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

Gabriele Nebe

Lehrstuhl B für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany (e-mail: gabi@math.rwth-aachen.de)

Abstract: Let p > 2 be a prime, R = ℤ_{p}[ζ_{p^{f} −1}], K = ℚ_{p}[ζ_{p^{f} −1}] and G = SL_{2}(p^{f}). The group ring RG is calculated nearly up to Morita equivalence: The projections of RG into the simple components of KG are given explicitly and the endomorphism rings and homomorphism bimodules between the projective indecomposable RG-lattices are described.

1991 Mathematics Subject Classification: 20C05, 20C11

1 Introduction

This paper investigates the group ring over p-adic integers of the special linear groups G := SL_{2}(p^{f}) of degree 2 for odd primes p and f > 2. For f = 2 the group ring ℤ_{p}SL_{2}(p^{2}) has been described up to Morita equivalence in [Neb 98]. The case p = 2 is slightly easier and has been treated in [Neb 00]. The main strategy is the same as for p = 2. There are additional technical difficulties due to the fact that for odd primes p, there are two pairs of ordinary characters of degree \frac{p^{f} ±1}{2} that have the same p-modular constituents.

In the first part of the paper the general method, already used to describe RG for p = 2 in [Neb 00] is repeated and adopted to the slightly more complicated situation here, that we do not work with a splitting field. The second part is devoted to the special situation G = SL_{2}(p^{f}), K = ℚ_{p}[ζ_{p^{f} −1}], R = ℤ_{p}[ζ_{p^{f} −1}], k = R/pR. Since details are needed, the technical description of kG given in [Kos 98] is repeated briefly in the first section. To describe RG it is useful to know some facts about the direct summands of K ⊗_{R} P for the projective indecomposable RG-lattices P. The necessary information is derived in section 3.2 from the description of the decomposition numbers in [HSW 82]. After all these preparations the section 3.3 gives a description of RG. The endomorphism rings of the projective indecomposable RG-lattices are calculated and the homomorphism lattices between two such modules are determined up to bimodule isomorphisms. Finally the projections of RG into the simple components of KG are described explicitly. This allows to read off the inclusion patterns for the irreducible RG-lattices as it is illustrated for the example G = SL_{2}(3^{3}).

This paper is intended to make one part of my habilitation thesis [Neb 99] available to a wider audience. I thank Dr. Alexander Zimmermann for giving me a copy of the preprint of [Kos 98].

2 General definitions.

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

Let Δ be an R-order in A. This section develops a language to describe Δ up to Morita equivalence, if it fulfills additional conditions. It does not weaken the tools to calculate Δ, if one makes unramified extensions of K. Therefore it is convenient to assume that K is big enough for Δ (cf. [Jac, Definition 9.10]) in the sense that k is a splitting field for k ⊗_{R} Δ and the division algebras K_{t} are totally ramified field extensions of K. So let K be big enough for Δ and 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.

Let V_{t} be the simple Aϵ_{t}-module and define the A-module

V := ⨁_{t=1}^{s} V_{t}.

The main assumption of this paper is that for 1 ≤ i ≤ h the endomorphism rings End_{Δ}(P_{i}) are commutative. This is equivalent to say that the multiplicities of the simple A-modules in K ⊗_{R} P_{i} are ≤ 1. Then V has a unique A-submodule isomorphic to K ⊗_{R} P_{i}. For all 1 ≤ i ≤ h choose an embedding

φ_{i} : P_{i} ↪ V.

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

V = (K ⊗_{R} φ_{i}(P_{i})) ⊕ Q_{i}.

Then the Δ-homomorphisms φ ∈ Hom_{Δ}(P_{i}, P_{j}) for 1 ≤ i, j ≤ h are considered as A-endomorphisms of V by letting φ_{|Q_{i}} = 0. So let

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

Definition 2.1 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.

Note that Λ_{ij} in general depends on the choice of φ_{i} and φ_{j} but Λ_{ii} does not. The primitive idempotents of E form a canonical “basis” of E in the sense

that the elements of E can be written uniquely as ∑_{t=1}^{s} a_{t}ϵ_{t} ∈ E with a_{t} ∈ K_{t}. If φ_{ij} = ∑_{t=1}^{s} a_{t}ϵ_{t} ∈ Λ_{ij} and φ_{ji} = ∑_{t=1}^{s} b_{t}ϵ_{t} ∈ Λ_{ji} then

φ := φ_{ij}φ_{ji} == ∑_{t=1}^{s} a_{t}b_{t}ϵ_{t} = φ_{ji}φ_{ij} ∈ Λ_{ii} ∩ Λ_{jj}.

Since φ corresponds to an endomorphism of P_{i} that factors through P_{j}, it is not a unit in Λ_{ii}. Therefore one concludes that a_{t}b_{t} ∈ π_{t}R_{t} lies in the maximal ideal π_{t}R_{t} of the ring of integers R_{t} in K_{t} for all t = 1, …, s.

To measure the difference of φ being a unit in Λ_{ii} we introduce a norm on E. To this purpose let L be the compositum of the fields K_{t} (1 ≤ t ≤ s) with ring of integers O and maximal ideal ℘.

Definition 2.2 Let 0 ≠ φ = ∑_{t=1}^{s} a_{t}ϵ_{t} ∈ E with a_{t} ∈ K_{t}. The fractional O-ideal ∑_{t=1}^{s} a_{t}O is called the norm of φ,

n(φ) := ∑_{i=t}^{s} a_{t}O ⊆ L.

It follows directly from the definition that

n(φ)n(ψ) divides n(φψ)

for all φ, ψ ∈ E − {0} and for i ∈ ℕ

n(φ^{i}) = n(φ)^{i}.

The unit groups Λ_{ii}^{∗} of the local rings Λ_{ii} consist precisely of the norm O elements:

Λ_{ii}^{∗} = {φ ∈ Λ_{ii} | n(φ) = O} for all 1 ≤ i ≤ h.

As seen above this implies that

n(φ_{ij}φ_{ji}) ⊆ ℘ for all 1 ≤ i ≠ j ≤ h, 0 ≠ φ_{ij} ∈ Λ_{ij}, 0 ≠ φ_{ji} ∈ Λ_{ji}.

In the applications Δ will always be a ring direct summand of a group ring RG for some finite group G. roup rings have the distinguished property that they are symmetric orders: For the regular trace tr_{reg} of A = KG one has tr_{reg}(g) = 0 for all 1 ≠ g ∈ G and tr_{reg}(1) = |G|. Hence {g^{−1} | g ∈ G} is the reciprocal basis of G with respect to the bilinear form T r_{reg} : A × A → K, T r_{reg}(a, b) := \frac{1}{|G|}tr_{reg}(ab) and RG = RG^{#} := {a ∈ KG | T r_{reg}(a, RG) ⊆ R}. This observation allows for instance to calculate the index of RG in any maximal overorder. One easily sees that also the blocks of RG are symmetric with respect to the restriction of T r_{reg} (cf. [Thé 95, Proposition (1.6.2)]). For calculations it is easier to work with the reduced trace of KG: Let χ_{t}(1) denote the dimension of an absolutely irreducible constituent of the simple KGϵ_{t}-module and let u_{t} := \frac{1}{|G|}χ_{t}(1)ϵ_{t} ∈ Z(KGϵ_{t})^{∗}, t = 1, …, s. Then \frac{1}{|G|}tr_{reg}(a) = ∑_{t=1}^{s} tr_{red}(u_{t}a) for all a ∈ KG.

Assume now that Δ is a symmetric order with respect to the associative bilinear form T r_{u} : (a, b) ↦ ∑_{t=1}^{s} tr_{red}(au_{t}ϵ_{t}b) with u = ∑_{t=1}^{s} u_{t}ϵ_{t} and u_{t} ∈ K_{t}. Since there is an idempotent e ∈ Δ, such that Λ ≅ eΔe [Thé 95, Proposition (1.6.2)] shows that Λ is also symmetric.

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}.

Let ν_{π} be the extension of the π-adic valuation of K to L (ν_{π}(π) = 1) and for fractional O-ideals I in L let ν_{π}(I) := min{ν_{π}(x) | x ∈ I}. For t = 1, …, s let D(R_{t}) be the inverse different of R_{t}, i.e. the dual of R_{t} with respect to the trace bilinear form over R.

Lemma 2.3 Let 1 ≤ j ≤ h. Let

D_{j} := ∏_{t∈c_{j}}[R_{t} : u_{t}^{−1}D(R_{t})]O.

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

2 ∑_{i=1}^{l}⌊ν_{π}(n(ψ_{i}))⌋ ≤ ν_{π}(D_{j}).

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

[M : Λ_{jj}]^{2} = [M : M^{#}] = D_{j}.

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

To describe the action of Δ on the irreducible A-modules we need some additional notation as it is developed in [Ple 83]. The main idea is not to consider norms of individual elements of Hom_{Δ}(P_{i}, P_{j}) but to consider the ideal generated by the coefficients a_{t} at ϵ_{t} (t ∈ c_{i} ∩ c_{j}) for all these homomorphisms. If the center of Δϵ_{t} is the ring of integers R_{t} in K_{t}, then this ideal is a certain power π_{t}^{m_{ij}^{(t)}}R_{t} of the maximal ideal in R_{t}. Then the order ⊕_{t=1}^{s}Δϵ_{t} is called a graduated order (see Definition 2.4) and determined by the matrices M^{(t)} := (m_{ij}^{(t)})_{i,j|t∈c_{i}∩c_{j}} up to Morita equivalence.

Definition 2.4 An R-order Γ in the simple K-algebra D^{n×n} is called graduated, if 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}.

where Ω the maximal R-order in the division algebra D with maximal ideal P. 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. An R-order Γ in a semisimple K-algebra is called graduated, if Γ is the direct sum of graduated orders in the simple components.

The Γ-lattices in the simple K ⊗_{R} Γ-modules can be easily described by means of its exponent matrix M, see [Ple 83, Remark (II.4)]. One can always conjugate Γ such that m_{1j} = 0 for all 1 ≤ j ≤ h.

From [Ple 83, Proposition (IV.1)] one finds

Lemma 2.5 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 2.5.

