Proposition 3.14 Let O_{∅} = ⊕_{T ⊂N}ℤ_{2}β_{∅,T} be as in the remark above. Then

O_{∅} ≅ ℤ_{2}(C_{2}^{f}).

In particular the endomorphism ring of P_{∅}

End_{RG}(P_{∅}) ≅ R(C_{2}^{f})

is isomorphic to the group ring over R of the Sylow 2-subgroup of G.

\underline{Proof.} The R-order End_{RG}(P_{∅}) ⊂ E is generated by β_{∅,∅} = id_{P_{∅}} = 1 ∈ E and x_{i} := β_{∅,i} = 2 ∑_{t∈c_{{i}}} ϵ_{t}, i = 1, …, f. The Frobenius automorphism F acts on End_{RG}(P_{∅}) mapping x_{i} to x_{i+1} (i = 1, …, f) where as usual the indices are taken modulo f. With the notation of [5], the ordinary irreducible characters in the principal block of RG are 1, η_{j}, δ_{i} where j = 1, …, 2^{f−1} − 1 and i = 1, …, 2^{f−1}. The Frobenius automorphism F acts on the irreducible characters η_{j} and δ_{i} by multiplying the indices with 2, where the indices of η are taken modulo 2^{f} − 1 and the ones of δ modulo 2^{f} + 1 and in both cases identified with the negative index. In the rest of the proof these characters are used to index the primitive idempotents of E. Then by [5]

x_{f} = 2( ∑_{j=1,j odd}^{2^{f−1}−1} ϵ_{η_{j}} + ϵ_{δ_{j}}).

Choose a generating set (b_{0}, …, b_{f−1}) of C_{2}^{f} and let 0 ≤ i = ∑_{j=0}^{f−1} z_{j}2^{j} < 2^{f} with z_{j} ∈ {0, 1}. Then define the linear character χ_{i} of C_{2}^{f} via χ_{i}(b_{j}) = (−1)^{z_{j}}. Let h(i) := ∑_{j=0}^{f−2} z_{j}2^{j} and define a bijection between the set of characters {χ_{0}, …, χ_{2^{f} −1}} and the set of characters in the principal block of G via

χ_{i} ↦ \{ \begin{matrix}1 & if i = 0 \\ η_{h(i)} & if i > 0 and z_{f−1} = 0 \\ δ_{h(i)} & if z_{f−1} = 1. \end{matrix}

This bijection induces a bijection of the primitive idempotents of E and KC_{2}^{f} and hence an isomorphism φ : E → KC_{2}^{f}. Clearly χ_{i}(b_{0}) = 1 if and only if z_{0} = 0. Therefore one finds φ(x_{f}) = 1 − b_{0} ∈ ℤ_{2}C_{2}^{f}.

The action of F on KC_{2}^{f} induced via φ is given by b_{i} ↦ b_{i+1} if i = 0, …, f −3, b_{f−2} ↦ ∑_{i=0}^{f−2} b_{i} and b_{f−1} ↦ b_{f−1}. One has φ(x_{1}) = 1 − b_{1}, …, φ(x_{f−3}) = 1 − b_{f−2}, φ(x_{f−2}) = 1 − ∑_{i=0}^{f−2} b_{i}, φ(x_{f−1}) = 1 − b_{f−1} ∈ ℤ_{2}C_{2}^{f}. Together with 1 = φ(1) these elements span ℤ_{2}C_{2}^{f}, therefore φ_{|O_{∅}} : O_{∅} → ℤ_{2}C_{2}^{f} is an isomorphism. □

Note that this proposition also shows that End_{RG}(P_{∅}) is also symmetric with respect to T r_{v} where v := 2^{−f} ∑_{t=1}^{s} ϵ_{t}.

Since the decomposition numbers of RG are ≤ 1, the projections Λϵ_{t} of Λ onto the simple components of K ⊗_{R} Λ are graduated orders. From Theorem 3.12 one gets a simple description of the graduated hull ⊕_{t=1}^{s}ϵ_{t}Λ of Λ, which also yields the theorem stated in the introduction.

Theorem 3.15 For 1 ≤ t ≤ s there is an exponent matrix M^{(t)} = (m_{H,I}^{(t)}) where H, I ⊂ N, t ∈ c_{H} ∩ c_{I} for Λϵ_{t} such that

m_{H,I}^{(t)} = |H − I| ( for all H, I such that t ∈ c_{H} ∩ c_{I}).

\underline{Proof.} Let M^{(t)} be an exponent matrix of Λϵ_{t} such that the row corresponding to M_{∅} consists of 0 only

m_{∅,H}^{(t)} = 0 for all H ⊂ N, t ∈ c_{H}.

From Lemma 3.9 it follows that

n(β′_{H⊃H∩I}β′_{H∩I⊂H}) = 2^{|H−I|}

and

n(β′_{I⊃H∩I}β′_{H∩I⊂I}) = 2^{|I−H|}.

Then Theorem 3.12 (b) implies that for I, H ⊂ N such that t ∈ c_{H} ∩ c_{I}

m_{H,I}^{(t)} + m_{I,H}^{(t)} = |H − I| + |I − H|.

