On blocks with cyclic defect group and their head orders.

Gabriele Nebe ^{∗} ^{†}

Abstract: It is shown that [Ple83, Theorem 8.5] describes blocks of cyclic defect group up to Morita equivalence. In particular such a block is determined by its planar embedded Brauer tree. Applying the radical idealizer process the head order of such blocks is calculated explicitly.

1 Introduction.

Blocks with cyclic defect group are very well understood. Despite of their very special structure these blocks are extensively used to study examples for the validity of various conjectures, since they are essentially described by combinatorial means. A detailed introduction to the theory of blocks with cyclic defect groups, that also deals with rationality questions of the involved characters, is given in [Fei82, Chapter 7]. Using the known character theoretic information and some new methods, essentially based on linear algebra, Plesken [Ple83, Chapter 8] gives a rather explicit description of blocks with cyclic defect group B = ℤ_{p}Gϵ of p-adic group rings. Theorem 2.5 shows that [Ple83, Theorem 8.5,8.10] determines the ring theoretic structure of such blocks up to isomorphism. In particular the planar embedded Brauer tree together with the character fields and the Galois action on the characters determine the Morita equivalence class of B.

The second part of this paper deals with the radical idealiser chain of B. This is a finite chain associated to an order Λ

Λ =: Λ_{0} ⊂ Λ_{1} ⊂ … ⊂ Λ_{N} = Λ_{N+1}

where Λ_{i+1} = Id(J(Λ_{i})) (i = 0, …, N) is the 2-sided idealizer of the Jacobson radical of Λ_{i} that necessarily ends in a hereditary order Λ_{N} called the head order of Λ. Section 3 investigates the radical idealiser chain of blocks with cyclic defect group and calculates the head order of such a block.

2 Blocks with cyclic defect group.

The aim of this section is to show, how far [Ple83, Theorem 8.5] determines blocks with cyclic defect group of group rings over discrete valuation rings. Moreover we provide the tiny extra bit of information to get a complete description up to isomorphism.

∗Abteilung Reine Mathematik, Universität Ulm, 89069 Ulm, Germany, nebe@mathematik.uniulm.de

†Radcliffe fellow

Throughout the paper let R be a (not necessarily commutative) discrete valuation ring with prime element π and residue class field k = R/πR and let K be the skew-field of fractions of R.

A convenient language to describe certain R-orders are exponent matrices.

Definition 2.1. (see [Rei75, Definition 39.2], [Ple83]) Let d = (d_{1}, …, d_{n}) ∈ ℕ^{n}, D := ∑_{i=1}^{n} d_{i} and M ∈ ℤ^{n×n}. Then

Λ(R, d, M) := {X = (x_{ij}) ∈ K^{D×D} | x_{ij} ∈ π^{m_{ij}}R^{d_{i}×d_{j}}}.

\underline{Example:} (hereditary orders, see [Rei75, Section 39]) Let Λ be a hereditary order and P be a projective Λ-lattice. Then \overline{P} := P/πP is uniserial. Let d_{1}, …, d_{n} be the dimensions of the simple Λ-modules in the order in which they occur in the radical series of \overline{P}. Then with respect to a suitable R-basis of P (adapted to this lattice chain),

Λ = Λ(R, d, H_{n}), where H_{n} = ( \begin{matrix}0 & 1 & … & … & 1 \\ 0 & 0 & 1 & … & 1 \\ ⋮ & ... & ... & ... & ⋮ \\ 0 & … & … & 0 & 1 \\ 0 & … & … & 0 & 0 \end{matrix} )

The description of blocks with cyclic defect given in [Ple83] will be repeated briefly. All the following results can be found in Chapter 8 of this lecture notes, so we omit the detailed citations. Let K be an unramified extension of ℚ_{p}, R its ring of integers and B be a block of RG with cyclic defect group of order p^{a}. Assume that k := R/pR is a splitting field of kB. By [Fei82, Chapter 7], the minimal choice of such a field K is the character field of any of the non-exceptional characters in B.

Let ϵ_{1}, …, ϵ_{h} be the central primitive idempotents in A := KB. Then h = a + e where e is the number of simple kB-modules and after a suitable permutation of the indices the centers Z(Aϵ_{s}) ≅ K for s = a + 1, …, a + e and Z(Aϵ_{s}) =: Z_{s} is a totally ramified extension of K of degree \frac{p^{s}−p^{s−1}}{e} for s = 1, …, a. The center of Bϵ_{s} is the maximal order R_{s} in Z_{s}.

The vertices in the Brauer tree are the exceptional vertex {1, …, a} and a+1, …, a+ e corresponding to the other simple A-modules. Let T_{odd} resp. T_{even} denote the set of vertices having an odd (resp. even) distance from the exceptional vertex.

For s ∈ {1, …, h} let r_{s} ⊂ {1, …, e} be the set of indices of the simple constituents of any Bϵ_{s} lattice. Then for the exceptional vertex r_{1} = … = r_{a} and the sets r_{s} are the orbits of certain permutations δ (if s ∈ T_{even}) resp. ρ (if s ∈ T_{odd}). Let d_{1}, …, d_{e} be the k-dimensions of the simple kB-modules and f_{1}, …, f_{e} be orthogonal idempotents of B that lift the corresponding central primitive idempotents of B/J(B).

Theorem 2.2. ([Ple83, Theorem 8.3]) With the notation above let i ∈ r_{s}. Then

(i) Bϵ_{s} ≅ Λ(R_{s}, (d_{i}, d_{δ(i)}, …, d_{δ^{|rs|−1}(i)}), H_{|r_{s}|}) for s = 1, …, a

and

(ii) Bϵ_{s} ≅ Λ(R, (d_{i}, d_{σ(i)}, …, d_{σ^{|rs|−1}(i)}), aH_{|r_{s}|}) for s = a + 1, …, a + e

where σ = δ if s ∈ T_{even} and σ = ρ if s ∈ T_{odd}.

It remains to describe how B sits inside the direct sum of the Bϵ_{s}, that is to describe the amalgamations between the Bϵ_{s}.

Theorem 2.3. ([Ple83, Theorem 8.5])

(i) For the exceptional vertex Γ_{a} := (ϵ_{1} +…+ϵ_{a})B one gets an inductive description: For s = 2, …, a let

X_{s} := Bϵ_{s}/J(Bϵ_{s})^{x_{s}} = Bϵ_{s}/π_{s}^{y_{s}}Bϵ_{s}

where x_{s} = |r_{1}|y_{s} and y_{s} = \frac{p^{s−1}−1}{e} and let ν_{s} : Bϵ_{s} → X_{s} be the natural epimorphism. Define R-orders Γ_{s} (s = 1, …, a) inductively by

Γ_{1} := ϵ_{1}B and Γ_{s} := {(x, y) ∈ Γ_{s−1} ⊕ Bϵ_{s} | φ_{s−1}(x) = ν_{s}(y)}

where φ_{s−1} is an epimorphism from Γ_{s−1} onto X_{s}.

(ii) Let Γ_{0} := (ϵ_{a+1} + … + ϵ_{a+e})B. Then for any i, j ∈ {1, …, e} − r_{1} with i ≠ j one

gets

f_{i}Γ_{0}f_{j} = ⊕_{s=a+1}^{a+e}f_{i}Bϵ_{s}f_{j}

and

f_{i}Γ_{0}f_{i} ≅ {(x, y) | x, y ∈ R^{d_{i}×d_{i}}, x ≡ y (mod p^{a})} ⊂ f_{i}ϵ_{s}Bf_{i}⊕f_{i}ϵ_{t}Bf_{i} ≅ R^{d_{i}×d_{i}}⊕R^{d_{i}×d_{i}}

if i ∈ r_{s} ∩ r_{t}.