Remark 2.6 Let M = (m_{ij})_{1≤i,j≤h} be an exponent matrix of a graduated order as in Lemma 2.5 such that m_{1,j} = 0 for all 1 ≤ j ≤ h. Let

m(i, j) := m_{ij} + m_{ji} for all 1 ≤ i, j ≤ h.

Then M is already determined by the m(i, j) since 2m_{ij} = m(i, j) − m(j, 1) + m(i, 1). The m(i, j) only depend on Γ and are called structure constants of Γ.

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

Throughout this chapter let p be an odd prime, f ∈ ℕ, and R := ℤ_{p}[ζ_{p^{f} −1}] the ring of integers in the unramified extension K of ℚ_{p} of degree f with residue class field ki := R/pR ≅ 𝔽_{p^{f}}. We want to describe the group ring RG where G = SL_{2}(k) is the group of 2 × 2 matrices over k with determinant 1. RG has three blocks, one of which is of defect 0 and the other two have the Sylow p-subgroups ≅ C_{p}^{f} of G as their defect groups.

The simple kG-modules can be obtained from Steinberg’s tensor product theorem: Let F be the Frobenius automorphism x ↦ x^{p} of k := 𝔽_{p^{s}}. For a vector space V let V^{(i)} be the vector space obtained by twisting V i-times with F, hence V^{(i)} = V with scalar multiplication x·v := x^{p^{i}}v (x ∈ k, v ∈ V). The group SL_{2}(k)

acts as group of automorphisms on the algebra k[X, Y] where ( \begin{matrix}a & b \\ c & d \end{matrix} ) sends X

and Y to dX − bY, −cX + aY, respectively. For 0 ≤ λ < p let M_{λ} ⊆ k[X, Y] be the subspace of homogenous polynomials of degree λ and for λ = (λ_{0}, …, λ_{f−1}) with 0 ≤ λ_{i} < p let

M_{λ} := M_{λ_{0}} ⊗_{k} M_{λ_{1}}^{(1)} ⊗_{k} … ⊗_{k} M_{λ_{f−1}}^{(f−1)}.

The modules k̄M_{λ} form a system of representatives of the isomorphism classes of simple SL_{2}(p^{f})-modules over the algebraic closure k̄ of k.

3.1 The group algebra kG.

In this section the description in [Kos 98] of the basic algebra belonging to the group algebra of G in characteristic p is repeated.

Let P := {0, …, p − 1}^{f} − {(p − 1, …, p − 1)}. The elements of P are in bijection to the simple kG-modules in one of the two blocks of positive defect. To be consistent with the notation in [HSW 82] for the simple kG-modules the elements λ ∈ P are indexed with the elements of N := {0, …, f − 1}. Then λ = (λ_{0}, …, λ_{f−1}) ∈ P corresponds to the simple kG-module M_{λ}.

The elements of N are considered modulo f.

[AJL 83] calculates the Ext-groups between the simple kG-modules: For i ∈ N and h = ±1 let

P (i, h) := {λ ∈ P | 0 ≤ λ_{i−1} ≤ p − 2 und 0 ≤ λ_{i} + h ≤ p − 1}

and f(i, h) : P (i, h) → P; λ ↦ γ, where γ_{j} = λ_{j} for all i, i−1 ≠ j ∈ N, γ_{i} = λ_{i} +h and γ_{i−1} = p − 2 − λ_{i−1}.

Theorem 3.1 ([AJL 83, Corollary 4.5]) Let λ, γ ∈ P. Then Ext_{kG}^{1}(M_{λ}, M_{γ}) ≠ 0 if and only if γ = f(i, h)(λ) for some i ∈ N, h ∈ ±1. If f ≥ 3, then the non-trivial Ext-groups are one dimensional.

For λ ∈ P let \overline{P_{λ}} be the projective kG-module with head M_{λ} and

\overline{Λ} := End_{kG}(⊕_{λ∈P} \overline{P_{λ}})

the basic algebra Morita equivalent to the sum of the two blocks of defect f of kG. [Kos 98] calculates defining relations between the generators of \overline{Λ} given by the Ext-quiver of kG:

For λ ∈ P let

S(λ) := {i ∈ N | λ_{i} = p − 1}.

Theorem 3.2 ([Kos 98, Theorem 2.2]) Let f ≥ 3 and Q be the quiver whose vertices are indexed by the elements of P with arrows

α_{i,h,λ} : f(i, h)(λ) → λ for all i ∈ N, h = ±1, λ ∈ P (i, h).

Let kQ be the path algebra of Q. Then for λ, γ ∈ P the paths in Q from λ to γ can be written as

(λ|(i_{1}, h_{1}), …, (i_{l}, h_{l})|γ) = α_{i_{1},h_{1},f(i_{1},−h_{1})(λ)}α_{i_{2},h_{2},I} … α_{i_{l},h_{l},γ}.

(λ||λ) is the idempotent in kQ, corresponding to the vertex λ. Then let X be the ideal of kQ, generated by

(f(j, r)(f(i, h)(λ))|(j, r), (i, h)|λ) − (f(j, r)(f(i, h)(λ))|(i, h)(j, r)|λ) (i, j ∈ N; j ≠ i − 1, i, i + 1; h, r = ±1; λ ∈ P (i, h) ∩ P (j, r))

(λ|(i, h), (i, h)|f(i, h)(f(i, h)(λ))) (i ∈ N; h = ±1; λ, f(i, h)(λ) ∈ P (i, h))

(λ|(i, 1), (i, −1)|λ) (i ∈ N; λ ∈ P (i, −1), i ∈ S(λ))

(λ|(i, 1), (i, −1)|λ) − (λ|(i, −1), (i, 1)|λ) (i ∈ N; λ ∈ P (i, −1) ∩ P (i, 1))

(f(i + 1, r)(f(i, h)(λ))|(i + 1, r), (i, h)|λ) − (f(j, r)(f(i, h)(λ)|(i, −h)(i + 1, r)|λ) (i ∈ N; h, r = ±1; λ ∈ P (i, h); λ, f(i, h)(λ) ∈ P (i + 1, r))

Then \overline{Λ} ≅ kQ/X.

\underline{Example.} The Ext-quiver of the principal block of kSL_{2}(3^{3}). The arrows α_{i,h,λ} and α_{i,−h,f(i,h)(λ)} are indicated by a single edge labeled with i.

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Definition 3.3 For i ∈ N, h = ±1 and λ ∈ P (i, h) the number i is called the direction of α_{i,h,λ} ∈ Q.

Let

Ψ : kQ → \overline{Λ}

be the epimorphism of [Kos 98] with Ker(Ψ) = X. The images under Ψ of the paths in Q from the vertex λ to the vertex γ generate the vector space Hom_{kG}(\overline{P_{λ}}, \overline{P_{γ}}) over k. [Kos 98, Proposition 9.2] constructs a k-basis for Hom_{kG}(\overline{P_{λ}}, \overline{P_{γ}}). To describe this basis we need to resume the notation of [Kos 98]:

Definition 3.4 (a) Let λ, γ ∈ P with S(λ) = S(γ) and ∅ ≠ T = {i_{1}, …, i_{l}} ⊆ N such that there is a path (λ|(i_{1}, h_{1}), …, (i_{l}, h_{l})|γ) ∈ Q from λ to γ for suitable h_{1}, …, h_{l} ∈ {±1}. Then

⟨λ ∼ γ|T ⟩ := Ψ((λ|(i_{1}, h_{1}), …, (i_{l}, h_{l})|γ))

denotes the image of such a path. ⟨λ ∼ λ|∅⟩ := Ψ((λ||λ)).

(b) If λ, γ ∈ P such that there is a path ω = (λ| … |γ) ∈ Q of length l(ω) = |S(λ) − S(γ)| then ⟨λ ⊃ γ⟩ denotes the image of such a path under Ψ.

(c) If λ, γ ∈ P such that there is a path ω = (λ| … |γ) ∈ Q of length l(ω) = |S(γ) − S(λ)| then ⟨λ ⊂ γ⟩ denotes the image of such a path under Ψ.

(d) Assume that for λ, γ ∈ P there exists some x ∈ P with S(x) = S(λ) ∩ S(γ) such that ⟨λ ⊃ x⟩ exists. Then x is unique and denoted by

λ ⊖ γ := x.

(e) Let λ ∈ P, i ∈ N − S(λ). If i − 1 ∈ S(λ), then let j − 1 ∉ S(λ) be such that {j, j + 1, …, i − 1} ⊆ S(λ). Then

[λ, i] := Ψ((λ|(j, 1), …, (i − 1, 1), (i, −1), (i, 1), (i − 1, −1), …, (j, −1)|λ)).

If i − 1 ∉ S(λ), then put

[λ, i] := Ψ((λ|(i, −1), (i, 1)|λ)).

For T ⊂ N − S(λ), the product ∏_{i∈T} [λ, i] does not depend on the ordering of the factors and is denoted by [λ, T].

Definition 3.5 Let λ, γ ∈ P, T_{1} ⊂ N − (S(λ) ∪ S(γ)) be such that

(1) λ ⊖ γ and γ ⊖ λ exist.

(2) ⟨(λ ⊖ γ) ∼ (γ ⊖ λ), T_{1}⟩ exists.

Then we define for T_{2} ⊂ N − (S(λ) ∪ S(γ) ∪ T_{1})

⟨⟨λ, T_{1}, T_{2}, γ⟩⟩ := ⟨λ ⊃ (λ ⊖ γ)⟩⟨(λ ⊖ γ) ∼ (γ ⊖ λ), T_{1}⟩[γ ⊖ λ, T_{2}]⟨(γ ⊖ λ) ⊂ γ⟩.

With this notation one has

Proposition 3.6 ([Kos 98, Proposition 9.2]) Let λ, γ ∈ P. The elements ⟨⟨λ, T_{1}, T_{2}, γ⟩⟩, for which λ, γ, T_{1} satisfy conditions (1) and (2) of Definition 3.5 and T_{2} ⊂ N − (S(λ) ∪ S(γ) ∪ T_{1}), form a basis of Hom_{kG}(\overline{P_{λ}}, \overline{P_{γ}}).

Remark 3.7 Let λ, γ be as in Definition 3.5. The directions of the arrows in the path corresponding to ⟨λ ⊃ (λ ⊖ γ)⟩ respectively ⟨(γ ⊖ λ) ⊂ γ⟩ are precisely the elements of S(λ) − S(γ) respectively S(γ) − S(λ). Therefore the path belonging to ⟨⟨λ, T_{1}, T_{2}, γ⟩⟩ is of the shape ⟨λ ∼ γ, T ⟩ for some subset T ⊂ N.

