The principal block of ℤ_{p}S_{2p}.

Gabriele Nebe

Abt. Reine Mathematik, Universität Ulm, 89069 Ulm, Germany nebe@mathematik.uni-ulm.de

1 Introduction

Let p be an odd prime. This note describes the principal block B of the group ring of the symmetric group S_{2p} over the p-adic integers ℤ_{p}.

The principal block \overline{B} of 𝔽_{p}S_{2p} has been considered in several papers. [Mar1] determines the Ext-quiver of \overline{B}, [Mar2] the Loewy series of the projective indecomposable \overline{B}-modules and finally [ErM] gives a presentation for the basic algebra that is Morita equivalent to \overline{B}, where some constants are not fully determined.

The present note does not depend on these results. The most important starting point used here is the decomposition matrix of B, which is calculated with the Jantzen-Schaper formula [Schap], [Jan]. Then the description of B is a very easy application of the general theory developed by Plesken [Ple1], [Ple2] and further in [Neb].

J. Scopes [Sco] shows that many properties of \overline{B} also hold for all other symmetric group blocks of defect 2. I am confident that the same methods used here can also be applied to describe these blocks up to Morita equivalence.

2 Notation.

Recall that the iirreducible ordinary respectively p-modular characters of the symmetric group S_{n} are indexed by the partitions respectively the p-regular partitions of n (see e.g. [Jam]).

To denote the partitions of 2p we use the ⟨2^{p}⟩-notation as explained in [Sco], [ErM], [Mar2]: If 1 ≤ j ≤ p then ⟨j⟩ is the ⟨2^{p}⟩-notation for the partition (p + j, 1^{p−j}) of 2p and for 1 ≤ j < k ≤ p, the symbol ⟨k, j⟩ stands for (k, j + 1, 2^{p−k}, 1^{k−j−1}) and ⟨k, k⟩ denotes (k, 1^{2p−k}). To distinguish between ordinary and modular characters in B, the ordinary characters are denoted by ⟨⟨j⟩⟩, ⟨⟨k, j⟩⟩ (1 ≤ j ≤ k ≤ p), whereas ⟨j⟩, (1 ≤ j ≤ p) and ⟨k, j⟩ (1 ≤ j < k ≤ p) are the modular characters in B.

3 The decomposition matrix of B

The decomposition matrix of B is given in [Mar2, Fig 8] (where a 1 is missing in row ⟨⟨k, k − 1⟩⟩ column ⟨k, k − 1⟩ and k runs from 4 (or even 3) to p − 1). It is easily calculated with the Jantzen-Schaper formula.

The decomposition matrix of B has a very regular shape: Let

X_{k} := ( \begin{matrix}1 & 0 & 0 & … & 0 \\ 1 & 1 & ... & ... & ⋮ \\ 0 & ... & ... & ... & 0 \\ ⋮ & ... & ... & ... & 0 \\ 0 & … & 0 & 1 & 1 \\ 0 & … & … & 0 & 1 \end{matrix} ) ∈ ℤ^{(k+1)×k}

and X′_{p} ∈ ℤ^{p×p} obtained from X_{p} by deleting the last row. Let

Y_{k} := ( \begin{matrix}0 & 1 & 0 & … & … & 0 \\ 1 & 1 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & 1 & ... & ⋮ \\ ⋮ & ... & ... & ... & ... & 0 \\ 0 & … & 0 & 0 & 1 & 1 \\ 0 & … & … & … & 0 & 1 \end{matrix} ) ∈ ℤ^{k×k}

and

Z_{k} := ( \begin{matrix}1 & 0 & … & 0 \\ 0 & 0 & … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & … & 0 \end{matrix} ) ∈ ℤ^{(k−1)×k}.

Theorem 1 If we order the partitions of 2p that belong to B lexicographically then the decomposition matrix of B has the following form:

\begin{matrix}X′_{p} & 0 & … & … & … & 0 \\ Y_{p} & X_{p−1} & 0 & … & … & 0 \\ Z_{p} & Y_{p−1} & X_{p−2} & 0 & … & ⋮ \\ 0 & Z_{p−1} & Y_{p−2} & X_{p−3} & ... & 0 \\ ⋮ & ... & ... & ... & ... & 0 \\ 0 & … & 0 & Z_{4} & Y_{3} & X_{2} \\ 0 & … & … & 0 & Z_{3} & Y_{2} \\ 0 & … & … & … & 0 & Z_{2} \end{matrix}

The ordinary characters that belong to the rows of X′_{p} are ⟨⟨p⟩⟩, ⟨⟨p − 1⟩⟩, … ⟨⟨1⟩⟩ and and the modular characters belonging to the columne of X′_{p} are ⟨p⟩, ⟨p − 1⟩, … ⟨1⟩. The ordinary characters that correspond to the rows of X_{k} are ⟨⟨k + 1, k⟩⟩, …, ⟨⟨k + 1, 1⟩⟩, ⟨⟨k + 1, k + 1⟩⟩, and the modular characters ⟨k + 1, k⟩, …, ⟨k + 1, 1⟩, ⟨k + 1, k⟩, belong to the columns of X_{k} (k = 0, …, p − 1).