In particular

m_{H,∅}^{(t)} = |H| for all H ⊂ N, t ∈ c_{H}.

Since the simple kG-modules are all self dual, one gets from Lemma 2.3

m_{H,I}^{(t)} = m_{I,H}^{(t)} − m_{I,∅}^{(t)} + m_{H,∅}^{(t)} = m_{I,H}^{(t)} − |I| + |H|.

Therefore the Theorem follows. □

Theorem 3.15 and Theorem 3.12 do not describe Λ up to Morita equivalence, because they do not give a system of simultaneous embeddings φ_{I} : P_{I} ↪ P_{∅} (I ⊂ N) that determines the multiplication in Λ. If the isomorphism type of Λ is uniquely determined by that of Λ/2Λ, then one can show that Λ is as given in the following conjecture.

\underline{Conjecture:} There are embeddings φ_{I} : P_{I} ↪ P_{∅}, I ⊂ N, such that for all I, H ⊂ N

Hom_{RG}(φ_{H}(P_{H}), φ_{I}(P_{I})) = 2^{|I−(H∩I)|}(pr_{H}(End_{RG}(φ_{H∩I}(P_{H∩I})))pr_{I}) ⊂ E.

One easily shows using the combinatorical description of the sets c_{I} in [5] that the R-order Λ = End_{RG}(⊕_{I⊂N}P_{I}) in the conjecture satisfies Λ/2Λ ≅ kQ/X as in Theorem 3.1.

Proposition 3.16 The conjecture is true for f ≤ 6.

\underline{Proof.} For all proper subsets ∅ ≠ I ⊂ N choose i ∈ I with i − 1 ∉ I. Let T be the set of such pairs (I, i). Via the monomorphism φ_{I−{i},i} : P_{I} → P_{I−{i}} in Lemma 3.10 one embeds P_{I} into P_{I−{i}} for all (I, i) ∈ T, such that recursively all projective indecomposable RG-modules are embedded into P_{∅}. For I ⊂ N let φ_{I} : P_{I} ↪ P_{∅} denote this embedding. Then for all (I, i) ∈ T it holds that

Hom_{RG}(φ_{I}(P_{I}), φ_{I−{i}}(P_{I−{i}})) = pr_{I}(End_{RG}(φ_{I−{i}}(P_{I−{i}}))).

One possibility for T in the case f = 5 is given in figure 1, which shows the Ext-quiver Q for the principal block of 𝔽_{2^{5}}SL_{2}(2^{5}). The vertices of the quiver are indexed by the projective indecomposable modules P_{I}, where only the elements of I are given and 0 stands for the empty set. The arrows P_{I} → P_{I+{i}} and P_{I+{i}} → P_{I} in Q are represented by one straight line. The mappings φ_{I} are constructed along the bold edges.

So to show the Proposition for the case f = 5, it remains to show that the homomorphism pr_{I} lies in Hom_{RG}(φ_{I}(P_{I}), φ_{I−{i}}(P_{I−{i}})) for all thin edges (P_{I}, P_{I−{i}}) with i ∈ I ⊂ N, i − 1 ∉ I and (I, i) ∉ T.

We prove this for the example I = {2, 4, 5} and i = 4. Let

h ∈ Hom_{RG}(φ_{{2,4,5}}(P_{{2,4,5}}), φ_{{2,5}}(P_{{2,5}}))

be a lift of Ψ(α_{4,{2,5}}). Theorem 3.1 says that there is an x ∈ End_{RG}(φ_{{5}}(P_{{5}})) with

h = hpr_{{2,5}} = pr_{{2,4,5}}pr_{{4,5}}(1 + 2x) = pr_{{2,4,5}}(1 + 2x).

Now

2End_{RG}(φ_{{5}}(P_{{5}})) ⊂ End_{RG}(φ_{{2,5}}(P_{{2,5}})),

so 1 + 2x is a unit in End_{RG}(φ_{{2,5}}(P_{{2,5}})). Therefore

pr_{{2,4,5}} ∈ Hom_{RG}(φ_{{2,4,5}}(P_{{2,4,5}}), φ_{{2,5}}(P_{{2,5}})).

The other cases are completely analogous. If f = 6, one additionally needs the action of the Galois group Gal(K/ℚ_{2}) ≅ C_{f} on the quiver Q. □

[figure: Figure 1: The Ext-quiver for the principal block of 𝔽_{2^{5}}SL_{2}(2^{5}); 2345; 345; 45; 5; 0; 2; 12; 125; 1245; 3; 23; 123; 1235; 1345; 145; 15; 1; 1234; 234; 34; 4; 25; 35; 24; 13; 14; 245; 124; 235; 135; 134]

Remark 3.17 If the conjecture is true, then one does not need irrationalities to describe RG. More precisely if KG ≅ ⨁_{t=1}^{s} K^{n_{t}×n_{t}} then there is a ℚ_{2}-algebra A ≅ ⨁_{t=1}^{s} ℚ_{2}^{n_{t}×n_{t}} and a (symmetric) ℤ_{2}-order Λ_{0} ⊂ A such that R ⊗_{ℤ_{2}} Λ_{0} = RG.

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Author’s address: Lehrstuhl B für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany e-mail: gabi@math.rwth-aachen.de