(iii) Finally there are epimorphisms ν and μ of Γ_{0} and Γ_{a} onto ⊕_{i∈r_{1}}(R/p^{a}R)^{d_{i}×d_{i}} such

that

B = {(x, y) ∈ Γ_{0} ⊕ Γ_{a} | ν(x) = μ(y)}.

Note that X_{s} ≅ Γ_{s−1}/J(Γ_{s−1})^{x_{s}}.

The possible ambiguity in this description is the choice of the epimorphisms in (i) and in (iii). It is clear that one can always fix one of the two epimorphisms. The choices for the other one correspond to the automorphisms of the image. So the question is, whether these automorphisms lift to automorphisms of B. This is clear for the maps in (iii). For (i) this is unfortunately not always the case.

To simplify notation, it is convenient to pass to the Morita-equivalent basic order. Let S be any discrete valuation ring with prime element π_{S} and Λ := Λ(S, (1, …, 1), H_{n}) the basic hereditary S-order of degree n. Let X := Λ/π_{S}Λ. Then Λ is generated by the idempotents e_{i} = diag(0, …, 0, 1, 0, … 0) (the 1 is on the i-th place), the elements

e_{i+1,i} = ( \begin{matrix}0 & … & … & 0 \\ ⋮ & ... & ... & ⋮ \\ ⋮ & 1 & ... & ⋮ \\ 0 & … & … & 0 \end{matrix} ) ∈ e_{i+1}Λe_{i} and g_{1,n} = ( \begin{matrix}0 & … & 0 & π_{S} \\ 0 & … & … & 0 \\ ⋮ & ... & ... & ⋮ \\ 0 & … & … & 0 \end{matrix} ) ∈ e_{1}Λe_{n}

where i = 1, …, n − 1. These generators map onto generators \overline{e_{i}}, \overline{e_{i+1,1}} and \overline{g_{1,n}} of X corresponding to the Ext-quiver of X which is a directed n-gon. They satisfy the relation that

\overline{g_{1,n}}\overline{e_{n,n−1}} ⋯ \overline{e_{2,1}} = 0

and similarly for any cyclic permutation of this product.

Lemma 2.4. Let Λ and X = Λ/π_{S}Λ be as above and let φ be an automorphism of X that fixes all the idempotents \overline{e_{i}}.

Then there are 0 ≠ \overline{λ_{i}} ∈ S/π_{S}S =: k_{S} with φ(\overline{e_{i+1,i}}) = \overline{λ_{i}}\overline{e_{i+1,i}} for i = 1, …, n − 1 and φ(\overline{g_{1,n}}) = \overline{λ_{n}}\overline{g_{1,n}}.

There is an automorphism ϕ of Λ that lifts φ if and only if the product \overline{λ_{1}} ⋯ \overline{λ_{n}} = 1. In particular, there is always an automorphism ϕ of Λ with

\overline{ϕ(e_{i+1,i})} = φ(\overline{e_{i+1,i}}), and ϕ(e_{j}) = e_{j} for all i = 1, …, n − 1, j = 1, …, n.

Proof. The automorphism φ maps the generator \overline{e_{i+1,i}} ∈ \overline{e_{i+1}}X\overline{e_{i}} = k_{S}\overline{e_{i+1,i}} to some other generator of this module (i = 1, …, n − 1) and similar for \overline{g_{1,n}}. Hence there are such units \overline{λ_{i}} ∈ k_{S}^{∗} as described in the lemma. Moreover any such tuple (\overline{λ_{1}}, …, \overline{λ_{n}}) ∈ (k_{S}^{∗})^{n} determines a unique automorphism of X fixing all the idempotents \overline{e_{i}}.

Choose units λ_{i} ∈ S^{∗} that map to \overline{λ_{i}} in k_{S}. Then the matrix

D := diag(1, λ_{1}, λ_{1}λ_{2}, …, λ_{1} ⋯ λ_{n−1}) ∈ Λ^{∗}

fixes all the e_{i} and conjugates e_{i+1,i} to λ_{i}e_{i+1,i} for all i = 1, …, n − 1 and g_{1,n} to (λ_{1} ⋯ λ_{n−1})^{−1}g_{1,n}. Hence if the product of the \overline{λ_{i}} is 1, then conjugation by D is the desired automorphism ϕ.

On the other hand it is easy to see that all automorphisms of Λ that fix the idempotents e_{i} are given by conjugation with a diagonal matrix D = diag(d_{1}, …, d_{n}) mapping the matrix units e_{ij} to \frac{d_{i}}{d_{j}} e_{ij}. □

Therefore there is a tiny bit missing in Theorem 2.3 (i) to describe the exceptional vertex Γ_{a} up to isomorphism. Since blocks of group rings are symmetric orders, however, the missing information can easily be obtained from the trace bilinear form.

Theorem 2.3 gives the center Z := Z(Γ_{a}) up to isomorphism. Instead of continuing with Plesken’s description, it seems to be easier to give generators for Γ_{a} over the center Z using the Ext-quiver of Γ_{a}. To this aim, we pass to the Morita equivalent basic order and assume that all simple Γ_{a}-modules are of dimension 1 over k. All projective Γ_{a}- lattices are uniserial when reduced modulo p, where the sequence of composition factors is given by the permutation δ. Therefore the Ext-quiver of Γ_{a} is again a directed ngon, where n = |r_{1}| is the number of simple Γ_{a}-modules. If n = 1, then Γ_{a} = Z(Γ_{a}) is already described completely by Theorem 2.3. So we will assume that n > 1. Let e_{1}, …, e_{n} ∈ Γ_{a} be orthogonal lifts of the central primitive idempotents of Γ_{a}/J(Γ_{a}) ordered in such a way that δ_{|r_{1}} = (1, …, n). Denote the corresponding matrix units in e_{i}KΓ_{a}ϵ_{s}e_{j} by e_{ij}^{s} (i, j ∈ {1, …, n}, s ∈ {1, …, a}). Then according to Theorem 2.3 and Lemma 2.4 after a choice of a suitable basis generators of Γ_{a} over its center Z can be chosen as

e_{1}, …, e_{n}, (e_{2,1}^{1}, …, e_{2,1}^{a}) =: e_{2,1} ∈ e_{2}Γ_{a}e_{1}, …, (e_{n,n−1}^{1}, …, e_{n,n−1}^{a}) =: e_{n,n−1} ∈ e_{n}Γ_{a}e_{n−1}

and

(x_{1}π_{1}e_{1,n}^{1}, …, x_{a}π_{a}e_{1,n}^{a}) =: g_{1,n} ∈ e_{1}Γ_{a}e_{n} for certain units x_{i} ∈ Z_{i} = Z(Γ_{a}ϵ_{i}).

Theorem 2.5. Let Z := Z(Γ_{a}) and let Z^{#} be the dual of Z with respect to the sum of usual the trace bilinear forms. Then there are units x_{i} ∈ Z_{i} = Z(Γ_{a}ϵ_{i}) (i = 1, …, a) such that

p^{a}Z^{#} = (x_{1}π_{1}, …, x_{a}π_{a})Z.

With the choice of these x_{i}, the order Γ_{a} is generated by

Z, e_{1}, …, e_{n}, e_{i+1,i} (i = 1, …, n − 1), and g_{1,n}

as defined above.