[Mar2] uses Scopes result, that the decomposition numbers in B are ≤ 1 to get the decomposition matrix from the Jantzen-Schaper formula [Schap], [Jan]. In general this formula only gives the positions of the non zero entries in the decomposition matrix and upper bounds for these entries. For B, the formula says that some of the 1 on the diagonal of Y_{k} might be 2 and the 1 in Z_{k} might also be 2. In these cases the assumptions of the following proposition which is only formulated and proved for the special case needed here, but holds in more general situations, are satisfied. This shows that one gets the decomposition matrix of B directly from the Jantzen-Schaper formula.

Proposition 2 Let p > 2 be an odd prime, n ∈ ℕ and λ be a p-regular partition of n, χ the irreducible ordinary character and φ the p-modular character of S_{n} that belongs to λ. Let L be the Specht lattice with head φ. Let φ′ be an other modular constituent of χ such that χ is the only irreducible character that has both φ and φ′ as a p-modular constituent. Let μ be the multiplicity of φ′ in L^{#}/L calculated by the Jantzen-Schaper formula and ν := ν_{p}(n!)/ν_{p}(χ(1)). Then the multiplicity of φ′ in L/pL is

d_{χ,φ′} = \frac{μ}{ν}.

\underline{Proof.} The group ring ℤ_{p}S_{n} has a canonical involution ^{∘} inverting the group elements. Since p is odd, one can find an involution invariant idempotent i = i^{∘} = i^{2} ∈ ℤ_{p}S_{n} such that iℤ_{p}S_{n}i =: Λ is a basic order ([Neb, Bemerkung 4.3.13]). Now χ is real, so Γ := Λ(χ), the projection of Λ into the simple component of ℚ_{p}Λ that belongs to χ, is also invariant under ^{∘}. The restriction of ^{∘} to Γ is an involution on a central simple algebra, it is of the form x ↦ F^{−1}x^{tr}F for some symmetric F ∈ ℚ_{p}Γ.

The lattice L corresponds to the projective Γ-lattice P = P_{φ} with head φ, so P = Γe for some primitive idempotent e = e^{∘} = e^{2} ∈ Γ. Since the multiplicity of φ in χ is 1, e is the lift of the central primitive idempotent in Γ/J(Γ) that corresponds to φ. Let f = f^{∘} = f^{2} ∈ Γ be a lift of the central primitive idempotent in Γ/J(Γ) that corresponds to φ′ and choose it orthogonal to e, i.e. ef = 0 = fe. Since the other modular constituents of χ do not play any role, we may assume that e+f = 1, by replacing Γ by (e+f)Γ(e+f). Again since p ≠ 2, we can choose a basis of P which is orthogonal with respect to F, such that the first basis vector spans eP ⊂ P, write everything with respect to this basis, and renormalize F such that this vector has length 1. The dual lattice of P with respect to F is P^{#} = F^{−1}P and μ = ν_{p}(det(F)).

On the other hand since χ is the only irreducible character that has both φ and φ′ as a p-modular constituent and Λ is a symmetric order, the dual of eΓf with respect to χ(1)/(n!)·trace is (eΓf)^{#} = fΓe (see [Thé, Proposition (1.6.2)]). Therefore one calculates

(eΓf)^{#} = fΓe = (eΓf)^{∘} = F^{−1}(eΓf)^{tr}F.

Since F is diagonal with first entry 1, one gets that

μ = ν_{p}(det(F)) = d_{χ,φ′}ν.
□

4 The degrees of the ordinary irreducible characters in B

The next easy lemma (which might be well known) shows that the degrees of the ordinary irreducible characters in B are ±1 or ±2 (mod p).

Lemma 3 The degrees of the ordinary irreducible characters that belong to B are: deg(⟨⟨k⟩⟩) ≡ deg(⟨⟨k, k⟩⟩) ≡ (−1)^{k−1} (mod p) for 1 ≤ k ≤ p, deg(⟨⟨k, j⟩⟩) ≡ 2 · (−1)^{j+k} (mod p) for 1 ≤ j < k ≤ p.

\underline{Proof.} This is an easy application of the hook-formula (cf. [Jam, Theorem 20.1]), which says that for a partition λ of n, the dimension dim(S^{λ}) is n! divided by the product of all hook lengths in λ. Note that (2p)!/(p^{2}) ≡ (p − 1)!^{2} (mod p). The notation ⟨⟨k⟩⟩ stands for the partition (p + k, 1^{p−k}). Applying the hook formula, one calculates deg(⟨⟨k⟩⟩) =

\frac{(2p)!}{(p − k)!(2p)(p + k − 1)!} ≡ \frac{(p − 1)!^{2}}{(p − k)!(p + k − 1) ⋯ (p + 1)(p − 1)!} ≡ (−1)^{k−1} (mod p).