Lemma 3.8 Let λ ∈ P and ∅ ≠ T ⊆ N such that ⟨λ ∼ λ, T ⟩ exists. Then T = N and there are signs ν_{0}, …, ν_{f−1} ∈ \{±1\}, such that λ_{i} = \frac{p−2−ν_{i}}{2} for i = 0, …, f −1.

\underline{Proof.} The operators f(i, h) and f(i′, h′) (i, i′ ∈ N, h, h′ = ±1) commute if one considers the entries of λ modulo 2. f(i, h) changes precisely the parity of λ_{i} and λ_{i−1}. Hence i ∈ T implies i − 1 ∈ T and therefore T = N. Moreover for all i = 0, …, f − 1 there is ν_{i} ∈ {±1} such that

p − 2 − λ_{i} + ν_{i} = λ_{i} ⇔ λ_{i} = \frac{p − 2 − ν_{i}}{2} . □

Definition 3.9 Let P̃ := \{λ ∈ P | λ_{i} = \frac{p−2−ν_{i}}{2} for ν_{i} = ±1 for all i ∈ N\}. For λ ∈ P̃, the element ⟨λ ∼ λ, N⟩ ∈ \overline{Λ} is called a circle through λ.

Corollary 3.10 (a) For λ ∈ P − P̃ the 2^{|N−S(a)|} endomorphisms [λ, T] with T ⊆ N − S(λ) form a k-basis of End_{kG}(\overline{P_{λ}}).

(b) If λ ∈ P̃ then there is a further basis element ⟨λ ∼ λ, N⟩.

Definition 3.11 Let λ, γ ∈ P be such that Hom_{kG}(\overline{P_{λ}}, \overline{P_{γ}}) ≠ 0. If λ or γ does not lie in P̃ then there is a unique T_{1} ⊂ N − (S(λ) ∪ S(γ)), such that ⟨(λ ⊖ γ) ∼ (γ ⊖ λ), T_{1}⟩ ∈ Q. Then let

d(λ, γ) := l(⟨λ ⊃ (λ ⊖ γ)⟩⟨(λ ⊖ γ) ∼ (γ ⊖ λ), T_{1}⟩⟨(γ ⊖ λ) ⊂ γ⟩)

denote the length of a shortest path from λ to γ in Q without repeated directions.

3.2 Decomposition numbers

To lift this description of kG to characteristic zero, we need some properties of the decomposition numbers of RG. In this section no central primitive idempotents are needed and the letter ϵ usually stands for ^{±}1. A useful description of the decomposition numbers and the Cartan invariants of the group SL_{2}(p^{f}) in characteristic p can be found in [HSW 82] (cf. also [Bur 76]).

In the notation of [HSW 82], the characters of the absolutely irreducible ℂSL_{2}(p^{f})- modules are 1, δ, δ′, η, η′, η_{i}, δ_{j} and St, where 1 ≤ i ≤ (p^{f} − 3)/2 and 1 ≤ j ≤ (p^{f} − 1)/2. The character degrees are η(1) = η′(1) = (p^{f} + 1)/2, δ(1) = δ′(1) = (p^{f} − 1)/2, δ_{i}(1) = p^{f} − 1, η_{i}(1) = p^{f} + 1 and St(1) = p^{f}. The action of F on the ordinary characters η_{i} and δ_{j} is given by multiplying the indices with p. δ_{j}^{F} = δ_{pj} and η_{i}^{F} = η_{pi} where the indices are considered modulo p^{f} + 1 respectively modulo p^{f} −1 and are identified with their negatives. δ_{j} and η_{i} belong to the principal block of RG, if and only if j and i are even.

To describe the decomposition numbers define for λ = (λ_{0}, …, λ_{f−1}) the set

W (λ) = {0 ≤ ν ≤ p^{f} − 1 | ν = ∑_{i=0}^{f−1} ϵ_{i}λ̃_{i}p^{i} for some ϵ_{0}, …, ϵ_{f−1} = ^{±}1}

where ˜ is the bijection of the set {0, …, p−1} defined by x̃ = p−1−x. Moreover let V^{ϵ}(λ) = W (λ) ∪ (p^{f} − ϵ − W (λ)) for ϵ = ^{±}1.

For j = 1, …, (p^{f} −3)/2 let d_{λ,j}^{(1)} := d_{λ,η_{j}} be the multiplicity of the Brauer character belonging to M_{λ} in the restriction of η_{j} to the p-regular classes of SL_{2}(p^{s}). Analogously let d_{λ,j}^{(−1)} := d_{λ,δ_{j}} (j = 1, …, (p^{f} − 1)/2), d_{λ,(p^{f} −1)/2}^{(1)} := d_{λ,η} = d_{λ,η′},

and d_{λ,(p^{f} +1)/2}^{(−1)} := d_{λ,δ} = d_{λ,δ′}. Then one gets

Theorem 3.12 ([HSW 82, Theorem 2.7] )

a) d_{λ,j}^{(ϵ)} = 1 if j ∈ V^{ϵ}(λ) and 0 otherwise.

b) d_{λ,St} = 1 if λ = p^{f} − 1 and 0 otherwise. d_{λ,1} = 1 if λ = 0 and 0 otherwise.

The elements of V^{ϵ}(λ) are treated like the indices of η (if ϵ = 1) respectively δ (if ϵ = −1), they are considered modulo p^{f} − ϵ and identified with their negatives.

The main reason why the case p > 2 is much more complicated than the case p = 2 is the existence of the exceptional characters η, η′, δ and δ′. If L is a projective indecomposable RG-lattice such that one of these four characters occurs as a constituent of the character of K⊗_{R}L, then there is an additional generator, called a circle (cf. Definition 3.9) of the endomorphism ring End_{kG}(L/pL). By Brauer reciprocity, the set of such L is in bijection to the set of modular constituents of one of the exceptional characters. [HSW 82, Corollary 2.4] determines these modular characters:

Lemma 3.13 η and η′ respectively δ and δ′ have the same p-modular constituents. The set of all these constituents is

M := \{(λ_{0}, …, λ_{f−1}) | λ_{i} = \frac{1}{2}(p − 2 + ϵ_{i}) for ϵ_{i} ∈ \{±1\}\}.

[HSW 82] even says that if λ ∈ M, then \frac{p^{f} −ϵ}{2} lies in W (λ) for ϵ = ^{±}1 with λ ≡ (p^{f} − ϵ)/2 (mod 2). Therefore \frac{p^{f} −ϵ}{2} ∈ V^{−ϵ}(λ) for all λ ∈ M with λ ≡ (p^{f} − ϵ)/2 (mod 2). Since the Frobenius automorphism F preserves the set of those modular characters λ one gets the following Lemma 3.14 Let λ ∈ M be a p-modular constituent of δ and ϵ := −1 or a p-modular constituent of η and ϵ := 1. Then

χ_{m}^{(−ϵ)} := \frac{1}{2}(p^{f} + ϵ) − p^{m} ∈ V^{−ϵ}(λ) for all 0 ≤ m ≤ f − 1.

\underline{Proof.} The argumentation above shows that \frac{1}{2}(p^{f} − ϵ)p^{m} ∈ V^{−ϵ}(λ) for m = 0, …, f − 1. One has \frac{p^{m}+1}{2}(p^{f} + ϵ) − \frac{p^{f+m}−ϵp^{m}}{2} = \frac{1}{2}(p^{f} + ϵ) + ϵp^{m} and (p^{f} + ϵ) − (\frac{1}{2}(p^{f} + ϵ) + ϵp^{m}) = \frac{1}{2}(p^{f} + ϵ) − ϵp^{m}. Therefore χ_{m}^{(−ϵ)} ∈ V^{−ϵ}(λ). □

Now the definition of a circle (Definition 3.9) is repeated in the language of Brauer characters.

Definition 3.15 Let Kr := (λ^{(1)}, …, λ^{(f)}) be a sequence of p-Brauer characters of G such that

λ^{(j)} = (λ_{0}^{(j)}, …, λ_{f−1}^{(j)}) with λ_{i}^{(j)} = \frac{1}{2}(p − 2 + ϵ_{i}^{(j)})p^{i} for some ϵ_{i}^{(j)} = ±1

(0 ≤ i ≤ f − 1) j = 1, …, f. Kr is called a circle, if there is a bijection σ : {1, …, f} → {0, …, f − 1} such that

ϵ_{i}^{(j)} = \{ \begin{matrix}ϵ_{i}^{(j−1)} & i ≠ σ(j), σ(j) − 1 \\ −ϵ_{i}^{(j−1)} & i = σ(j) or σ(j) − 1 \end{matrix} for j = 2, …, f

where the indices i ∈ {0, …, f − 1} are considered modulo f.

Lemma 3.16 Let Kr := (λ^{(1)}, …, λ^{(f)}) be a circle and ϵ = ±1. Assume that there is some a ∈ X := ⋂_{j=1}^{f} V^{ϵ}(λ^{(j)}) with 0 ≤ a < (p^{f} − ϵ)/2. Then there exists 0 ≤ m ≤ f − 1 such that a = χ_{m}^{(ϵ)}.

\underline{Proof.} By definition there are ν_{0}^{(j)}, …, ν_{f−1}^{(j)} ∈ \{±1\} such that λ_{i}^{(j)} = \frac{1}{2}(p−2+ν_{i}^{(j)})

(j = 1, …, f, i = 0, …, f − 1). Then λ̃_{i}^{(j)} = \frac{p−ν_{i}^{(j)}}{2} . Now let a ∈ X with 0 ≤ a ≤ \frac{p^{f} −ϵ}{2}. Then there are ϵ_{i}^{(j)} ∈ \{±1\} (1 ≤ j ≤ f, 0 ≤ i ≤ f − 1) with ϵ_{f−1}^{(j)} = 1 for all 1 ≤ j ≤ f, such that for all 1 ≤ j ≤ f either

a = ∑_{i=0}^{f−1} ϵ_{i}^{(j)} \frac{p − ν_{i}^{(j)}}{2} p^{i} = \frac{p^{f} − ϵ_{0}^{(j)}ν_{0}^{(j)}}{2} + ∑_{i=1}^{f−1} \frac{ϵ_{i−1}^{(j)} − ϵ_{i}^{(j)}ν_{i}^{(j)}}{2} p^{i}

or

a = (p^{f} − ϵ) − ∑_{i=0}^{f−1} ϵ_{i}^{(j)} \frac{p − ν_{i}^{(j)}}{2} p^{i} = \frac{p^{f} − 2ϵ + ϵ_{0}^{(j)}ν_{0}^{(j)}}{2} − ∑_{i=1}^{f−1} \frac{ϵ_{i−1}^{(j)} − ϵ_{i}^{(j)}ν_{i}^{(j)}}{2} p^{i}.