Proof. We may assume that n > 1. B is a symmetric order with respect to the associative symmetric bilinear form

(x, y) ↦ \frac{1}{|G|}trace_{reg}(xy) = trace_{red}(xyz) =: Tr_{z}(x, y)

where trace_{reg} and trace_{red} denote the regular respectively reduced trace of KB and z = ∑_{s=1}^{a+e} \frac{χ_{s}(1)}{|G|}ϵ_{s}, where ϵ_{1}, …, ϵ_{a+e} are the central primitive idempotents of KB and χ_{1}, …, χ_{a+e} corresponding absolutely irreducible (complex) characters of G.

Let f_{1}, …, f_{n} denote orthogonal idempotents in B that map onto the central primitive idempotents of B/J(B) such that

e_{i} = f_{i}(ϵ_{1} + … + ϵ_{a}) (i = 1, …, n).

Since n > 1

⟨g_{1,n}⟩_{Z} = e_{1}Γ_{a}e_{n} = f_{1}Bf_{n} = (f_{n}Bf_{1})^{#} = (e_{n}Γ_{a}e_{1})^{#} = ⟨e_{n,n−1} ⋯ e_{2,1}⟩_{Z}^{#}

can be calculated via the symmetrizing form above. Since the character degrees of the absolutely irreducible characters belonging to the exceptional vertex are all equal the dual with respect to Tr_{z} is as stated in the theorem, yielding the remaining generator g_{1,n} for Γ_{a}. □

Note that the x_{i} do not depend on the degrees of the irreducible complex characters in B, since all exceptional absolutely irreducible characters have the same degree. Therefore one gets

Corollary 2.6. Let B_{i} (i = 1, 2) be two blocks with cyclic defect group ≅ C_{p^{a}} and assume that R is an unramified extension of ℤ_{p} that is large enough so that k is a splitting field for kB_{i}. Then B_{1} and B_{2} are Morita equivalent if and only if their Brauer trees (including the permutations δ and ρ) and the character fields Z_{1}, …, Z_{a} coincide.

Also, symmetric orders remain symmetric orders, when one extends the ground ring. Therefore the explicit description in [Ple83, Theorem 8.5] shows that the Brauer tree determines a block of cyclic defect up to Morita equivalence (over an algebraically closed field). This is also shown in [Lin96, Theorem 2.7(ii)] with completely different methods.

Corollary 2.7. Let B_{i} (i = 1, 2) be two blocks with isomorphic cyclic defect group and assume that R is large enough so that k and K are splitting fields for kB_{i} and KB_{i}. (Here we drop the assumption that K is unramified over ℚ_{p}.) Then B_{1} and B_{2} are Morita equivalent if and only if their planar embedded Brauer trees coincide.

2.1 Galois descent.

We now perform the Galois descent to obtain a description over ℤ_{p} (see [Ple83, Chapter 8]). So let B be a block of ℤ_{p}G such that B is a summand of R ⊗ B. We assume that K is chosen to be minimal, i.e. K = ℚ_{p}[χ_{a+1}] = … = ℚ_{p}[χ_{a+e}] is the character field of any non-exceptional absolutely irreducible Frobenius character that belongs to B. The maximal unramified subfield K̃ of the character field Z̃_{s} := ℚ_{p}[χ_{s}] (s = 1, …, a) of any exceptional absolutely irreducible Frobenius character in B does not depend on the character and is a subfield of K. Let m := [K : K̃] denote its index.

If R̃ denotes the ring of integers in K̃, then R̃ embeds into the center of B such that B can be viewed as an R̃-order and R ⊗R̃ B ≅ B.

The Galois group Gal(K/K̃) = Gal(k/k̃) ≅ C_{m} (where k̃ := R̃/pR̃) acts on the simple B-modules and the corresponding idempotents f_{1}, …, f_{e} with orbits of length m. Therefore orthogonal lifts of the central primitive idempotents of B/J(B) can be chosen as f̃_{1}, …, f̃ẽ ∈ B where ẽ := \frac{e}{m} is the number of simple 𝔽_{p}B-modules, each of which has character field k = R/pR.

The central primitive idempotents in A := ℚ_{p} ⊗ B are ϵ̃_{1}, …, ϵ̃_{a}, ϵ̃_{a+1}, …, ϵ̃_{a+ẽ} indexed in such a way that Z̃_{s} is a totally ramified extension of K̃ of degree \frac{p^{s}−p^{s−1}}{e} for s = 1, …, a.

For an appropriate ordering of the index set {1, …, e} the k-dimensions of the simple 𝔽_{p}B-modules are d_{1}, …, dẽ and the set of indices of the simple ϵ̃_{s}B-modules is r̃_{s} = r_{s} ∩ {1, …, ẽ}.

For s = a + 1, …, a + ẽ, the center of Bϵ̃_{s} is isomorphic to R and Bϵ̃_{s} is isomorphic to one of the R-orders in Theorem 2.2 (ii).

Let n′ := |r̃_{1}| = \frac{|r_{1}|}{m}. For s = 1, …, a let D_{s} be a central Z̃_{s}-division algebra of index m and Ω_{s} be its maximal order with prime element ℘_{s}. Then

Bϵ̃_{s} ≅ Λ(Ω_{s}, (d_{i}, d_{δ(i)}, …, d_{δ^{n′−1}(i)}), H_{n′}).

Then the Hasse invariant of D_{s} (as defined in [Rei75, (31.7)]) is independent of s and can be read off from the planar embedded Brauer tree together with the Galois action of Gal(k/k̃) ≅ Gal(K/K̃) on the modular constituents of any exceptional character in B:

Theorem 2.8. Let ψ be a p-modular constiuent of any of the exceptional characters in B. Let F denote the Frobenius automorphism of k/k̃. Then there is some r ∈ ℤ prime to m such that

δ^{n′}(ψ) = F^{r}(ψ) where n′ = |r̃_{1}| = \frac{|r_{1}|}{m}.

Let t = r^{−1} ∈ ℤ/mℤ. Then for all s ∈ \{1, …, a\} the Hasse invariant of D_{s} is \frac{t}{m}.

Proof. To simplify notation we again assume that all the character degrees d_{i} are equal to 1. Then for s ∈ {1, …, a} the order Bϵ_{s} = Λ(Ω_{s}, (1, …, 1), H_{n′}) and

P = ( \begin{matrix}0 & … & 0 & ℘_{s} \\ 1 & 0 & … & 0 \\ 0 & ... & ... & ⋮ \\ 0 & … & 1 & 0 \end{matrix} )

is a generator of J(Bϵ̃_{s}). Then P also generates the Jacobson radical of Bϵ_{s} = R⊗R̃Bϵ_{s}. Let L_{ψ} be a Bϵ_{s}-lattice whose head has character ψ. Then the head of L_{ψ}P^{n′} has character δ^{n′}(ψ) which is Galois conjugate to ψ and hence of the form F^{r}(ψ) for some r. Therefore conjugation by P^{n′} = diag(℘_{s}, …, ℘_{s}) induces the Galois automorphism F^{r} on the inertia subfield K of D_{s}. By the general theory of division algebras over local fields (see [Rei75]) r is prime to m and the Hasse invariant of D_{s} is \frac{t}{m} as stated in the theorem. □

The amalgamations in B are described as in Theorem 2.3 (see [Ple83, Theorem 8.10]), where now the epimorphisms in (i) are only mappings between R̃-orders. For (iii) one should note that R/pR ≅ Ω_{s}/℘_{s}Ω_{s} for all s = 1, …, a.