⟨⟨k, k⟩⟩ stands for the hook (k, 1^{2p−k}). There the product of the hook lengths is (2p − k)!(2p)(k − 1)! ≡ 2p^{2}(p − 1)!^{2}(−1)^{k−1} (mod p^{3}). If k > j then ⟨⟨k, j⟩⟩ belongs to the partition (k, j + 1, 2^{p−k}, 1^{k−j−1}). The product of all hook lengths in this partition is (k − j − 1)!(k − j + 1) ⋯ (p − j)(p − k)!(j − 1)!(p + j − k)p(k − j − 1)!(k − j + 1) ⋯ (k − 1)p(p + k − j). Since p + k − j ≡ k − j (mod p) this is p^{2}(p−j)!(p−k)!(j −1)!(−1)(k −1)! (mod p^{3}). Now (j −1)! is (p−j +1)...(p−1)(−1)^{j−1} modulo p, therefore one gets p^{2}(p − 1)!^{2}(−1)^{k+j} modulo p^{3}. □

5 The exponent matrices.

If χ is an ordinary irreducible character in B, then let B(χ) denote the projection of B into the simple summand of ℚ_{p}B that corresponds to χ. Since the decomposition numbers of B are 0 or 1, the order B(χ) can be described using the language of exponent matrices as developed in [Ple2]. If φ_{1}, …, φ_{h} are the p-modular constituents of χ, of degree n_{i} := φ_{i}(1), then there is a matrix M(χ) = (m_{ij}(χ)) ∈ ℤ^{h×h}, such that B(χ) is conjugate to

Λ(φ_{1}, …, φ_{l}, M(χ)) :=
{X = (x_{ij})_{i,j=1,...,h} ∈ ℤ_{p}^{n×n} | x_{ij} ∈ (p^{m_{ij}(χ)}ℤ_{p})^{n_{i}×n_{j}} for all 1 ≤ i, j ≤ h}.

Here n = n_{1} + … + n_{h} = χ(1) is the degree of χ.

Since 1 ∈ B(χ), one clearly has m_{ii}(χ) = 0 for all i. All exponent matrices can be normalized such that the first column consists of zeros only and all entries of M are ≥ 0, by writing the matrices with respect to a suitable basis of the projective B(χ)-lattice that corresponds to φ_{1}.

Let c_{i} denote the set of all ordinary irreducible characters of B, that have φ_{i} as a p-modular constituent:

c_{i} := {χ ∈ Irr(B) | d_{χ,φ_{i}} = 1}.

The next lemma follows from [Ple2, Theorem (III.8)] and [Ple2, Corollary (IV.7)] (applied to the special situation here, that ℚ_{p} is a splitting field and all characters in B have defect 2).

Lemma 4 Let 1 ≤ i ≠ j ≤ l.

a) 0 < m_{ij}(χ) + m_{ji}(χ) ≤ 2 is bounded by the defect.

b) If c_{i} ∩ c_{j} = {χ} then m_{ij}(χ) + m_{ji}(χ) = 2.

c) If c_{i} ∩ c_{j} = {χ, ψ} then m_{ij}(χ) + m_{ji}(χ) = m_{ij}(ψ) + m_{ji}(ψ).

The exponent matrices for the hook representations (which are ⟨⟨j⟩⟩ and all non pregular partitions ⟨⟨k, k⟩⟩ in our particular situation) are given in [Ple2, Theorem (VI.2)]. If χ belongs to a p-regular partition, then M(χ) is determined by the Jantzen-Schaper formula (see [Schap], [Mar2, Appendix], [MaR, Section 1.2]), as stated in [Ple3, Proposition IV.1]: Order the p-modular constituents of χ such that φ_{1} belongs to the same partition as χ. Then the Specht lattice of χ is the projective B(χ) lattice L corresponding to φ_{1}. The Jantzen-Schaper formula gives the multiplicities m_{1j}(χ) of φ_{j} in L^{#}/L. It is easy to see that these form the first line of M(χ). Since all characters in B are self dual, one has

m_{ij}(χ) = m_{ji}(χ) + m_{1j}(χ) − m_{1i}(χ).

Since the m_{ij}(χ) are nonnegative and m_{ij}(χ) + m_{ji}(χ) is either 1 or 2 (for i ≠ j), this equation has only one solution.

Even without using the additional information on the first rows, the entries in the exponent matrices can be calculated successively by the same strategy as explained in [Ple2, Example (VI.3)] for p = 5.