Let K_{0} denote the set of 1 ≤ j ≤ f such that the first equation holds and K_{1} the set of the other 1 ≤ j ≤ f. We now show that if K_{1} = ∅, then a = \frac{p^{f} ±ϵ}{2}.

Assume that K_{1} = ∅. Then there is some ν ∈ {±1} with 2a ≡ −ν (mod p) and ϵ_{0}^{(j)}ν_{0}^{(j)} = ν for all 1 ≤ j ≤ f. Therefore a − \frac{p^{f} −ν}{2} = ∑_{i=1}^{f−1} a_{i}p^{i} where

a_{i} := \frac{ϵ_{i−1}^{(j)} − ϵ_{i}^{(j)}ν_{i}^{(j)}}{2} ∈ \{−1, 0, 1\} for all 1 ≤ j ≤ f

and it remains to show that a_{i} = 0 for all 1 ≤ i ≤ f −1. Seeking a contradiction we let m ∈ {1, …, f −1} be maximal such that a_{m} ≠ 0. Then for n = m+1, …, f −1 and all 1 ≤ j ≤ f

ϵ_{n−1}^{(j)} = ϵ_{n}^{(j)}ν_{n}^{(j)} = ∏_{i=n}^{f−1} ν_{i}^{(j)}

and for n = m one has

0 ≠ a_{m} = ϵ_{m−1}^{(j)} = −ϵ_{m}^{(j)}ν_{m}^{(j)} = − ∏_{i=m}^{f−1} ν_{i}^{(j)} for all 1 ≤ j ≤ f.

If j := σ^{−1}(m) then σ(j) − 1 = m − 1 ∈ {0, …, f − 1} and the condition that Kr is a circle yields ∏_{i=m}^{f−1} ν_{i}^{(j)} = − ∏_{i=m}^{f−1} ν_{i}^{(j−1)}, contradicting the equality for a_{m} above. Therefore all a_{i} are zero and a is as stated. Analogously one proves: If K_{0} = ∅, then a = \frac{p^{f} −ϵ}{2} or a = \frac{p^{f} −3ϵ}{2} = (p^{f} −ϵ)− \frac{p^{f} +ϵ}{2}. Assume now that both sets K_{0}, K_{1} ≠ ∅. Then a ≡ (p^{f} − ϵ_{0}^{(j)}ν_{0}^{(j)})/2 (mod p) and a ≡ (p^{f} − 2ϵ + ϵ_{0}^{(j)}ν_{0}^{(j)})/2 (mod p). Since p > 2 this implies that ϵ_{0}^{(j)}ν_{0}^{(j)} = ϵ for all 1 ≤ j ≤ f. Let j ∈ K_{0} and a_{i} be as above (1 ≤ i ≤ f − 1). The uniqueness of the p-adic expansion of a gives

\frac{ϵ_{i−1}^{(j)} − ϵ_{i}^{(j)}ν_{i}^{(j)}}{2} = a_{i} for all j ∈ K_{0}

and

\frac{ϵ_{i−1}^{(j)} − ϵ_{i}^{(j)}ν_{i}^{(j)}}{2} = −a_{i} for all j ∈ K_{1}.

Again let m be maximal such that a_{m} ≠ 0. Assume that there is another 1 ≤ l < m such that a_{l} ≠ 0 and choose l maximal with this condition. Then for all 1 ≤ j ≤ f one has

ϵ_{n−1}^{(j)} = ϵ_{n}^{(j)}ν_{n}^{(j)} = ∏_{i=n}^{f−1} ν_{i}^{(j)} for all n = m + 1, …, f − 1

and

ϵ_{n−1}^{(j)} = ∏_{i=n}^{m−1} ν_{i}^{(j)} = − ∏_{i=n}^{f−1} ν_{i}^{(j)} for all n = l + 1, …, m − 1.

For m and l one gets for all j ∈ K_{h}, h = 0, 1

0 ≠ (−1)^{h}a_{m} = ϵ_{m−1}^{(j)} = −ϵ_{m}^{(j)}ν_{m}^{(j)} = − ∏_{i=m}^{f−1} ν_{i}^{(j)}

and

0 ≠ (−1)^{h}a_{l} = ϵ_{l−1}^{(j)} = −ϵ_{l}^{(j)}ν_{l}^{(j)} = ∏_{i=l}^{f−1} ν_{i}^{(j)}.

In particular the quotient

a_{l}/a_{m} = − ∏_{i=l}^{m−1} ν_{i}^{(j)}

is constant on K_{0} ∪ K_{1} = {1, …, f}. But the condition that Kr is a circle implies that ∏_{i=l}^{m−1} ν_{i}^{(j)} = − ∏_{i=l}^{m−1} ν_{i}^{(j−1)} for j := σ^{−1}(m). This is a contradiction, hence there is no such m with 1 ≤ l < m and a_{l} ≠ 0. Since p^{f}−ϵ−(\frac{p^{f} −ϵ}{2}+p^{m}) = \frac{p^{f} −ϵ}{2}−p^{m}, a = χ_{m}^{(ϵ)} as claimed in the Lemma. □

The next lemma determines the p-modular constituents of the ordinary character associated with χ_{m}^{(ϵ)}.

Lemma 3.17 Let ϵ = ±1 and 0 ≤ m ≤ f − 1. Let λ = (λ_{0}, …, λ_{f−1}) denote a p-Brauer character of G with ∑_{i=0}^{f−1} λ_{i} ≡ χ_{m}^{(ϵ)} (mod 2). Then χ_{m}^{(ϵ)} ∈ V^{ϵ}(λ) if and only if there are signs ϵ_{i} ∈ {±1} such that

λ_{i} = \frac{p − 2 + ϵ_{i}}{2} for all 0 ≤ i ≠ m ≤ f − 1

and

λ_{m} ∈ \{\frac{p − 5}{2}, \frac{p − 3}{2}, \frac{p − 1}{2}, \frac{p + 1}{2}\}.

\underline{Proof.} Let λ be as in the Lemma. If λ_{m} = \frac{p−3}{2} or λ_{m} = \frac{p−1}{2}, then \frac{p^{f} −ϵ}{2} +p^{i} ∈ V^{ϵ}(λ) for all 0 ≤ i ≤ f − 1 by Lemma 3.14. Otherwise there is ν = ±1, such that λ_{m} + ν = \frac{p−3}{2} or \frac{p−1}{2}. For λ′ = (λ_{0}, …, λ_{m−1}, λ_{m} + ν, λ_{m+1}, …, λ_{f−1}) one has by [HSW 82, Corollary 2.4] that \frac{p^{f} −ϵ}{2} ∈ W (λ′). Therefore \frac{p^{f} −ϵ}{2} ± p^{m} ∈ V^{ϵ}(λ) and all sets V^{ϵ}(λ) in the Lemma contain χ_{m}^{(ϵ)}.

The sum of the degrees of these Brauer characters if p^{f} +ϵ, which is the degree of the character belonging to χ_{m}^{(ϵ)}. □

The other important structure of the Ext-quiver of kG is given by the mstrings. Let m ∈ N be fixed and λ := λ^{(p−1)} ∈ P (m, −1) be such that λ_{m} = p − 1. For j = p−2, …, 0 define λ^{(j)} := f(m, −1)(λ^{(j+1)}). Then there is a sequence of relations (λ|(m, 1), (m, −1)|λ), (λ^{(j)}|(m, 1), (m, −1)|λ^{(j)})−(λ^{(j)}|(m, −1), (m, 1)|λ^{(j)}) for j = p − 2, …, 1 in X.

Definition 3.18 Let λ ∈ P, m ∈ N with λ_{m} = 0, λ_{m−1} < p − 1. Define λ^{(0)} = λ and λ^{(j+1)} = f(m, 1)(λ^{(j)}) for j = 0, …, p − 2. The sequence

ST (λ, m) : = (λ^{(0)}, λ^{(1)}, …, λ^{(p−1)})

is called the m-string with origin λ or simply an m-string.

Lemma 3.19 Let (λ^{(0)}, λ^{(1)}, …, λ^{(p−1)}) be an m-string. Then W (λ^{(p−1)}) ⊂ W (λ^{(p−2)}) and for 1 ≤ j ≤ p − 2 one has W (λ^{(j)}) ⊂ W (λ^{(j−1)}) ∪ W (λ^{(j+1)}). Moreover W (λ^{(j−1)}) − W (λ^{(j)}) ≠ ∅.

\underline{Proof.} This follows from the equality

∑_{i=0}^{f−1} ϵ_{i}λ̃_{i}p^{i} = ∑_{i=0,i≠j,j−1}^{f−1} ϵ_{i}λ̃_{i}p^{i} − ϵ_{j−1}(p − λ̃_{j−1})p^{j−1} + ϵ_{j}(λ̃_{j} − ϵ_{j}ϵ_{j−1})p^{j}.

If λ_{j} = p − 1, then λ̃_{j} = 0 and one may choose ϵ_{j} = −ϵ_{j−1}. □ We now consider the intersection V^{ϵ}(λ^{(j)}) ∩ V^{ϵ}(λ^{(j−2)}) for m-strings.

Lemma 3.20 Let l := \frac{p−1}{2} and (λ^{(0)}, …, λ^{(p−1)}) be an m-string such that there is 0 ≤ i ≤ p − 3 and ν = ^{±}1 with V^{ν}(λ^{(i)}) ∩ V^{ν}(λ^{(i+2)}) ≠ ∅. Then i = l − 2 or i = l − 1 and

{χ_{m}^{(ν)}} = V^{ν}(λ^{(l−2)}) ∩ V^{ν}(λ^{(l−1)}) ∩ V^{ν}(λ^{(l)}) ∩ V^{ν}(λ^{(l+1)})

where one has to omit λ^{(l−2)} if p = 3.