Similarly as in Theorem 2.5 one shows:

Theorem 2.9. The description above (see [Ple83, p. 140ff]) determines B up to isomorphism. More precisely let Γ̃_{a} := (ϵ̃_{1}+…+ϵ̃_{a})B. and let ẽ_{1}, …, ẽ_{n′} (n′ = |r̃_{1}| = \frac{|r_{1}|}{m}) be lifts of the central primitive idempotents in Γ̃_{a}/J(Γ̃_{a}). Again we assume that the k-dimensions of the simple Γ̃_{a}-modules are 1. Then ẽ_{i}Γ̃_{a}ẽ_{i} is generated as a Z(Γ̃_{a})-order by (ζ_{1}, …, ζ_{a}) and (℘_{1}, …, ℘_{a}), where ζ_{s} ∈ Ω_{s} is a primitive (q^{m}−1)st root of unity (q := |k̃| = |R̃/pR̃|) and the prime elements ℘_{s} ∈ Ω_{s} are chosen such that ζ_{s}^{℘_{s}} = F^{r}(ζ_{s}) = ζ_{s}^{q^{r}} where r is as in Theorem 2.8 (i.e. \frac{t}{m} is the Hasse invariant of D_{s} where rt ≡ 1 (mod m)). The remaining generators of Γ̃_{a} are ẽ_{i+1,i} ∈ ẽ_{i+1}Γ̃_{a}ẽ_{i} (i = 1, …, n′ − 1) and g̃_{1,n′} ∈ ẽ_{1}Γ̃_{a}ẽ_{n′} defined analogously to the ones in Theorem 2.5.

Proof. Let L_{i} = ẽ_{i}Γ̃_{a} be any projective indecomposable Γ̃_{a}-lattice (i = 1, …, n′). Then, by the above, the endomorphism ring of L_{i} is a successive amalgam of the orders Ω_{s}, s = 1, …, a. Since k̃-automorphisms of Ω_{s}/℘_{s}Ω_{s} lift to (inner) R̃-automorphisms of Ω_{s}, this ring is uniquely determined by [Ple83, Theorem 8.10] up to isomorphism and

ẽ_{i}Γ̃_{a}ẽ_{i} = EndΓ̃_{a}(L_{i}) = ⟨(ζ_{1}, …, ζ_{a}), (℘_{1}, …, ℘_{a}), Z(Γ̃_{a})⟩.

To generate Γ̃_{a}, by Nakayama’s lemma, it is enough to choose additional elements of ẽ_{i}Γ̃_{a}ẽ_{j} (i ≠ j ∈ {1, … n′}) that generate

ẽ_{i}Γ̃_{a}ẽ_{j}/(ẽ_{i}J(Γ̃_{a})^{2}ẽ_{j} + pẽ_{i}Γ̃_{a}ẽ_{j})

as an ẽ_{i}Γ̃_{a}ẽ_{i}-module. The same arguments as in the proof of Theorem 2.5 now imply the theorem. □

Corollary 2.10. The planar embedded Brauer tree together with the character fields K, Z̃_{1}, …, Z̃_{a} and the Galois action on the modular constituents of the exceptional characters determine the block B of ℤ_{p}G up to Morita equivalence.

3 The radical idealizer chain for blocks with cyclic defect groups

In this section we will investigate the radical idealizer chain for blocks with cyclic defect group, where we mainly concentrate on describing the head order. Head orders are hereditary orders and hence they are the maximal elements for the “radically covering” relation, where an order Γ radically covers and order Λ, Γ ≻ Λ, if Γ ⊇ Λ and J(Γ) ⊇ J(Λ). Then for all orders in the idealizer chain Λ_{i} ≻ Λ_{i−1} (see [Rei75, Section 39]). Moreover it is easy to see that if Γ ≻ Λ then every simple Γ module is semi-simple as a Λ-module (see [Neb04, Lemma 2.2]). In particular the simple Λ_{N}-modules are semi-simple Λ-modules.

We will use the notation introduced in the last section and perform the calculations for the block B of RG. The block B of ℤ_{p}G can be treated similarly and the head order of B is easily derived from the one of B (see Remark 3.17). However, it is crucial for the whole process that R is an unramified extension of ℤ_{p}.

For the radical idealizer process we treat the exceptional vertex Γ_{a} and Γ_{0} separately always keeping track of the amalgamations between them, which are controlled by the following lemma.

Lemma 3.1. Let S be a discrete valuation ring with prime element π and let Λ_{i} (i = 1, 2) be S-orders. Given epimorphisms φ_{i} : Λ_{i} → X := S^{s×s}/π^{t}S^{s×s} let

Λ := {(x_{1}, x_{2}) ∈ Λ_{1} ⊕ Λ_{2} | φ_{1}(x_{1}) = φ_{2}(x_{2})}.

Then

Id(J(Λ)) ⊇ \{(x_{1}, x_{2}) ∈ Λ_{1} ⊕ Λ_{2} | \overline{φ_{1}(x_{1})} = \overline{φ_{2}(x_{2})}\} =: Γ

where : X → S^{s×s}/π^{t−1}S^{s×s} is the natural epimorphism.

Proof. Clearly J(Λ) = {(y_{1}, y_{2}) ∈ J(Λ_{1}) ⊕ J(Λ_{2}) | φ_{1}(y_{1}) = φ_{2}(y_{2})} and φ_{i}(J(Λ_{i})) = J(X) = πX for i = 1, 2. Let (x_{1}, x_{2}) ∈ Γ and (y_{1}, y_{2}) ∈ J(Λ). Then clearly x_{i}y_{i} and y_{i}x_{i} are in J(Λ_{i}) (i = 1, 2). Since φ_{1} is surjective, there is z_{1} ∈ Λ_{1} with πφ_{1}(z_{1}) = φ_{1}(y_{1}). Choose z_{2} ∈ Λ_{2} with φ_{2}(z_{2}) = φ_{1}(z_{1}). Then

φ_{1}(y_{1}x_{1}) = φ_{1}(z_{1})πφ_{1}(x_{1}) = φ_{2}(z_{2})πφ_{2}(x_{2}) = φ_{2}(y_{2}x_{2})

and similarly φ_{1}(x_{1}y_{1}) = φ_{2}(x_{2}y_{2}). Hence (x_{1}, x_{2}) ∈ Id(J(Λ)). □ The following trivial lemma suffices to deduce the head order of Γ_{a}.

Lemma 3.2. Let Λ be an order in A and ϵ a central idempotent of A. Then

Λϵ ⊆ Id(J(Λ))ϵ ⊆ Id(J(Λϵ)).

Corollary 3.3. The head order of Γ_{a} is ⊕_{s=1}^{a}Bϵ_{s}. Similarly the head order of Γ̃_{a} is ⊕_{s=1}^{a}Bϵ̃_{s}.

Proof. The orders Bϵ_{s} = Id(J(Bϵ_{s})) and Bϵ̃_{s} are already hereditary for s = 1, …, a.

□

Note that this corollary is not true, when R is replaced by a ramified extension of ℤ_{p}.

3.1 The first steps.

The main task to calculate the idealizer chain for Γ_{0} is to calculate the one of ϵ_{s}Γ_{0} for s = a + 1, …, a + e. These orders have a certain symmetry with respect to a cyclic permutation of their simple modules, and therefore can be encoded in a simple way. All orders in this radical idealizer chain share this symmetry.