Theorem 5

B(⟨⟨p⟩⟩) ≅ Λ(⟨p⟩, (0))

For j = p − 1, …, 1:

B(⟨⟨j⟩⟩) ≅ Λ(⟨j⟩, ⟨j + 1⟩, A)
B(⟨⟨p, p − 1⟩⟩) ≅ Λ(⟨p, p − 1⟩, ⟨p − 1⟩, A)
B(⟨⟨p, p − 2⟩⟩) ≅ Λ(⟨p, p − 2⟩, ⟨p, p − 1⟩, ⟨p − 2⟩, ⟨p − 1⟩, ⟨p⟩, B)

For j = p − 3, …, 1:

B(⟨⟨p, j⟩⟩) ≅ Λ(⟨p, j⟩, ⟨p, j + 1⟩, ⟨j⟩, ⟨j + 1⟩, C)
B(⟨⟨p, p⟩⟩) ≅ Λ(⟨p, 1⟩, ⟨1⟩, A)

For k = p − 1, …, 3:

B(⟨⟨k, k − 1⟩⟩) ≅ Λ(⟨k, k − 1⟩, ⟨k + 1, k − 1⟩, ⟨k + 2, k + 1⟩, D), if k < p − 1

B(⟨⟨k, k − 1⟩⟩) ≅ Λ(⟨k, k − 1⟩, ⟨k + 1, k − 1⟩, ⟨p⟩, D), if k = p − 1

B(⟨⟨k, k − 2⟩⟩) ≅ Λ(⟨k, k − 2⟩, ⟨k, k − 1⟩, ⟨k + 1, k − 2⟩, ⟨k + 1, k − 1⟩, ⟨k + 1, k⟩, B) and for j = k − 3, …, 1:

B(⟨⟨k, j⟩⟩) ≅ Λ(⟨k, j⟩, ⟨k, j + 1⟩, ⟨k + 1, j⟩, ⟨k + 1, j + 1⟩, C)
B(⟨⟨k, k⟩⟩) ≅ Λ(⟨k, 1⟩, ⟨k + 1, 1⟩, A)
B(⟨⟨2, 1⟩⟩) ≅ Λ(⟨3, 1⟩, ⟨4, 3⟩, A)
B(⟨⟨2, 2⟩⟩) ≅ Λ(⟨3, 1⟩, ⟨3, 2⟩, A)
B(⟨⟨1, 1⟩⟩) ≅ Λ(⟨3, 2⟩, (0))

A := ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ), B := ( \begin{matrix}0 & 1 & 1 & 2 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 \end{matrix} ), C := ( \begin{matrix}0 & 1 & 1 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix} ), D := ( \begin{matrix}0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ).

From these exponent matrices one can read off the lattice of invariant ℤ_{p}S_{2p}-lattices in the corresponding irreducible ℚ_{p}S_{2p}-module as explained in [Ple2, Remark II.4]. There are four different pictures, corresponding to the 4 different exponent matrices A, D, C, B:

[figure: (0,0); (0,1); (1,1)=p(0,0); (1,1,0); (2,1,1)=p(1,0,0); (1,0,0); (0,0,0); (0,0,0,0); (1,0,0,0); (1,1,0,0); (1,1,1,1)=p(0,0,0,0); (2,1,1,1)=p(1,0,0,0); (1,1,1)=p(0,0,0); (1,1,1,0); (2,1,0); (2,1,1,0); (1,0,1,0); A; D; C]

[figure: B; (0,0,0,0,0); (1,0,0,0,0); (1,1,0,0,0); (1,1,0,0,1); (1,1,1,0,1); (1,0,1,0,1); (1,0,1,0,0); (2,1,1,0,1); (2,1,1,1,1)=p(1,0,0,0,0); (1,1,1,1,1)=p(0,0,0,0,0)]

6 The basic order of B

This section describes the basic order Λ of B. Let P_{1}, …, P_{h} represent the isomorphism classes of projective indecomposable B right modules.

Then B is Morita equivalent to

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

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

Inspired by the way to give generators for the basic algebra Λ/pΛ using the Ext-quiver of \overline{B} := B/pB, we give lifts of those generators in Λ, which clearly generate the ℤ_{p}-order Λ.

Since the decomposition numbers of B are ≤ 1, the endomorphism rings End_{B}(P_{i}) are commutative (1 ≤ i ≤ h).

Let

V := ⨁_{χ} V_{χ}

be the sum over a system of representatives of the isomorphism classes of simple ℚ_{p}Bmodules and

E := End_{ℚ_{p}B}(V) ≅ Z(ℚ_{p}B).

For all 1 ≤ j ≤ h choose an embedding

φ_{j} : P_{j} ↪ V.

Let Q_{j} be the unique ℚ_{p}B-invariant complement of ℚ_{p}φ_{j}(P_{j}) in V,

V = ℚ_{p}φ_{j}(P_{j}) ⊕ Q_{j}.

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

φ_{|Q_{j}} = 0.

Definition 6 For i = 1, …, h let φ_{i}^{−1} : V → ℚ_{p}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_{B}(P_{i}, P_{j}) ↪ E, φ ↦ φ_{i}^{−1}φφ_{j}.

Via these embeddings Hom_{B}(P_{i}, P_{j}) is viewed as a subset of E:

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

Note that Λ is determined, when one knows the Λ_{ij} for a fixed system of embeddings φ_{i} : P_{i} ↪ V.

To describe elements in Λ_{ij}, we use the canonical ℚ_{p}-basis consisting of the primitive idempotents ϵ_{χ} of E.