\underline{Proof.} The lemma is only proved for m = 0. The other cases follow by applying the Frobenius automorphism F. Let λ^{(i)} = (λ_{0}, …, λ_{f−1}). Then λ^{(i+2)} = (λ′_{0}, …, λ′_{f−1}) with λ′_{i} = λ_{i} for i = 1, …, f − 1 and λ′_{0} := λ_{0} + 2 ≤ p − 1. Since c_{λ,λ′} > 0, there is ϵ = ±1 such that

(W (λ) ∩ W (λ′)) ∪ (W (λ) ∩ (p^{f} − ϵ − W (λ′))) ≠ ∅.

Assume that there is some α ∈ W (λ) ∩ W (λ′). Then there are signs ϵ_{i}, ϵ′_{i} (0 ≤ i ≤ f − 1) with ∑_{i=0}^{f−1} ϵ_{i}λ̃_{i}p^{i} = ∑_{i=0}^{f−1} ϵ′_{i}λ̃′_{i}p^{i}. Hence

ϵ_{0}λ̃_{0} − ϵ′_{0}(λ̃_{0} − 2) = p(∑_{i=1}^{f−1}(ϵ_{i}λ̃_{i} − ϵ′_{i}λ̃_{i})p^{i−1}).

The left hand side of the equation is an even number between −2(p − 1) and 2(p − 1). The right hand side is divisible by p. Therefore ϵ_{0}λ̃_{0} − ϵ′_{0}(λ̃_{0} − 2) = 0. This has no solution λ̃_{0} ≥ 2. Therefore there is some ϵ = ^{±}1 and α ∈ W (λ) ∩ (p^{f} − ϵ) − W (λ′) with 0 ≤ α ≤ p^{f} − ϵ. Then α can be written as

α = ∑_{i=0}^{f−1} ν_{i}λ̃_{i}p^{i} = p^{f} − ϵ − ∑_{i=0}^{f−1} ν′_{i}λ̃′_{i}p^{i},

for suitable ν_{i}, ν′_{i} ∈ {^{±}1} with ν_{f−1} = ν′_{f−1} = 1. Hence

ν_{0}λ̃_{0} + ν′_{0}(λ̃_{0} − 2) + ϵ = p^{f} − p(∑_{i=1}^{f−1}(ν_{i}λ̃_{i} + ν′_{i}λ̃_{i})p^{i−1}).

The left hand side is an odd number between −2p and 2p and the right had side is divisible by p. Therefore both sides equal ν_{0}p and

λ̃_{0} = \frac{1}{2}(p + 2 − ϵν_{0})

and i = l − 2 or i = l − 1.

For the other λ_{i} the equality

∑_{i=1}^{f−1} ν_{i}λ̃_{i}p^{i−1} = p^{f−1} − ν_{0} − ∑_{i=1}^{f−1} ν′_{i}λ̃_{i}p^{i−1}

follows. Now considering \overline{λ} := (λ_{1}, …, λ_{f−1}) as a Brauer character of SL_{2}(p^{f−1}) one obtains an element in W (\overline{λ}) ∩ p^{f−1} − ν_{0} − W (\overline{λ}). By [HSW 82, Lemma 2.3] this intersection contains at most one element namely \frac{1}{2}(p^{f−1} − ν_{0}). Lemma 3.13 therefore implies that λ_{f−1} = λ′_{f−1} < p − 1. Moreover \frac{1}{2}(p^{f−1} − ν_{0}) is even if and only if ∑_{i=1}^{f−1} λ̃_{i} is even. Therefore the numbers λ_{i} with i ≥ 1 determine the sign ν_{0} and the parity of λ_{0} determines ϵν_{0}.

The element in V^{ϵ}(λ)∩V^{ϵ}(λ′) is \frac{1}{2}p(p^{f−1}−ν_{0})+ν_{0}λ̃_{0} = \frac{1}{2}(p^{f} −ν)+ν_{0}. Therefore χ_{0}^{(ϵ)} ∈ V^{ϵ}(λ) ∩ V^{ϵ}(λ′). With Lemma 3.16 and Lemma 3.17 the Lemma follows.

□

We call an m-string as in the Lemma tangent, since the m-string satisfies the condition of the Lemma if and only if λ^{(l)} and λ^{(l−1)} lie on a circle. The arrows (λ^{(l)}|(m, −1)|λ^{(l−1)}) and (λ^{(l−1)}|(m, 1)|λ^{(l)}) in Q are called exceptional arrows in direction m.

\underline{Example.} The tangent strings in the quiver of the principal 5-block of SL_{2}(5^{3}). The circles (which are triangles here) and therefore also the exceptional arrows are the ones in the triangle (indicated by thick lines).

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.; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; 411; 220; 114; 123; 202; 011; 422; 321; 222; 211; 101; 312; 242; 132; 121; 213; 224; 231; 022; 110; 141; 112]

3.3 The group ring RG.

In this section we want to describe the basic order that is Morita equivalent to RG. The group ring RG has three blocks, one of which is of defect 0. The two other blocks are of defect f and will be treated separately. So let B denote either the principal block of G or the block containing the faithful irreducible characters and y := 0 or y := 1 in the respective cases. Let

P′ := {λ ∈ P | ∑_{i=0}^{f−1} λ_{i} ≡ y (mod 2)}

be the set of indices of the simple B-modules.

For λ ∈ P′ let P_{λ} be the projective indecomposable RG-module with head M_{λ} and let

Λ := End_{RG}(⊕_{λ∈P′}P_{λ}).

For i ∈ N, h = ±1 and λ ∈ P (i, h) ∩ P′ let

φ_{i,h,λ} ∈ Hom_{RG}(P_{f(i,h)(λ)}, P_{λ})

be a lift of Ψ(α_{i,h,λ}). If T ⊂ N −S(λ) define the endomorphism β′_{λ,T} ∈ End_{RG}(P_{λ}) as the lift of [λ, T] obtained by replacing the Ψ(α_{i,h,γ}) in all definitions by φ_{i,h,γ} (γ ∈ P′ ∩ P (i, h)) and let β′_{λ} ∈ End_{RG}(P_{λ}) be such a lift of ⟨λ ∼ λ, N⟩ for all λ ∈ P̃ ∩ P′.

Corollary 3.21 The lifts β′_{λ,T} , β′_{λ} ∈ End_{RG}(P_{λ}) of the basis vectors [λ, T], ⟨λ ∼ λ, N⟩ of End_{kG}(\overline{P_{λ}}) form an R-basis of the lattice End_{RG}(P_{λ}).

As in section 2 let V be the direct sum over a system of representatives of isomorphism classes of simple K⊗_{R}B-modules and E := End_{KG}(V). For λ, γ ∈ P′ the vector space Hom_{RG}(P_{λ}, P_{γ}) is considered as embedded in E. Let ϵ_{1}, …, ϵ_{s} be the primitive idempotents of E.

If f is even, then K is a splitting field for KG. Then the ϵ_{t} are numbered in such a way that ϵ_{1} corresponds to δ or η and ϵ_{2} corresponds to δ′ or η′. In this case put O := R and A := {1, 2}.

If f is odd, let

μ := (−1)^{(p−1)/2} and O := R[\sqrt{μp}].

Then the KG-modules with character δ + δ′ and η + η′ are irreducible and O is isomorphic to the maximal order in the endomorphism ring of these modules. Then we order the idempotents such that O ≅ Z(ϵ_{1}B) and put A := {1}.

Let ℘ be the maximal ideal of O.

If δ belongs to B, then let χ_{m} denote the index of η_{χ_{m}^{(1)}} ∈ B and if η belongs to B, then χ_{m} is the index of δ_{χ_{m}^{(−1)}} ∈ B.

For λ ∈ P′ let

c_{λ} := {1 ≤ t ≤ s | ϵ_{t}P_{λ} ≠ 0}

be the set of the indices of the simple KG-modules isomorphic to a direct summand of K ⊗_{R} P_{λ}.

From Lemma 3.13 one gets Remark 3.22 Let λ ∈ P′. Then λ ∈ P̃ if and only if 1 ∈ c_{λ}.

Since End_{RG}(P_{λ}) is a symmetric order, one finds:

Remark 3.23 Let λ ∈ P′ and put n := |N − S(λ)|.

(a) If λ ∉ P̃, then |c_{λ}| = 2^{n} = dim_{R}(End_{RG}(P_{λ})) and End_{RG}(P_{λ}) is a sublattice of ⊕_{t∈c_{λ}}Rϵ_{t} of index p^{2^{n−1}f}R.

(b) Let λ ∈ P̃. If f is even then |c_{λ}| = 2^{n} + 1 = dim_{R}(End_{RG}(P_{λ})) and End_{RG}(P_{λ}) is a sublattice of ⊕_{t∈c_{λ}}Rϵ_{t} of index p^{(2^{n}+1)f/2}R. If f is odd, then |c_{λ}| + 1 = 2^{n} + 1 = dim_{R}(End_{RG}(P_{λ})) and End_{RG}(P_{λ}) is a sublattice of Oϵ_{1} ⊕ ⊕_{t∈c_{λ},t≠1}Rϵ_{t} of index p^{((2^{n}+1)f−1)/2}R.

The next lemma is the crucial observation in the determination of Λ. In the moment, we can only prove it for endomorphisms of projective modules P_{λ} with λ ∉ P̃.

Lemma 3.24 Let λ ∈ P′ − P̃ and T ⊆ N − S(λ) and β′_{λ,T} be as above. Then

n(β′_{λ,T}) = p^{l(λ,T)}O where l(λ, T) := ∑_{i∈T} \frac{1}{2}l(λ, i).

\underline{Proof.} Let i ∈ N − S(λ) and l := \frac{1}{2}l(λ, i). Then

β′_{λ,i} = f_{1} … f_{l}g_{l} … g_{1}

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

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

Since the ϵ_{1}-coefficient of β′_{λ,i} is 0, the properties of the norm (see section 2) imply that p^{l} divides n(β′_{λ,i}) and therefore p^{l(λ,T)} divides n(β′_{λ,T}).