Definition 3.4. For v = (v_{0}, …, v_{n−1}) ∈ ℤ^{n} and d = (d_{1}, …, d_{n}) ∈ ℕ^{n} define

Λ(d, v) := Λ(R, d, M) := {X = (x_{ij}) ∈ K^{D×D} | x_{ij} ∈ π^{m_{ij}}R^{d_{i}×d_{j}}}

where

m_{ij} = \{ \begin{matrix}v_{j−i} & if j ≥ i \\ v_{n+j−i} − v_{n−1} & if j < i \end{matrix}

and D = ∑_{i=1}^{n} d_{i}.

Remark 3.5. Since the dimension vector d will be fixed most of the time, we will omit it and let Λ(v_{0}, …, v_{n−1}) := Λ(d, v).

The order Γ_{0} is an amalgam of the orders ϵ_{s}B = ϵ_{s}Γ_{0} (s = a + 1, …, a + e) of the form Λ(R, d, aH_{n}) ≅ Λ(0, \underbrace{a, …, a}_{n−1}) for some dimension vector d and n = |r_{s}|. The

amalgamations in Γ_{0} are only on the diagonal, more precisely, the part of B belonging to ϵ_{s}B is of the form

Λ(R, d, ( \begin{matrix}\underline{0}_{a} & a & … & … & a \\ 0 & \underline{0}_{a} & a & … & a \\ ⋮ & ... & ... & ... & ⋮ \\ 0 & … & … & \underline{0}_{a} & a \\ 0 & … & … & 0 & \underline{0}_{a} \end{matrix} )) = Λ(\underline{0}_{a}, a^{n−1})

where all the underlined entries obey a certain congruence modulo p^{a} to a diagonal entry in some other ϵ_{t}B (t ≠ s) which is indicated by underlining the 0 and the index a. By Lemma 3.1 these amalgamations will decrease by 1 in each step until after a steps the order B_{a} contains the central primitive idempotents ϵ_{a+1}, …, ϵ_{a+e}.

In the following we fix some s ∈ {a + 1, …, a + e}, put n := |r_{s}|, and let

Λ := Λ_{0} := Bϵ_{s} ⊆ Λ_{1} := B_{1}ϵ_{s} ⊆ … ⊆ Λ_{N} := B_{N}ϵ_{s}

where

B =: B_{0} ⊂ B_{1} ⊂ … ⊂ B_{N} = B_{N+1}

is the radical idealizer chain of B. Together with the structure of Λ_{i} we keep track of the additional information, how B_{i} is embedded into the direct sum of the B_{i}ϵ_{s} using the notation above.

Lemma 3.6. If a ≥ n then

Λ_{n} ≅ Λ(\underline{0}_{a−n}, (a − n)^{n−1}).

Proof. An easy induction on j shows that for j = 1, …, n

Λ_{j} = Λ(\underline{0}_{a−j}, (a − j + 1), (a − j + 2), …, a − 1, a^{n−j}).

Then

Λ_{n} = Λ(\underline{0}_{a−n}, (a − n + 1), (a − n + 2), …, a − 1) ≅ Λ(\underline{0}_{a−n}, (a − n)^{n−1})

by conjugation with the diagonal matrix diag(1, π, π^{2}, …, π^{n−1}). □ Inductively we get

Corollary 3.7. Let a = z_{s}n + b with 0 ≤ b < n and m_{0} := z_{s}n + 1. Then

Λ_{m_{0}−1} = Λ(\underline{0}_{b}, b^{n−1}).

If b = 0 then Λ_{m_{0}−1} is already a maximal order and we are done.

Lemma 3.8. Assume that b > 0 and define l_{0}, x_{0} by n = l_{0}b + x_{0} with 0 < x_{0} ≤ b. Then

Λ_{m_{0}} = Λ(\underline{0}_{b−1}, b^{n−1}).

If 0 ≤ m < n − l_{0} then

(a) Λ_{m+m_{0}} is of the form Λ_{m+m_{0}} = Λ(\underline{0}_{f(m)}, v_{1}, …, v_{n−1}) with 0 < v_{1} ≤ v_{2} ≤ … ≤ v_{n−1} = b, where f(m) = max{0, b − m − 1}.

(b) If m = l(b − 1) + y with 0 ≤ y < b − 1 then

Λ_{m+m_{0}} = Λ(\underline{0}_{f(m)}, 1^{l}, 2^{l}, …, (b − y − 1)^{l}, (b − y)^{l+1}, …, (b − 1)^{l+1}, b^{n−m−1})

i.e. Λ_{m+m_{0}} = Λ(\underline{0}_{f(m)}, v_{1}, …, v_{n−1}) with

v_{j} = \{ \begin{matrix}⌊\frac{j−1}{l}⌋ + 1 & if 1 ≤ j ≤ (b − y − 1)l \\ b − y + ⌊\frac{j−1−(b−y−1)l}{l+1}⌋ & if (b − y − 1)l < j ≤ (b − 1)l + y \\ b & if j > (b − 1)l + y \end{matrix}

(c) The radical J(Λ_{m+m_{0}}) = Λ(\underline{1}_{f(m)}, v_{1}, …, v_{n−1}).

Proof. The form of Λ_{m_{0}} is clear. For the other statements we argue by induction on m, where the case m = 0 is trivial. Assume that m < n − l_{0} − 1 and that Λ_{m+m_{0}} = Λ(\underline{0}_{f(m)}, v_{1}, …, v_{n−1}) has the properties (a), (b), (c). Then Λ_{m+m_{0}+1} is of the form

Λ_{m+m_{0}+1} = Λ(\underline{0}_{f(m+1)}, ṽ_{1}, … ṽ_{n−1}),

since the inequalities on the entries of the exponent matrix preserve the symmetry conditions in (a). The form of the amalgamations follows from Lemma 3.1. Clearly ṽ_{i} ≤ v_{i} for all i and ṽ_{0} = 0. The remaining conditions in (a) and the property (c) follow once we have shown (b). Let v′_{i} := v_{i} for i > 0 and v′_{0} := 1 = v_{0} + 1. Then the conditions on m_{1j} = ṽ_{j−1} (j > 1) that Λ_{m+m_{0}+1} lies in the left idealizer of J(Λ_{m+m_{0}}) read as

ṽ_{j−1} ≥ max{v_{k−1} − v′_{k−j} | k = j, …, n} =: max_{1}

and

ṽ_{j−1} ≥ max{b + v′_{k−1} − v_{k+n−j} | k = 1, …, j − 1} =: max_{2}.

The inequalities for the right idealizer of J(Λ_{m+m_{0}}) read as

ṽ_{j−1} ≥ max{v_{j−1}−1, v′_{j−k_{1}}−v_{n+1−k_{1}}+b, v_{n+j−k_{2}}−v_{n+1−k_{2}} | k_{1} = 2, …, j, k_{2} = j+1, …, n}

and agree with the conditions above after an easy variable transformation. Hence right and left idealizer of J(Λ_{m+m_{0}}) coincide and are equal to Id(J(Λ_{m+m_{0}})).