Whereas End_{B}(P_{j}) is canonically (i.e. independent of the choice of φ_{j}) embedded into E, the embedding Hom_{B}(P_{i}, P_{j}) ↪ E for i ≠ j depends on the choice of φ_{i} and φ_{j}. In each component χ of E, φ_{i} can be multiplied by a scalar d_{i}^{(χ)}. Then the χ-component of φ_{i}^{−1}φφ_{j} ∈ Λ_{ij} gets multiplied by (d_{i}^{(χ)})^{−1}d_{j}^{(χ)}.

Since the valuation of the projection of Λ_{ij} into the components of the maximal order of E is already fixed by the exponent matrices, one may assume that the d_{i}^{(χ)} are units in ℤ_{p}.

So changing the system of embeddings φ_{i} without changing the exponent matrices means conjugation by integral invertible diagonal matrices in each component.

An inspection of the decomposition matrix shows:

Lemma 7 The ordinary and modular constituents of the projective B-modules are given in the following tables:

⟨p⟩:

ord. char. deg  modular constituents of this ord. char.
⟨⟨p⟩⟩  1 ⟨p⟩
⟨⟨p − 1⟩⟩  −1 ⟨p⟩ ⟨p − 1⟩
⟨⟨p, p − 2⟩⟩  2 ⟨p⟩ ⟨p − 1⟩ ⟨p − 2⟩ ⟨p, p − 1⟩ ⟨p, p − 2⟩
⟨⟨p − 1, p − 2⟩⟩ −2 ⟨p⟩  ⟨p, p − 2⟩ ⟨p − 1, p − 2⟩

⟨p − 1⟩:

ord. char. deg  modular constituents of this ord. char.
⟨⟨p − 1⟩⟩ −1 ⟨p⟩ ⟨p − 1⟩
⟨⟨p − 2⟩⟩ 1  ⟨p − 1⟩ ⟨p − 2⟩
⟨⟨p, p − 1⟩⟩ −2 ⟨p − 1⟩ ⟨p, p − 1⟩
⟨⟨p, p − 2⟩⟩ 2 ⟨p⟩ ⟨p − 1⟩ ⟨p − 2⟩ ⟨p, p − 1⟩ ⟨p, p − 2⟩

For j = 2, …, p − 2: ⟨j⟩:

ord. char. deg modular constituents of this ord. char.
⟨⟨j⟩⟩  (−1)^{j−1} ⟨j + 1⟩ ⟨j⟩
⟨⟨j − 1⟩⟩ (−1)^{j}  ⟨j⟩ ⟨j − 1⟩
⟨⟨p, j⟩⟩ 2(−1)^{j−1} ⟨j + 1⟩ ⟨j⟩  ⟨p, j + 1⟩ ⟨p, j⟩
⟨⟨p, j − 1⟩⟩ 2(−1)^{j}  ⟨j⟩ ⟨j − 1⟩  ⟨p, j⟩ ⟨p, j − 1⟩

⟨1⟩:

ord. char. deg modular constituents of this ord. char.
⟨⟨1⟩⟩  1 ⟨2⟩ ⟨1⟩
⟨⟨p, 1⟩⟩  2 ⟨2⟩ ⟨1⟩ ⟨p, 2⟩ ⟨p, 1⟩
⟨⟨p, p⟩⟩  1  ⟨1⟩  ⟨p, 1⟩ ⟨p, j − 1⟩

For k = 4, …, p: ⟨k, k − 1⟩:

ord. char. deg  modular constituents of this ord. char.
⟨⟨k, k-1⟩⟩ -2 ⟨k + 1, k-1⟩ ⟨k, k-1⟩  ⟨k + 2, k + 1⟩
⟨⟨k, k-2⟩⟩ 2 ⟨k + 1, k-1⟩ ⟨k, k-1⟩ ⟨k, k-2⟩  ⟨k + 1, k⟩ ⟨k + 1, k-2⟩
⟨⟨k-1, k-3⟩⟩ 2  ⟨k, k-1⟩ ⟨k, k-2⟩ ⟨k-1, k-3⟩  ⟨k, k-3⟩  ⟨k-1, k-2⟩
⟨⟨k-2, k-3⟩⟩ -2  ⟨k, k-1⟩  ⟨k-1, k-3⟩ ⟨k-2, k-3⟩

where ⟨p + 1, j⟩ has to be read as ⟨j⟩ and ⟨p + 2, p + 1⟩ as ⟨p⟩.