On the other hand

∑_{T ⊆N−S(λ)} l(λ, T) = \frac{1}{2} ∑_{T ⊆N−S(λ)} ∑_{i∈T} l(λ, i) = \frac{1}{2} ∑_{i∈N−S(a)} l(λ, i)2^{n−1} = f2^{n−1}.

The Lemma now follows from Remark 3.23 and Lemma 2.3. □ In particular one finds

Corollary 3.25 Let λ ∈ P′, i ∈ S(λ), i − 1 ∉ S(λ) and γ := f(i, −1)(λ). Then φ_{i,1,γ}φ_{i,−1,λ} ∈ pEnd_{RG}(P_{λ})^{∗}.

\underline{Proof.} Since i−1, i ∉ S(γ) one has [γ, i] = Ψ((γ|(i, −1), (i, 1)|γ)). Therefore β′_{γ,i} = φ_{i,−1,λ}φ_{i,1,γ} is an element of norm p according to Lemma 3.24. By Theorem 3.2 φ_{i,1,γ}φ_{i,−1,λ} ∈ pEnd_{RG}(P_{λ}). So there is a unit u ∈ End_{RG}(P_{λ}) with φ_{i,1,γ}φ_{i,−1,λ} = pu. □

To calculate the other β′_{λ,m} we need to consider m-strings. In the moment we only know the β′_{λ,m} corresponding to the ends of the m-strings.

Lemma 3.26 Let (λ^{(0)}, …, λ^{(p−1)}) be an m-string with origin λ^{(0)} ∈ P′. For j = 1, …, p − 1 define

f_{j} := φ_{λ^{(j)},m,−1}, g_{j} := φ_{λ^{(j−1)},m,1} ∈ E.

(i) g_{p−1}f_{p−1} ∈ pEnd_{RG}(P_{λ^{(p−1)}})^{∗}.

(ii) If 1 ≤ j < p − 1 then

g_{j}f_{j} = ∑_{i=1}^{s} a_{ji}ϵ_{i}

with a_{ji} ∈ pO (1 ≤ i ≤ s) and a_{ji} ∉ p℘ for all i ∈ c_{λ^{(j)}} − c_{λ^{(j+1)}}.

(iii) If 1 ≤ j < p − 1 then g_{j}f_{j} − f_{j+1}g_{j+1} ∈ pEnd_{RG}(P_{λ^{(j)}})^{∗}.

\underline{Proof.} (i) is the statement of Corollary 3.25. (ii) and (iii) is shown by induction. Let j = p − 2. Theorem 3.2 says that

g_{j}f_{j} − f_{j+1}g_{j+1} ∈ pEnd_{RG}(P_{λ^{(j)}}).

Because of (i) the norm of this endomorphism is pO. Hence \frac{1}{p}(g_{j}f_{j} − f_{j+1}g_{j+1}) ∈ End_{RG}(P_{λ^{(j)}})^{∗} which implies (iii) and (ii) for j = p − 2. Now assume that 0 ≤ j < p − 2 and (ii) holds for j + 1. By Theorem 3.2 one has g_{j}f_{j} − f_{j+1}g_{j+1} ∈ pEnd_{RG}(P_{λ^{(j)}}). Since (ii) holds for j + 1 and c_{λ^{(j+1)}} − c_{λ^{(j+2)}} ≠ ∅ the norm of this endomorphism is pO. Therefore (iii) and hence also (ii) holds for j. □

Now we want to show an analogous statement to Lemma 3.24 also for λ ∈ P̃. To this purpose we recall what is already shown in section 3.2.

Remark 3.27 Let (λ^{(1)}, …, λ^{(f)}) be a circle in Q with λ^{(i)} ∈ P′. Then

c_{λ^{(1)}} ∩ … ∩ c_{λ^{(f)}} = A ∪ {χ_{0}, …, χ_{f−1}}.

Lemma 3.28 Let λ ∈ P̃ ∩ P′, T ⊆ N − S(λ) and β′_{λ,T}, β′_{λ} be as above. Then

n(β′_{λ,T}) = p^{l(λ,T)}O and n(β′_{λ}) = \{ \begin{matrix}p^{f/2}R & f even \\ ℘^{f} & f odd \end{matrix} .

\underline{Proof.} As in the proof of Lemma 3.24 one finds that n(β′_{λ,T}) is divisible by p^{l(λ,T)}O if one uses Lemma 3.26 (ii). By Lemma 2.3 the norm of the last basis vector β′_{λ} divides p^{f/2}R respectively ℘^{f} = p^{\frac{f−1}{2}} ℘ depending on whether f is even or odd. So it suffices to show that p^{f/2} respectively ℘^{f} divides the coefficients of β′_{λ}. Then the Lemma follows similarly as Lemma 3.24.

Let (λ^{(1)}, …, λ^{(f)}) be a circle through λ =: λ^{(1)}. Let f_{i} ∈ Hom_{RG}(P_{λ^{(i)}}, P_{λ^{(i+1)}}) denote a lift of the image of Ψ(λ^{(i)}, (m_{i}, ν_{i}), λ^{(i+1)}) (1 ≤ i ≤ f) such that

β′_{λ} = ∏_{i=1}^{f} f_{i} = ∑_{t=1}^{s} b_{t}ϵ_{t}

with b_{t} ∈ O. Remark 3.27 says that

(1) b_{t} = 0, if t ∉ A ∪ {χ_{0}, …, χ_{f−1}}.

Since λ^{(i)} and λ^{(i+1)} are neighbours on an m_{i}-string there is γ^{(i)} ∈ P′, such that (λ^{(i+1)}|(m_{i}, ν_{i})|γ^{(i)}) ∈ Q or p = 3 and there is γ^{(i)} with (γ^{(i)}|(m_{i}, ν_{i})|λ^{(i)}) ∈ Q. We assume that for all 1 ≤ i ≤ f the condition (2) holds:

(2) There is γ^{(i)} ∈ P′, such that (λ^{(i+1)}|(m_{i}, ν_{i})|γ^{(i)}) ∈ Q.

With analogous considerations one can also treat the other case. For 1 ≤ i ≤ f let h_{i} be a lift of Ψ(λ^{(i+1)}|(m_{i}, ν_{i})|γ^{(i)}) and let g_{i} ∈ Hom_{RG}(P_{λ^{(i+1)}}, P_{λ^{(i)}}) be a lift of Ψ(λ^{(i+1)}, (m_{i}, −ν_{i}), λ^{(i)}). Theorem 3.2 states that f_{i}h_{i} ∈ pHom_{RG}(P_{λ^{(i)}}, P_{γ^{(i)}}). More precisely there are e_{i} ∈ End_{RG}(P_{λ^{(i+1)}}) such that

(3) f_{i}h_{i} = p ∏_{j=1,j≠i}^{f} g_{j}e_{i}h_{i} (1 ≤ i ≤ f)

by Proposition 3.6. Lemma 3.26 yields that the coefficient of h_{i} at ϵ_{χ_{m_{i}}} is not zero. For j = 1, …, f let x_{j} and y_{j} be the coefficients of f_{j}, and g_{j} and e the coefficient of e_{i} at ϵ_{χ_{m_{i}}}. Because of (3)

x_{i} = p ∏_{j=1,j≠i}^{f} y_{j}e.

The coefficient of β′_{λ} at ϵ_{χ_{m_{i}}} is

b_{χ_{m_{i}}} = ∏_{j=1}^{f} x_{j} = p ∏_{j=1,j≠i}^{f} x_{j}y_{j}e.

By Lemma 3.26 (ii) x_{j}y_{j}R = pR therefore

(∗) p^{f} divides b_{χ_{m_{i}}} for all 1 ≤ i ≤ f.

Let β′′_{λ} = ∏_{j=f}^{1} g_{j} ∈ End_{RG}(P_{λ}) be defined analogously to β′_{λ} going the other way around the circle. Then n(β′_{λ}β′′_{λ}) is divisible by p^{f} by Lemma 3.26 (iii). With the same Lemma (ii) one finds

(4) If t ∈ A, then the coefficients x′_{j} and y′_{j} of f_{j} and g_{j} at ϵ_{t} are not 0.

Let b_{t} = ∏_{j=1}^{f} x′_{j} and b′_{t} := ∏_{j=1}^{f} y′_{j}. Then p^{f} | b_{t}b′_{t}. If f is odd, then ℘^{f} | b_{1}O or ℘^{f} | b′_{1}O. Since one may take β′′_{λ} instead of β′_{λ} as basis vector of End_{RG}(P_{λ}) it follows that b_{1}O = b′_{1}O = ℘^{f} = n(β′_{λ}). Assume now that f is even. Let Φ = T r_{u} be the symmetrizing form of End_{RG}(P_{λ}). Then Φ(β′_{λ}, id_{|P_{λ}}) ∈ R. But p^{f}Φ(β′_{λ}, id_{|P_{λ}}) ≡ b_{1} + b_{2} (mod p^{f}R). Therefore b_{1} ≡ −b_{2} (mod p^{f}R). As in the case f odd one concludes that both b_{1} and b_{2} have p-adic valuation \frac{f}{2} and n(β′_{λ}) = p^{f/2}R. □

The statement (*) of the proof above allows to conclude the following:

Corollary 3.29 There is a unit x ∈ R^{∗}, such that β′_{λ} ≡ xβ_{λ} (mod p^{f} ⊕_{i∈c_{λ}} Rϵ_{i}), where

β_{λ} := \{ \begin{matrix}p^{f/2}(ϵ_{1} − ϵ_{2}) & f even \\ p^{(f−1)/2}\sqrt{μp}ϵ_{1} & f odd . \end{matrix}

We now define generators β_{λ,T} of End_{RG}(P_{λ}) that will replace the old β′_{λ,T}.

Definition 3.30 (a) Let j ∈ {0, …, p − 2}. If the m-string (λ^{(0)}, …, λ^{(p−1)}) is not tangent or j ≠ \frac{p−3}{2}, then define

pr_{λ^{(j)},m} := ∑_{t∈c_{λ^{(j)}}∩c_{λ^{(j+1)}}} ϵ_{t} ∈ E.

Otherwise let χ_{m} be as in Lemma 3.20 and define

pr_{λ^{(j)},m} := ∑_{t∈c_{λ^{(j)}}∩c_{λ^{(j+1)}}−{χ_{m}}} ϵ_{t} ∈ E.