By the induction assumption for all 1 ≤ i ≤ n

v_{i−1} = \{ \begin{matrix}⌊\frac{i−2}{l}⌋ + 1 & if 1 ≤ i − 1 ≤ (b − y − 1)l \\ b − y + ⌊\frac{i−2−(b−y−1)l}{l+1}⌋ & if (b − y − 1)l < i − 1 ≤ (b − 1)l + y \\ b & if i − 1 > (b − 1)l + y. \end{matrix}

Since the ‘slope’ of v is decreasing v_{k−1} − v′_{k−j} is maximal if v′_{k−j} is the last 1 in v′, hence if k − j = l i.e. k = l + j. If k := min(l + j, n) then

max_{1} = v_{k−1}−1 = \{ \begin{matrix}⌊\frac{j−2}{l}⌋ + 1 & if 1 < j − 1 ≤ (b − y − 2)l \\ b − y + ⌊\frac{j−2−(b−y−2)l}{l+1}⌋ − 1 & if (b − y − 2)l < j − 1 ≤ (b − 2)l + y \\ b − 1 & if (b − 2)l + y < j − 1. \end{matrix}

This implies that max_{1} = v_{j−1} if j −1 ≤ (b−y −2)l. If (b−y −2)l < j −1 ≤ (b−y −1)l then

v_{j−1} = ⌊\frac{j − 2}{l}⌋ + 1 = b − y − 1 = max_{1} = b − y + ⌊\frac{j − 2 − (b − y − 2)l}{l + 1}⌋ − 1.

If (b − y − 1)l < j − 1 ≤ (b − 2)l + y then max_{1} = b − y − 1 + ⌊\frac{j−2−(b−y−2)l}{l+1}⌋ and v_{j−1} = b − y + ⌊\frac{j−2−(b−y−1)l}{l+1}⌋. Therefore max_{1} < v_{j−1} if and only if j − 3 − y + b is divisible by l + 1, i.e.

j − 1 = (b − 2)l + y − x(l + 1), x = 0, 1, …, y − 2

when v_{j−1} = b − x − 1 is the first occurrence of b − x − 1 in v.

If (b−2)l+y < j −1 ≤ (b−1)l+y then v_{j−1} = b−1 = max_{1} and if j −1 > (b−1)l+y then max_{1} = b − 1 < v_{j−1} = b.

For max_{2} one finds that b + v′_{k−1} − v_{k+(n−j)} is maximal if k = j − 1 since the ‘slope’ of v is decreasing. Hence

max_{2} = b + v′_{j−2} − v_{n−1} = v′_{j−2}.

Combining these conditions one finds that

ṽ_{j−1} = \{ \begin{matrix}v_{j−1} − 1 & if j − 1 = (b − 2)l + y − x(l + 1) for x = 0, …, y − 1 \\ v_{j−1} & otherwise \end{matrix} .

With these ṽ_{j} the multiplication by Λ_{m+m_{0}+1} = Λ(\underline{0}_{f(m+1)}, ṽ_{1}, …, ṽ_{n−1}) preserves the congruences in J(Λ_{m+m_{0}}) given in (c), since

b − m − 1 ≤ f(m) ≤ ṽ_{j−1} + v_{n+1−j} − b and b − m − 1 ≤ f(m) ≤ ṽ_{n+1−j} − b + v_{j−1}

for all j. This implies part (b) of the lemma. □

Corollary 3.9. Let m_{1} := n − l_{0} − 1 + m_{0} and y = x_{0} − 1. Then

Λ_{m_{1}} = Λ(0, 1^{l_{0}}, …, (b − y − 1)^{l_{0}}, (b − y)^{l_{0}+1}, …, (b − 1)^{l_{0}+1}, b^{l_{0}}) =: Λ(v^{(1)}).

Corollary 3.10. For all s ∈ {1, …, a + e} the s-th component of the head order of B is equal to the head order of the projection ϵ_{s}B.

3.2 The head order.

If n = (l_{0} +1)b is divisible by b, then the order Λ_{m_{1}} as defined in Corollary 3.9 is already hereditary. More precisely we have the following

Lemma 3.11. Let b be a factor of n = lb. Then Λ_{m_{1}} = Λ(0, 1^{l}, …, (b − 1)^{l}, b^{l−1}) is hereditary, Λ_{m_{1}} ∼ Λ(R, (D_{1}, …, D_{l}), H_{l}) where D_{i} = ∑_{j=0}^{b−1} d_{jl+i}.

Proof. Λ_{m_{1}} = Λ(R, d, M) where

m_{ij} = ⌊\frac{j − i − 1}{l}⌋ + 1.

Let t_{i} := m_{i1} = ⌊\frac{−i}{l}⌋−1. Conjugating by the diagonal matrix T := diag(π^{t_{i}}) one obtain the conjugate order Λ_{m_{1}}^{T} = Λ(R, d, M̃), where

m̃_{ij} = m_{ij} − t_{i} + t_{j} = ⌊\frac{j − i − 1}{l}⌋ + 1 − ⌊\frac{−i}{l}⌋ + ⌊\frac{−j}{l} ⌋.

Writing j = j_{1}l + j_{2} and i = i_{1}l + i_{2} with 0 < j_{2}, i_{2} ≤ l one gets

m̃_{ij}⌊\frac{j_{2} − i_{2} − 1}{l}⌋ + 1 − ⌊\frac{−i_{2}}{l} ⌋ + ⌊\frac{−j_{2}}{l} ⌋ = ⌊\frac{j_{2} − i_{2} − 1}{l}⌋ + 1 = \{ \begin{matrix}0 & if j_{2} ≤ i_{2} \\ 1 & if j_{2} > i_{2} \end{matrix} .

Hence after reordering the constituents Λ_{m_{1}} has the form as claimed in the lemma. □

We now assume that 0 < x_{0} < b. Continuing to trace down the radical idealizer process like in Lemma 3.8 seems to be a rather tedious work. If x_{0} ≥ \frac{b}{2} then after m_{2} = b − x_{0} − 1 steps one arrives at an order

Λ_{m_{1}+m_{2}} = Λ(0, 1^{l_{0}}, 2^{l_{0}+1}, 3^{l_{0}}, 4^{l_{0}+1}, …, z^{l_{0}}, (z + 1)^{l_{0}+1}, (z + 2)^{l_{0}+1}, …, (b − 1)^{l_{0}+1}, b^{l_{0}})

where z = 2b − 2x_{0} − 1. If x_{0} ≤ \frac{b}{2} then after m_{2} = x_{0} − 1 steps one arrives at an order

Λ_{m_{1}+m_{2}} = Λ(0, 1^{l_{0}}, 2^{l_{0}}, …, z^{l_{0}}, (z + 1)^{l_{0}+1}, (z + 2)^{l_{0}}, …, (b − 2)^{l_{0}+1}, (b − 1)^{l_{0}}, b^{l_{0}})

where z = b − 2x_{0} + 1. If x = \frac{b}{2} = gcd(n, b) = d then Λ_{m_{1}+m_{2}} is again hereditary

Λ_{m_{1}+m_{2}} ∼ Λ(R, (D_{1}, …, D_{2l_{0}+1}), H_{2l_{0}+1})

where D_{j} = ∑_{i≡−l_{0}j} d_{i}, where the congruence is modulo 2l_{0} + 1 = \frac{n}{d}.

Instead of continuing like this, we prefer to calculate the head order Λ_{N} = ϵ_{s}B_{N}, which is also the head order of Λ_{m_{1}}, directly where we need the following trivial lemma:

Lemma 3.12. Let Λ ⊂ Γ be two orders with J(Λ) ⊂ J(Γ). If e ∈ Λ is an idempotent then J(eΛe) ⊂ J(eΓe).

Proof.

J(eΛe) = eJ(Λ)e ⊆ eJ(Γ)e = J(eΓe).
□

The head order Λ_{N} of Λ_{m_{1}} has the following properties:

Properties 3.13. 0) Λ_{N} is of the form Λ(w) for some w ∈ ℤ_{≥0}^{n}.