⟨3, 2⟩:

ord. char. deg modular constituents of this ord. char.
⟨⟨3, 2⟩⟩ −2 ⟨5, 4⟩  ⟨4, 2⟩  ⟨3, 2⟩
⟨⟨3, 1⟩⟩  2  ⟨4, 3⟩ ⟨4, 2⟩ ⟨4, 1⟩ ⟨3, 2⟩ ⟨3, 1⟩
⟨⟨2, 2⟩⟩ −1  ⟨3, 2⟩ ⟨3, 1⟩
⟨⟨1, 1⟩⟩  1  ⟨3, 2⟩

For k = 4, …, p: ⟨k, k − 2⟩:

ord. char. deg  modular constituents of this ord. char.
⟨⟨k, k-2⟩⟩ 2 ⟨k + 1, k⟩ ⟨k + 1, k-2⟩ ⟨k, k-1⟩ ⟨k, k-2⟩  ⟨k + 1, k-1⟩
⟨⟨k, k-3⟩⟩ -2  ⟨k + 1, k-2⟩  ⟨k, k-2⟩ ⟨k, k-3⟩  ⟨k + 1, k-3⟩
⟨⟨k-1, k-2⟩⟩ -2 ⟨k + 1, k⟩  ⟨k, k-2⟩  ⟨k-1, k-2⟩
⟨⟨k-1, k-3⟩⟩ 2  ⟨k, k-1⟩ ⟨k, k-2⟩ ⟨k, k-3⟩ ⟨k-1, k-2⟩ ⟨k-1, k-3⟩

⟨3, 1⟩:

ord. char. deg modular constituents of this ord. char.
⟨⟨3, 1⟩⟩  2 ⟨4, 3⟩ ⟨4, 2⟩ ⟨4, 1⟩ ⟨3, 2⟩ ⟨3, 1⟩
⟨⟨3, 3⟩⟩ 1 ⟨4, 1⟩ ⟨3, 1⟩
⟨⟨2, 1⟩⟩ −2  ⟨3, 1⟩
⟨⟨2, 2⟩⟩ −1  ⟨3, 2⟩ ⟨3, 1⟩

For k = 4, …, p and j = 2, …, k − 3: ⟨k, j⟩:

ord. char.  deg  modular constituents of this ord. char.
⟨⟨k, j⟩⟩  2(-1)^{k+j} ⟨k + 1, j⟩ ⟨k, j + 1⟩ ⟨k, j⟩  ⟨k + 1, j + 1⟩
⟨⟨k, j-1⟩⟩ 2(-1)^{k+j−1} ⟨k + 1, j⟩  ⟨k, j⟩ ⟨k, j-1⟩  ⟨k + 1, j-1⟩
⟨⟨k-1, j⟩⟩ 2(-1)^{k+j−1}  ⟨k, j + 1⟩ ⟨k, j⟩  ⟨k-1, j⟩ ⟨k-1, j + 1⟩ ⟨k, k-1⟩^{∗}
⟨⟨k-1, j-1⟩⟩ 2(-1)^{k+j}  ⟨k, j⟩ ⟨k, j-1⟩ ⟨k-1, j⟩ ⟨k-1, j-1⟩

where ⟨k, k-1⟩^{∗} only occurs for j = k − 3.

For k = 4, …, p: ⟨k, 1⟩:

ord. char. deg modular constituents of this ord. char.
⟨⟨k, 1⟩⟩  2(−1)^{k+1} ⟨k + 1, 2⟩ ⟨k + 1, 1⟩ ⟨k, 2⟩ ⟨k, 1⟩
⟨⟨k, k⟩⟩  (−1)^{k−1}  ⟨k + 1, 1⟩  ⟨k, 1⟩
⟨⟨k − 1, 1⟩⟩  2(−1)^{k}  ⟨k, 2⟩ ⟨k, 1⟩ ⟨k − 1, 2⟩ ⟨k − 1, 1⟩
⟨⟨k − 1, k − 1⟩⟩ (−1)^{k}  ⟨k, 1⟩  ⟨k − 1, 1⟩

The tables are organized as follows: The head denotes the projective indecomposable B-lattice L. The first column lists the irreducible characters that occur in ℚ_{p}L, then the degree of these characters modulo p is given and the last columns give the B-composition factors of L/pL.

Theorem 8 There are embeddings φ_{i} : P_{i} ↪ V, such that Λ is generated by the idempotents id_{|P_{⟨p⟩}}, …, id_{|P_{⟨3,1⟩}} and the following elements φ_{x,y} ∈ Λ_{x,y}:

φ_{⟨p⟩,⟨p−1⟩} = ϵ_{⟨⟨p−1⟩⟩} + pϵ_{⟨⟨p,p−2⟩⟩}, φ_{⟨p−1⟩,⟨p⟩} = pϵ_{⟨⟨p−1⟩⟩} + \frac{1}{2}ϵ_{⟨⟨p,p−2⟩⟩}, φ_{⟨p⟩,⟨p,p−2⟩} = ϵ_{⟨⟨p,p−2⟩⟩} + ϵ_{⟨⟨p−1,p−2⟩⟩}, φ_{⟨p,p−2⟩,⟨p⟩} = pϵ_{⟨⟨p,p−2⟩⟩} + pϵ_{⟨⟨p−1,p−2⟩⟩}, φ_{⟨p−1⟩,⟨p−2⟩} = ϵ_{⟨⟨p−2⟩⟩} + ϵ_{⟨⟨p,p−2⟩⟩}, φ_{⟨p−2⟩,⟨p−1⟩} = pϵ_{⟨⟨p−2⟩⟩} − \frac{p}{2}ϵ_{⟨⟨p,p−2⟩⟩}, φ_{⟨p−1⟩,⟨p,p−1⟩} = ϵ_{⟨⟨p,p−1⟩⟩} + ϵ_{⟨⟨p,p−2⟩⟩}, φ_{⟨p,p−1⟩,⟨p−1⟩} = pϵ_{⟨⟨p,p−1⟩⟩} + pϵ_{⟨⟨p,p−2⟩⟩}. For j = 2, …, p − 2 φ_{⟨j⟩,⟨j−1⟩} = ϵ_{⟨⟨j−1⟩⟩} + ϵ_{⟨⟨p,j−1⟩⟩}, φ_{⟨j−1⟩,⟨j⟩} = pϵ_{⟨⟨j−1⟩⟩} − \frac{p}{2}ϵ_{⟨⟨p,j−1⟩⟩}, φ_{⟨j⟩,⟨p,j⟩} = ϵ_{⟨⟨p,j⟩⟩} + ϵ_{⟨⟨p,j−1⟩⟩}, φ_{⟨p,j⟩,⟨j⟩} = pϵ_{⟨⟨p,j⟩⟩} + pϵ_{⟨⟨p,j−1⟩⟩}, φ_{⟨1⟩,⟨p,1⟩} = ϵ_{⟨⟨p,1⟩⟩} + ϵ_{⟨⟨p,p⟩⟩}, φ_{⟨p,1⟩,⟨1⟩} = pϵ_{⟨⟨p,j⟩⟩} − 2pϵ_{⟨⟨p,p⟩⟩}. For k = 4, …, p φ_{⟨k,k−1⟩,⟨k,k−2⟩} = ϵ_{⟨⟨k,k−2⟩⟩} + ϵ_{⟨⟨k−1,k−3⟩⟩}, φ_{⟨k,k−2⟩,⟨k,k−1⟩} = pϵ_{⟨⟨k,k−2⟩⟩} − pϵ_{⟨⟨k−1,k−3⟩⟩}, φ_{⟨k,k−1⟩,⟨k−1,k−3⟩} = ϵ_{⟨⟨k−1,k−3⟩⟩} + ϵ_{⟨⟨k−2,k−3⟩⟩}, φ_{⟨k−1,k−3⟩,⟨k,k−1⟩} = pϵ_{⟨⟨k−1,k−3⟩⟩} + pϵ_{⟨⟨k−2,k−3⟩⟩}, φ_{⟨k,k−2⟩,⟨k,k−3⟩} = ϵ_{⟨⟨k,k−3⟩⟩} + ϵ_{⟨⟨k−1,k−3⟩⟩}, φ_{⟨k,k−3⟩,⟨k,k−2⟩} = pϵ_{⟨⟨k,k−3⟩⟩} + pϵ_{⟨⟨k−1,k−3⟩⟩}, φ_{⟨k,k−2⟩,⟨k−1,k−2⟩} = ϵ_{⟨⟨k−1,k−2⟩⟩} + ϵ_{⟨⟨k−1,k−3⟩⟩}, φ_{⟨k−1,k−2⟩,⟨k,k−2⟩} = pϵ_{⟨⟨k−1,k−2⟩⟩} + pϵ_{⟨⟨k−1,k−3⟩⟩}, and for j = 2, …, k − 3 φ_{⟨k,j⟩,⟨k−1,j⟩} = ϵ_{⟨⟨k−1,j⟩⟩} + ϵ_{⟨⟨k−1,j−1⟩⟩}, φ_{⟨k−1,j⟩,⟨k,j⟩} = pϵ_{⟨⟨k−1,j⟩⟩} + pϵ_{⟨⟨k−1,j−1⟩⟩}, φ_{⟨k,j⟩,⟨k,j−1⟩} = ϵ_{⟨⟨k,j−1⟩⟩} + ϵ_{⟨⟨k−1,j−1⟩⟩}, φ_{⟨k,j−1⟩,⟨k,j⟩} = pϵ_{⟨⟨k,j−1⟩⟩} + pϵ_{⟨⟨k−1,j−1⟩⟩}, φ_{⟨k,1⟩,⟨k−1,1⟩} = ϵ_{⟨⟨k−1,1⟩⟩} + ϵ_{⟨⟨k−1,k−1⟩⟩}, φ_{⟨k−1,1⟩,⟨k,1⟩} = pϵ_{⟨⟨k−1,1⟩⟩} − 2pϵ_{⟨⟨k−1,k−1⟩⟩} and φ_{⟨3,2⟩,⟨3,1⟩} = ϵ_{⟨⟨3,1⟩⟩} + ϵ_{⟨⟨2,2⟩⟩}, φ_{⟨3,1⟩,⟨3,2⟩} = pϵ_{⟨⟨3,1⟩⟩} + \frac{1}{2}pϵ_{⟨⟨2,2⟩⟩}.