(b) For λ ∈ P′ let pr_{λ} := ∑_{t∈c_{λ}} ϵ_{t} ∈ E be the unit element of End_{RG}(P_{λ}) ⊂ E.

(c) Let λ ∈ P′, m ∈ N − S(λ). If m − 1 ∉ S(λ), then λ lies on an m-string.

Then let

β_{λ,m} := p · pr_{λ,m}.

If m − 1 ∈ S(λ), then let j − 1 ∉ S(λ) be the unique element such that J := {j, j + 1, …, m − 1} ⊆ S(λ). Then define γ := f(m − 1, 1)(f(m − 2, 1)(… f(j + 1, 1)(f(j, 1)(λ)) …)) and

β_{λ,m} := p^{|J|+1}pr_{γ,m}pr_{λ}.

(d) For T ⊂ N − S(λ) let β_{λ,T} := ∏_{m∈T} β_{λ,m}, where β_{λ,∅} := pr_{λ}.

In the same spirit as Lemma 3.26 and using the notation there one shows Lemma 3.31 For j = 0, …, p − 2

p · pr_{λ^{(j)},m} ∈ End_{RG}(P_{λ^{(j)}}) ∩ End_{RG}(P_{λ^{(j+1)}}).

More precisely there are units u_{j}, v_{j} ∈ End_{RG}(P_{λ^{(j)}})^{∗} such that

p · pr_{λ^{(j)},m} ≡ u_{j}f_{j+1}g_{j+1} ≡ g_{j+1}f_{j+1}v_{j+1} (mod ⊕_{l=0}^{f−1} p^{f+1}Rϵ_{χ_{l}}).

With this lemma one now can give a precise description of the endomorphism rings of the projective indecomposable RG-lattices:

Theorem 3.32 Let λ ∈ P′. If λ ∉ P̃, then

End_{RG}(P_{λ}) = ⟨β_{λ,T} | T ⊂ N − S(λ)⟩_{R}.

If λ ∈ P̃, then

End_{RG}(P_{λ}) = ⟨β_{λ,T} , β_{λ} | T ⊂ N − S(λ)⟩_{R}.

\underline{Proof.} From Lemma 3.31 and Lemma 3.26 it follows that β_{λ,m} ∈ End_{RG}(P_{λ}). If λ ∈ P̃, then Corollary 3.29 states that β_{λ} ∈ End_{RG}(P_{λ}). Write β_{λ,T} = ∑_{t∈c_{λ}} a(λ, T)_{t}ϵ_{t} and β′_{λ,T} = ∑_{t∈c_{λ}} a(λ, T)′_{t}ϵ_{t}. Since the β_{λ,m} are obtained from β′_{λ,m} by multiplication with units in local rings, there are k_{T} ∈ R^{∗} with k_{T} a(λ, T)_{t} ≡ a(λ, T)′_{t} (mod pn(β_{λ,T})) for all t ∈ c_{λ}. A relation between the β_{λ,T} (modulo ℘) increases the index of the order generated by the β′_{λ,T}. Therefore the β_{λ,T} generate the same R-order as the β′_{λ,T} and the Theorem follows from Corollary 3.21. □

The remainder of the paper is devoted to the description of the homomorphism spaces between different projective indecomposable RG-lattices and the determination of the order Γ := ⨁_{t=1}^{s} Λϵ_{t}. The order Γ is a graduated order. Since all p-modular characters of RG are self dual, one may calculate exponent matrices for Γ from the structure constants m^{(t)}(λ, γ) (λ, γ ∈ P′, t ∈ c_{λ} ∩ c_{γ}) by Remark 2.6.

Proposition 3.33 Let λ, γ ∈ P′ such that λ or γ does not lie in P̃ and Hom_{RG}(P_{λ}, P_{γ}) ≠ {0}. Let pr_{λ,γ} := ∑_{t∈c_{λ}∩c_{γ}} ϵ_{t}. Then

Hom_{RG}(P_{λ}, P_{γ}) ≅ (pr_{λ,γ}(End_{RG}(P_{γ⊖λ}))

as End_{RG}(P_{λ}) − End_{RG}(P_{γ})-bimodule. The non trivial structure constants (see Remark 2.6) are

m^{(t)}(λ, γ) = d(λ, γ),

where d(λ, γ) is as in Definition 3.11.

\underline{Proof.} Let T_{1} ⊂ N − (S(λ) ∪ S(γ)) be as in Definition 3.11. Proposition 3.6 says that Hom_{RG}(P_{λ}, P_{γ}) is generated as End_{RG}(P_{γ⊖a})-module of any lift of

w_{λ,γ} := ⟨λ ⊃ (λ ⊖ γ)⟩⟨(λ ⊖ γ) ∼ (γ ⊖ λ), T_{1}⟩⟨(γ ⊖ λ) ⊂ γ⟩

whence Hom_{RG}(P_{λ}, P_{γ}) ≅ (pr_{λ,γ}(End_{RG}(P_{γ⊖λ})).

Let β_{λ,γ} be the product of the lifts of the φ_{i,h,λ′} that occur in w_{λ,γ}. Analogously define β_{γ,λ}. Since one of λ and γ, say λ, does not lie in P̃ one has A ∩ c_{λ} = ∅. Let m ∈ {0, …, f − 1} such that χ_{m} ∈ c_{λ} ∩ c_{γ}. Then by Lemma 3.17 λ_{m} ∈ \{l − 2, l + 1\} where l := \frac{p−1}{2}

Assume that the path to w_{λ,γ} contains an exceptional arrow (λ′|(m, ±1)|γ′) in direction m. Then λ′_{m} = l − 1 and γ′_{m} = l or λ′_{m} = l and γ′_{m} = l − 1. Since w_{λ,γ} is of the shape ⟨λ ∼ γ, T ⟩ for some T ⊂ N, the entry λ_{m} is changed at most once again to p − 2 − λ_{m}. Since p − 2 − (l − 2) = l + 1 and p − 2 − (l + 1) = l − 2 ≠ l, l − 1, this is a contradiction.

Therefore β_{λ,γ}β_{γ,λ} = ∑_{t∈c_{λ}∩c_{γ}} x_{t}ϵ_{t}, with x_{t}R = p^{d(λ,γ)}R by Lemma 3.31. The proposition follows. □

If both elements λ and γ lie in P̃, then the description of Hom_{RG}(P_{λ}, P_{γ}) is more complicated, since this module is not a cyclic End_{RG}(P_{γ})-module.

Proposition 3.34 Let λ, γ ∈ P̃ ∩ P′. Then there are T_{1}, T_{2} ⊂ N with N = T_{1} ∪^{.} T_{2} such that ⟨λ ∼ γ|T_{i}⟩ ∈ Q for i = 1, 2. Let d_{i} := |T_{i}| denote the length of these paths, where we assume w.l.o.g. that d_{1} ≤ d_{2}. For i = 1, 2 let C_{i} be the set of indices of the simple KG-modules that lie in c_{x}, for all x in the path ⟨λ ∼ γ|T_{i}⟩ and R_{i} := {χ_{m} | m ∈ T_{i}} ⊂ C_{i}. Let t ∈ c_{λ} ∩ c_{γ}. Then

m^{(t)}(λ, γ) = \{ \begin{matrix}d_{1} & t ∈ C_{1} − C_{2} \\ d_{2} & t ∈ C_{2} − C_{1} \\ f − d_{i} & t ∈ R_{i} \\ md_{1} & t ∈ A \end{matrix}

where m := 1, if f is even and m := 2, if f is odd. To get the isomorphism type of Hom_{RG}(P_{λ}, P_{γ}) define

β_{1} := ∑_{t∈C_{1}−R_{1}} ϵ_{t} and β_{2} := ∑_{t∈C_{2}−R_{2}−A} ϵ_{t} + b_{2}

with

b_{2} := \{ \begin{matrix}p^{\frac{f}{2} −d_{1}}(ϵ_{1} − ϵ_{2}), & if f is even \\ p^{\frac{f−1}{2} −d_{1}} \sqrt{μp}ϵ_{1}, & if f is odd \end{matrix}

Then

Hom_{RG}(P_{λ}, P_{γ}) ≅ β_{1}End_{RG}(P_{γ}) + β_{2}End_{RG}(P_{γ}) + ∑_{t∈R_{1}} p^{d_{2}}Rϵ_{t} + ∑_{t∈R_{2}} p^{d_{1}}Rϵ_{t}

as End_{RG}(P_{λ}) − End_{RG}(P_{γ})-bimodule.

\underline{Proof.} By Proposition 3.6 and Lemma 3.27

c_{λ} ∩ c_{γ} = C_{1} ∪ C_{2} and C_{1} ∩ C_{2} = A ∪ R_{1} ∪ R_{2} = A ∪ {χ_{0}, …, χ_{f−1}}.

For j = 1, 2 let β_{j} be the product of the lifts of the φ_{i,h,λ′} that occur in ⟨λ ∼ γ|T_{j}⟩. Then β_{1} and β_{2} generate the End_{RG}(P_{γ})-module Hom_{RG}(P_{λ}, P_{γ}) by Proposition 3.6.

For j = 1, 2 write β_{j} = ∑_{t∈C_{j}} x_{t}^{(j)}ϵ_{t} with x_{t}^{(j)} ∈ O.

Analogously define β′_{1}, β′_{2} to the paths ⟨γ ∼ λ|T_{1}⟩ and ⟨γ ∼ λ|T_{2}⟩. Let ^{∘} be the involution on Λ induced by the R-linear mapping on RG defined by g ↦ g^{−1} for all g ∈ G. Then ^{∘} maps End_{RG}(P_{λ})β_{i}End_{RG}(P_{γ}) onto End_{RG}(P_{γ})β′_{i}End_{RG}(P_{λ}). Hence there are a_{λ}^{(i)} ∈ End_{RG}(P_{λ}), a_{γ}^{(i)} ∈ End_{RG}(P_{γ}) and f_{t} ∈ O (t ∈ C_{i}) with

β_{i}^{∘} = ∑_{t∈C_{i}} f_{t}(x_{t}^{(i)})^{∘}ϵ_{t} = a_{γ}^{(i)}β′_{i}a_{λ}^{(i)}.