1) Λ_{N} is an order, i.e. for all i < j < k one has

(i) b − w_{n+j−k} ≤ w_{k−i} − w_{j−i} ≤ w_{k−j} (ii) b − w_{n+i−k} ≤ w_{k−j} − w_{n+i−j} + b ≤ w_{k−i} (iii) b − w_{n+i−j} ≤ w_{n+j−k} − w_{n+i−k} ≤ w_{j−i}

which just expresses the fact that the entries m_{ij} in the exponent matrix of Λ_{N} satisfy m_{ik} + m_{kj} ≥ m_{ij} for all i, j, k ∈ {1, …, n}.

2) Λ_{N} is hereditary, i.e.

w_{j−1} + w_{n+1−j} − b ∈ {0, 1} for all j > 1.

3) Λ_{N} radically covers the order Λ_{m_{1}} = Λ(v^{(1)}) defined in Corollary 3.9. This property implies with Lemma 3.12 that w_{j−1} = v_{j−1}^{(1)} and w_{n−j+1} = v_{n−j+1}^{(1)} whenever v_{j−1}^{(1)} + v_{n+1−j}^{(1)} − b = 1. In particular

3’) w_{1} = … = w_{l_{0}} = 1, w_{n−1} = … = w_{n−l_{0}} = b, and w_{n−l_{0}−1} = b − 1.

Lemma 3.14. Λ_{N} is uniquely determined by Properties 3.13 0), 1), 2), and 3’). More precisely let n = l_{0}b + x_{0} be as above and assume that 1 ≤ x_{0} ≤ b − 1. Then

(i) Λ_{N} = Λ(w) where w = (0, 1^{l_{1}}, 2^{l_{2}} …, b^{l_{b}}) with l_{1} = l_{0} = l_{b} and l_{j} ∈ {l_{0}, l_{0} + 1} for all j = 1, …, b.

(ii) Let e := (e_{1}, …, e_{b}), where e_{k} = l_{k} − l_{0} ∈ {0, 1} for k = 1, … b − 1 and e_{b} := 1. For all j let a_{j} := ∑_{k=1}^{j} e_{k}. Let d := gcd(n, b) = \frac{b}{i}. Then x_{0} = \frac{a_{i}}{i} b, e_{i} = 1 and

e = (e_{1}, …, e_{i})^{d} = (e_{1}, …, e_{i}, e_{1}, …, e_{i}, …, e_{1}, …, e_{i}).

The entries of w are uniquely determined by

a_{j} = ⌊\frac{x_{0} · j}{b} ⌋ for all j = 1, …, b.

Proof. (i) Property 3.13 3’) together with Property 3.13 1) (i) (for i = 1) show that for 0 < k − j ≤ l_{0}

0 ≤ w_{k−1} − w_{j−1} ≤ 1

and if k − j ≥ l_{0} + 1, then w_{k−1} − w_{j−1} ≥ 1. This implies (i). (ii) Put d = gcd(b, x_{0}) = \frac{b}{i}. Then i = min\{j ∈ \{1, …, b\} | \frac{b}{j} divides x_{0}\}. We now show by induction on j that a_{j} = ⌊\frac{x_{0}·j}{b} ⌋ and l_{j} = l_{b−j+1} for j = 1, …, i − 1. This is clear for j = 1 since l_{1} = l_{0} = l_{b} and a_{1} = 0 = ⌊\frac{x_{0}}{b} ⌋. Assume that 1 < j ≤ i − 1 and that a_{k} = ⌊\frac{x_{0}·k}{b}⌋ and l_{k} = l_{b−k+1} for k = 1, …, j − 1. Let

X_{1} := ((t − 1), t^{l_{t}}, …, (t + j − 1)^{l_{t+j−1}}, (t + j))

be a subsequence of w. Then the difference between the first and the last entry of X_{1} is j + 1 and the distance between these entries is ∑_{q=t}^{t+j−1} l_{q} + 1. Since w_{l_{0}j+a_{j}} = j, Property 3.13 1) (i) implies that

∑_{q=t}^{t+j−1} l_{q} = l_{0}j + ∑_{q=t}^{t+j−1} e_{q} ≥ l_{0}j + a_{j} for all 1 ≤ t < b + 1 − j.

Similarly for subsequences of (w, w) of the form

X_{2} := ((b − t − 1), (b − t)^{l_{b−t}}, …, b^{l_{b}}, 0, 1^{l_{1}}, …, (j − t − 1)^{l_{j−t−1}}, (j − t))

Property 3.13 1) (ii) implies that

∑_{q=b−t}^{b} l_{q} + 1 + ∑_{q=1}^{j−t−1} l_{q} = l_{0}j + ∑_{q=b−t}^{b} e_{q} + ∑_{q=1}^{j−t−1} e_{q} ≥ l_{0}j + a_{j} for all 0 ≤ t ≤ j.

This implies that for every subsequence of length j of the sequence (e, e), the sum over the entries in this subsequence is ≥ a_{j} and therefore

x_{0} = ∑_{t=1}^{b} e_{t} ≥ \frac{b}{j}a_{j}.

Let b_{j} := ∑_{t=b−j+1}^{b} e_{t}. Similar arguments as above, using the second and second last entries of the sequences X_{1} and X_{2} above and the fact that w_{n−jl_{0}−b_{j}} = b − j, show that

∑_{q=t}^{t+j−1} l_{q} = l_{0}j + ∑_{q=t}^{t+j−1} e_{q} ≤ l_{0}j + b_{j} for all 1 ≤ t ≤ b + 1 − j

and

∑_{q=b−t}^{b} l_{q} + 1 + ∑_{q=1}^{j−t−1} l_{q} = l_{0}j + ∑_{q=b−t}^{b} e_{q} + ∑_{q=1}^{j−t−1} e_{q} ≤ l_{0}j + b_{j} for all 0 ≤ t ≤ j

which yields

\frac{b}{j}a_{j} ≤ x_{0} ≤ \frac{b}{j}b_{j}.

By induction hypothesis, we have b_{j} = a_{j} +1 (if l_{b−j+1} = l_{j}) or b_{j} = a_{j} (if l_{b−j+1} = l_{0} and l_{j} = l_{0} + 1). Note that the case l_{b−j+1} = l_{0} + 1 and l_{j} = l_{0} is not possible since then w_{n−jl_{0}−b_{j}} + w_{jl_{0}+b_{j}} = b − j + 1 + j + 1 = 2 contradicting Property 3.13 2). If b_{j} = a_{j} then x_{0} = \frac{b}{j}a_{j} and \frac{x_{0}j}{b} is an integer showing that j ≥ i. If b_{j} = a_{j} + 1 then l_{b−j+1} = l_{j} and

\frac{jx_{0}}{b} − 1 ≤ a_{j} ≤ \frac{jx_{0}}{b}

which give a_{j} = ⌊\frac{jx_{0}}{b} ⌋ as claimed, since j ≤ i − 1 and hence \frac{jx_{0}}{b} is not an integer.

It remains to show that if j = i, i.e. \frac{b}{j} = gcd(b, x_{0}) = gcd(b, n), then a_{j} = a_{i} = \frac{ix_{0}}{b} and e and Λ_{N} are as claimed. For this it is enough to show that a_{i} = b_{i}, since then every subsequence of e of length i contains exactly a_{i} times 1. Applying this to (e_{1}, …, e_{i}) and (e_{2}, …, e_{i+1}) this shows that e_{i+1} = e_{1}. Repeating it follows that e = (e_{1}, …, e_{i}, e_{1}, …, e_{i}, … e_{1}, …, e_{i}) as claimed.