\underline{Proof.} Let x, y be two distinct Brauer characters in B such that the Cartan invariant c_{x,y} is two. Let {χ, ψ} = c_{x} ∩ c_{y} be the two ordinary irreducibles in B, in which x and y occur both as modular constituents. Then the rank of Λ_{x,y} is also two, and, since m_{x,y}(χ)+m_{y,x}(χ) = m_{x,y}(ψ)+m_{y,x}(ψ) = 1 by Theorem 5, the ℤ_{p}-lattice Λ_{x,y} is generated by

ap^{m_{x,y}(χ)}ϵ_{χ} + bp^{m_{x,y}(ψ)}ϵ_{ψ} and p^{m_{x,y}(ψ)+1}ϵ_{ψ}

with a, b ∈ ℤ_{p}^{∗}. Changing the basis one can always choose a = 1 and b ∈ {1, …, p − 1}. But in fact one can choose the embeddings φ_{i}, such that

(⋆) a = b = 1 if x < y.

Since Λ is a symmetric order, Λ_{y,x} is the dual lattice of Λ_{x,y}. So in the normalization of (⋆) one calculates Λ_{y,x} for x < y to be generated by

p^{1−m_{x,y}(χ)}ϵ_{χ} + (−\frac{χ(1)}{ψ(1)})p^{1−m_{x,y}(ψ)}ϵ_{ψ} and p^{2−m_{x,y}(ψ)}ϵ_{ψ}.

It remains to prove (⋆): Let x < y and c_{x} ∩ c_{y} = {χ, ψ} with χ < ψ (say). An inspection of the decomposition matrix shows that then the characters y and ψ belong to the same p-regular partition: y = ψ. So starting with the last ordinary character χ = ⟨1, 1⟩ in B one may choose the χ-component of the generators of the Λ_{x,y} (x < y ≠ χ) by changing the basis of Λ_{x,y} and adopt the χ-component of the generators of Λ_{x,χ} by conjugation in Λ(χ).

The following decorated exponent matrices show, which entries below the diagonal one may choose by base change (^{∗}) and which one by conjugation (^{+}).

A := (\begin{matrix}0 & 1 \\ 0^{+} & 0 \end{matrix}), B := (\begin{matrix}0 & 1 & 1 & 2 & 1 \\ 0^{+} & 0 & 1 & 1 & 1 \\ 0^{+} & 1 & 0 & 1 & 1 \\ 0 & 0^{∗} & 0^{∗} & 0 & 0 \\ 0^{+} & 1 & 1 & 1^{∗} & 0 \end{matrix}), C := (\begin{matrix}0 & 1 & 1 & 2 \\ 0^{+} & 0 & 1 & 1 \\ 0^{+} & 1 & 0 & 1 \\ 0 & 0^{∗} & 0^{∗} & 0 \end{matrix}), D := (\begin{matrix}0 & 1 & 2 \\ 0^{+} & 0 & 1 \\ 0 & 0^{∗} & 0 \end{matrix}).

□

This investigation of the block B shows that the projections of B into the simple components of ℚ_{p}B are as big as possible which also means that the amalgamations between these projections are maximal:

Corollary 9 Let φ, ϑ denote two different irreducible Brauer characters in B. Then the Cartan invariant c_{φ,ϑ} is two if and only if the dimension of the Ext-group between the two simple 𝔽_{p}B-modules is one.

7 Quiver and relations for \overline{B}

Relations for the Ext-quiver for \overline{B} = B/pB are given in [ErM] where they could not determine all the constants. Taking the explicit generators of the basic order of B from Theorem 8 instead of the ones given in [ErM], it is easy to compute the exact relations, even for B and not only modulo p.

Corollary 10 Using the notation of [ErM], we denote by ϵ̃ the generator of Λ given in Theorem 8 that maps to a multiple of the generator for Λ/pΛ denoted by ϵ in [ErM]. id denotes the identity on the respective projective indecomposable Λ-lattice. Then the following relations hold in Λ:

1) ϵ̃_{i}ϵ̃_{i+1} = 0 = η̃_{i}η̃_{i+1}, ϵ̃_{i}η̃_{i}ϵ̃_{i} = pϵ̃_{i}, η̃_{i}ϵ̃_{i}η̃_{i} = pη̃_{i}, ϵ̃_{i}η̃_{i} + η̃_{i+1}ϵ̃_{i+1} = p · id.

2) These paths are also 0 in Λ.

3) If e = ⟨j⟩:

γ̃_{i+1}δ̃_{i+1} + δ̃_{i}γ̃_{i} + u_{i}η̃ϵ̃ = p · id, where u_{i} = \frac{3}{2} if e ≠ ⟨p − 1⟩ and u_{i} = 1, if e = ⟨p − 1⟩. If e = ⟨k, 1⟩: γ̃_{i+1}δ̃_{i+1} + δ̃_{i}γ̃_{i} + 3η̃ϵ̃ = −2p · id.

4) φ_{⟨p,j−1⟩,⟨p,j⟩}φ_{⟨p,j⟩,⟨j⟩} = −2φ_{⟨p,j−1⟩,⟨j−1⟩}φ_{⟨j−1⟩,⟨j⟩}. All other squares commute.

5) λ̃κ̃ = η̃ϵ̃− ρ̃_{1}ζ̃_{1}.

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