Since β_{i}^{∘} also generates the bimodule End_{RG}(P_{γ})β′_{i}End_{RG}(P_{λ}) the elements a_{γ}^{(i)} and a_{λ}^{(i)} are units in the local rings End_{RG}(P_{γ}) and End_{RG}(P_{λ}) ⊂ E. In particular the coefficients of β_{i}^{∘} and β′_{i} have the same p-adic valuation.

Now the structure constants are determined. To this purpose let t ∈ c_{λ} ∩ c_{γ}. First assume that t ∉ C_{1} ∩ C_{2} and let t ∈ C_{i} (i = 1 or 2). Then by Lemma 3.31

ϵ_{t}Hom_{RG}(P_{λ}, P_{γ})Hom_{RG}(P_{γ}, P_{λ}) = p^{d_{i}}Rϵ_{t}

whence m^{(t)}(λ, γ) = d_{i}. Now let t = χ_{m} ∈ R_{i} for some 0 ≤ m ≤ f − 1. As in the proof of Lemma 3.28 (3) one sees that f_{t}(x_{t}^{(i)})^{2} ∈ p^{f}R. Therefore the p-adic valuation ν_{p}(x_{t}^{(i)}) ≥ \frac{1}{2}(f − ν_{p}(f_{t})). For \{i, l\} = \{1, 2\} one finds for x_{t}^{(l)}

(1) f_{t}(x_{t}^{(l)})^{2}R = p^{d_{l}}R.

Therefore m^{(t)}(λ, γ) = d_{l} = f − d_{i}. Now assume t ∈ A. Then ν_{p}(f_{t}(x_{t}^{(i)})(x_{t}^{(i)})^{∘}) = d_{i}. Therefore m^{(t)}(λ, γ) = m · min(d_{1}, d_{2}) = m · d_{1}. The coefficients of β_{1} and β_{2} can be determined as follows: Applying an isomorphism one may assume that x_{t}^{(i)} = 1 for all t ∈ C_{i} − R_{i} − A and i = 1, 2 and that x_{t}^{(1)} = 1 for all t ∈ A. Let i = 1 or i = 2. First it is shown:

(2) If t ∈ R_{i}, then p^{d_{i}} divides x_{t}^{(i)}.

By Lemma 3.31 there are units u_{1}, …, u_{d_{i}} in the endomorphism rings of the projective indecomposable RG-lattices occurring in the path ⟨λ ∼ γ|T_{i}⟩ such that β_{i}β_{i}^{∘}u = ∑_{t∈C_{i}−R_{i}} p^{d_{i}}ϵ_{t} + p^{f+d_{i}} ∑_{t∈R_{i}} a_{t}ϵ_{t}, where a_{t} ∈ O and u := ∏_{l=1}^{d_{i}} u_{l} = ∑_{t∈C_{i}} v_{t}ϵ_{t} with v_{t} ∈ O^{∗}. Let {i, l} = {1, 2}. Then

f_{t}x_{t}^{(i)}(x_{t}^{(i)})^{∘}v_{t} = \{ \begin{matrix}p^{f+d_{i}}a_{t} & t ∈ R_{i} \\ p^{d_{i}} & t ∈ C_{i} − R_{i}. \end{matrix}

If t ∈ R_{i} then ν_{p}(f_{t}) = d_{l} by (1) because x_{t}^{(l)} = 1. Since the p-adic valuations of x^{(i)} and (x^{(i)})^{∘} are the same, ν_{p}(x_{t}^{(i)}) ≥ \frac{1}{2}(f + d_{i} − ν_{p}(f_{t})) = \frac{1}{2}(f + d_{i} − d_{l}) = d_{i} for all t ∈ R_{i}. Hence (2) is proven. If t ∈ R_{i} then x_{t}^{(l)} is chosen to be 1 and p^{f−d_{l}}ϵ_{t}β_{l} ∈ Hom_{RG}(P_{λ}, P_{γ}). Therefore one can assume that x_{t}^{(i)} = 0 for all t ∈ R_{i}. It suffices to determine the coefficients x_{t}^{(2)} for t ∈ A. To this purpose consider β_{1}^{∘}β_{2} = ∑_{t∈A} f_{t}x_{t}^{(2)}ϵ_{t} ∈ End_{RG}(P_{λ})∩End_{RG}(P_{γ}). By Lemma 3.31 there is u ∈ R^{∗}, such that f_{t} = up^{d_{1}} for all t ∈ A. Since the norm of β_{1}^{∘}β_{2} is p^{f/2}R (if f is even) respectively ℘^{f} (if f is odd) by Lemma 3.28, one gets with Corollary 3.29, that β_{1}^{∘}β_{2} is an R^{∗}-multiple of p^{d_{1}}b_{2}. Replacing β_{2} by an R^{∗}-multiple and adjusting the coefficients of β_{2} at ϵ_{t} with t ∈ C_{2} − C_{1} one gets the proposition. □

3.4 The principal block of SL_{2}(3^{3})

Let G := SL_{2}(3^{3}) and R := ℤ_{3}[ζ_{26}]. The decomposition matrix of the principal block of RG:

000 200 020 002 011 101 110 022 202 220 211 121 112
1 3 3 3 4 4 4 9 9 9 12 12 12
1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
δ 13 1 0 0 0 1 1 1 0 0 0 0 0 0
δ′ 13 1 0 0 0 1 1 1 0 0 0 0 0 0
δ_{4} 26 0 1 0 1 1 1 0 0 0 0 0 0 1
δ_{12} 26 0 1 1 0 0 1 1 0 0 0 1 0 0
δ_{8} 26 0 0 1 1 1 0 1 0 0 0 0 1 0
δ_{2} 26 1 0 0 0 0 1 0 1 0 0 0 0 1
δ_{6} 26 1 0 0 0 0 0 1 0 1 0 1 0 0
δ_{10} 26 1 0 0 0 1 0 0 0 0 1 0 1 0
η_{4} 28 1 0 0 1 1 1 1 0 0 0 0 0 1
η_{12} 28 1 1 0 0 1 1 1 0 0 0 1 0 0
η_{10} 28 1 0 1 0 1 1 1 0 0 0 0 1 0
η_{2} 28 0 1 0 0 0 1 0 1 0 0 0 0 1
η_{6} 28 0 0 1 0 0 0 1 0 1 0 1 0 0
η_{8} 28 0 0 0 1 1 0 0 0 0 1 0 1 0

Corollary 3.35 The action of RG on the simple KG-module with character χ is given by the graduated order Λ_{χ}, where, in contrary to Definition 2.4, not the dimensions but the names of the modular constituents of χ are given.

Λ_{1} = Λ(R, 000, (0)), Λ_{δ+δ′} = Λ(R[\sqrt{−3}], 000, 011, 101, 110, V), Λ_{δ_{4}} = Λ(R, 200, 101, 112, 011, 002, N), Λ_{δ_{2}} = Λ(R, 000, 101, 112, 022, U), Λ_{η_{2}} = Λ(R, 200, 101, 112, 022, U), Λ_{η_{4}} = Λ(R, 000, 101, 110, 011, 112, 002, M), where

V := ( \begin{matrix}0 & 0 & 0 & 0 \\ 2 & 0 & 1 & 1 \\ 2 & 1 & 0 & 1 \\ 2 & 1 & 1 & 0 \end{matrix} )), N := ( \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 2 & 1 & 0 & 1 & 0 \\ 2 & 1 & 1 & 0 & 0 \\ 3 & 2 & 1 & 1 & 0 \end{matrix} )), U := ( \begin{matrix}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 3 & 2 & 1 & 0 \end{matrix} ) ,

and M := ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 \\ 2 & 1 & 1 & 0 & 1 & 0 \\ 2 & 1 & 2 & 1 & 0 & 0 \\ 3 & 2 & 2 & 1 & 1 & 0 \end{matrix} )).

The other projections of RG to the simple components of KG in the principal block are calculated by applying the Galois automorphism F.

These exponent matrices allow to read off the inclusion patterns of the irreducible RG-lattices as explained in [Ple 83, Remark (II.4)], from which we also use the notation. The inclusions of the lattices corresponding to δ_{2}, δ_{6}, δ_{10}, η_{2}, η_{6}, and η_{8} (left picture) respectively to δ + δ′ (right picture) are given as follows:

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1000 1100 2100  1110 1111 = 3L 2111 2110 3210 2210 3211 2211 0000 =: L

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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.} 1110  1000  2001 2011 1100  2111 2000 1001 1011  2100 2110 1101  2101 2010 0000 =: M  1010 3M = 1111

\underline{Proof.} To get a nice exponent matrix for Λ_{χ} one looks at the subgraph of the Ext-quiver on those vertices that correspond to the modular constituents of the character χ. If χ is Galois conjugate to δ_{2} or η_{2}, then this subgraph is a straight line. If one orders the constituents along this line, one always finds an exponent matrix U = U_{ij} with U_{ij} = max(0, i − j) (cf. [Neb 99, Satz 5.6.1]). We only show how to get the exponent matrix in the most complicated case η_{4}. To order the modular constituents we start at one extremal point λ of the subgraph, say λ = 000. Note that the vertex 002 has distance 3 from 000, since only paths without repeated directions are allowed. We then list the vertices according to their distance to λ. Let M be the exponent matrix of Λ_{η_{4}} corresponding to the ordering of the modular constituents as above such that m_{1,j} = 0 for all j. Now η_{4} corresponds to the character with number χ_{2}. The minimum of the length of the paths from 000 to 110, 101, or 011 is 1, but the edge (000, 011) has direction 2. So by Proposition 3.34 the first 4 entries in the first column of M are 0, 1, 1, 2. Since 112, 002 ∉ P̃, Proposition 3.33 gives the remaining entries in the first column of M. The distance between 101 and 110 in Q is 1. But the edge joining the two vertices has direction 2. So by Proposition 3.34, the sum of the two entries m_{23}+m_{32} is 2. Now Lemma 2.5 yields m_{23} = m_{32} = 1. Analogously one calculates the remaining entries of M. □

References

[AJL 83] H.H. Andersen, J. Jørgensen, P. Landrock, The projective indecomposable modules of SL_{2}(K), Proc. Lond. Math. Soc., III. Ser. 46, (1983) 38-52.

[Bur 76] R. Burkhardt, Die Zerlegungsmatrizen der Gruppen P SL(2, p^{f}). J. Algebra 40, (1976) 75-96.

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