Assume that a_{i} ≠ b_{i}. Then b_{i} = a_{i} + 1 and either a_{i} = \frac{x_{0}}{d} and b_{i} = \frac{x_{0}}{d} + 1 or a_{i} = \frac{x_{0}}{d} − 1 and b_{i} = \frac{x_{0}}{d} (where d := \frac{b}{i} = gcd(n, b)). Assume the latter, then

x_{0} = ∑_{j=1}^{b} e_{j} = ∑_{k=0}^{d−1} ∑_{j=ki+1}^{ki+i} e_{j} ≤ db_{i} = x_{0}.

Hence for all k the sum ∑_{j=ki+1}^{ki+i} e_{j} = b_{i}, in particular a_{i} = ∑_{j=1}^{i} e_{j} = b_{i}. In the other case one argues similarly using a_{i} instead of b_{i}. □

Theorem 3.15. The head order of B is

B_{N} = ⨁_{s=1}^{a+e} Δ_{s}

where Δ_{s} = Bϵ_{s} for s = 1, …, a. If s ∈ \{a + 1, …, a + e\} let d := gcd(|r_{s}|, a), t := \frac{|r_{s}|}{d} and c := (\frac{a}{d})^{−1} ∈ (ℤ/tℤ)^{∗}. As in Theorem 2.2 let σ := δ_{|r_{s}} if s ∈ T_{even} and σ := ρ_{|r_{s}} if s ∈ T_{odd}. Then the order of σ is |r_{s}| and we define τ := σ^{t} and γ := σ^{c} and choose i ∈ r_{s} arbitrarily. Then

Δ_{s} ≅ Λ(R, (D_{i}, D_{γ(i)}, …, D_{γ^{t−1}(i)}), H_{t})

where D_{j} = ∑_{l=0}^{d−1} d_{τ^{l}(j)}.

Proof. For 1 ≤ s ≤ a the theorem follows from Corollary 3.3. For a + 1 ≤ s ≤ a + e let n := |r_{s}|, a = μn + b with 0 ≤ b < n. If b = 0, then Δ_{s} ≅ Λ_{m_{1}} as defined in Corollary 3.9 is already a maximal order and the theorem follows from Lemma 3.11.

So assume that 1 ≤ b ≤ n − 1. Then d = gcd(a, n) = gcd(b, n) and we write n = lb + x with 0 ≤ x < b and put n = n′d, b = b′d, x = x′d. Then there is k ∈ ℤ with cb′ = 1 + n′k where c is as defined in the theorem. For j ∈ ℤ put

f(j) := 1 + ⌊\frac{j − 1 − ⌊\frac{xj}{n} ⌋}{l}⌋ = 1 + ⌊\frac{−1 − ⌊\frac{(x′−n′)j}{n′} ⌋}{l}⌋.

Since x′ − n′ = −b′l is divisible by l, one finds that

f(j + n′) = f(j) + b′ for all j ∈ ℤ.

Let

Λ := Λ(f(0), …, f(n − 1)) = Λ(R, d, M) where m_{ij} = f(j − i).

We claim that Λ = Δ_{s}. By Lemma 3.14 it is enough to show that Λ is a hereditary order that has property 3.13 3’). The latter is checked by a straightforward calculation. We show that Λ is hereditary, by establishing an isomorphism with the hereditary order in the theorem.

Put t_{i} := m_{i1} = f(1 − i). Conjugating by the diagonal matrix T := diag(π^{t_{i}}) one obtains the conjugate order

Λ^{T} = Λ(R, d, M̃), where m̃_{ij} = m_{ij} − t_{i} + t_{j} = f(j − i) − f(i) + f(j).

Writing j = 1 + cj_{2} + n′j_{1} and i = 1 + ci_{2} + n′i_{1} with 0 ≤ j_{2}, i_{2} < n′ one gets m̃_{ij} = f(c(j_{2} − i_{2})) − f(−ci_{2}) + f(−cj_{2})

= 1 + ⌊\frac{−1 − ⌊\frac{−cb′l(j_{2}−i_{2})}{n′}⌋}{l}⌋ − ⌊\frac{−1 − ⌊\frac{cb′li_{2}}{n′} ⌋}{l}⌋ + ⌊\frac{−1 − ⌊\frac{cb′lj_{2}}{n′} ⌋}{l}⌋.

Since cb′ = 1 − kn′ one gets

m̃_{ij} = 1 + ⌊\frac{−1 − ⌊\frac{l(i_{2}−j_{2})}{n′}⌋}{l}⌋ − ⌊\frac{−1 − ⌊\frac{li_{2}}{n′} ⌋}{l}⌋ + ⌊\frac{−1 − ⌊\frac{lj_{2}}{n′} ⌋}{l}⌋.

Now 0 ≤ i_{2} < n′ implies that 0 ≤ ⌊\frac{li_{2}}{n′} ⌋ ≤ l − 1 and therefore ⌊\frac{−1−⌊ \frac{li_{2}}{n′} ⌋}{l}⌋ = −1. Similarly

⌊\frac{−1−⌊ \frac{lj_{2}}{n′} ⌋}{l}⌋ = −1. For the first term we have 1 − n′ ≤ i_{2} − j_{2} ≤ n′ − 1 implying that

⌊\frac{−1 − ⌊\frac{l(i_{2}−j_{2})}{n′}⌋}{l}⌋ ∈ \{0, −1\}.

More precisely this yields

m̃_{ij} = \{ \begin{matrix}0 & if i_{2} ≥ j_{2} \\ 1 & if i_{2} < j_{2}. \end{matrix}

In particular Λ is a hereditary order and hence Λ = Δ_{s}. After a suitable reordering of the constituents the order Λ^{T} ≅ Δ_{s} has the form as claimed in the theorem. □

Remark 3.16. Let s ∈ {a + 1, …, a + e} and Δ_{s} := ϵ_{s}B_{N}. Let n := |r_{s}| = n′d, d = gcd(a, n), a = a′d, and ca′ ≡ 1 (mod n′). Let ν : ℤ/nℤ → ℤ/n′ℤ be the natural epimorphism. Assume that the simple Bϵ_{s}-modules are labeled S_{i} with i ∈ ℤ/nℤ such that σ(S_{i}) = S_{i+1}, where σ is as in Theorem 3.15.

Then the simple Δ_{s}-modules are T_{j} with j ∈ ℤ/n′ℤ can be labeled such that

(T_{j})_{|Bϵ_{s}} = ⨁_{i∈ν^{−1}(cj)} S_{i}

The Δ_{s}-lattices in the simple Aϵ_{s}-module form a chain

… ⊃ L_{1} ⊃ L_{2} ⊃ … ⊃ L_{n′} ⊃ pL_{1} =: L_{n′+1} ⊃ …

where L_{j}/L_{j+1} ≅ T_{j} for j = 1, …, n′.

Remark 3.17. Theorem 3.15 also holds when the block B of RG is replaced by the block B of ℤ_{p}G from Theorem 2.9.

Remark 3.18. Replacing R by a ramified extension of ℤ_{p} in Remark 3.16 and a by the π-adic valuation of p^{a} still yields a description of the head order of the non exceptional vertex Γ_{0}.

References

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[Lin96] M. Linckelmann, The isomorphism problem for cyclic blocks and their source algebras. Invent. math. 125 (1996) 265-283.

[Neb04] G. Nebe, On the radical-idealizer chain of symmetric orders. (preprint, math.RT/0310191)

[Ple83] W. Plesken, Group rings of finite groups over the p-adic integers. Springer Lecture Notes in Mathematics 1026 (1983).

[Rei75] I. Reiner, Maximal Orders. Academic Press, 1975.