On representations of twisted group rings

Matthias Künzer

April 5, 2003

Abstract

We generalize certain parts of the theory of group rings to the twisted case. Let G be a finite group acting (possibly trivially) on a field L of characteristic coprime to the order of the kernel of this operation. Let K ⊆ L be the fixed field of this operation, let S be a discrete valuation ring with field of fractions K, maximal ideal generated by π and integral closure T in L. We compute the colength of T ≀ G in a maximal order in L≀G. Moreover, if S/πS is finite, we compute the S/πS-dimension of the center of T ≀ G/Jac(T ≀ G). If this quotient is split semisimple, this yields a formula for the number of simple T ≀ G-modules, generalizing Brauer’s formula.

Contents

0 Introduction  2
0.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.3 Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
0.6 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Rational  6
1.1 The Plancherel formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 A Morita equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 The center of L ≀ G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 (Counter)examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 The central Plancherel formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Locally integral  17
2.1 Projectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The Wedderburn colength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Some estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 The central colength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Modular  24
3.1 Brauer-Nesbitt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Counting simple modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

AMS subject classification: 16S35, 20C05.

0 Introduction

0.1 Outline

Let G be a finite group acting on a field L with fixed field K. The twisted group ring L ≀ G carries the multiplication (σy)(τz) = στy^{τ}z, where σ, τ ∈ G, y, z ∈ L (^{1}).

We denote by N the kernel of the operation of G on L. If N = G, then K = L and L ≀ G is nothing but the untwisted classical group ring KG. If N = 1, then L ≀ G has the only simple module L, and is isomorphic to End_{K}L.

If the characteristic of L and the order of N are coprime, then L ≀ G is semisimple, and even separable over K. Many of the apparent difficulties we shall encounter vanish if the characteristic of L and the order of G are coprime. For instance, in this case the Plancherel formula (1.16) is immediate.

The Wedderburn isomorphism sends an element of L ≀ G to the tuple of its operations on the simple modules, such an operation being considered as an endomorphism over the respective L ≀ G-linear endomorphism skew field. The Plancherel formula yields a Fourier inversion formula for this isomorphism. Schur relations may then be deduced from composition of the Wedderburn isomorphism with its Fourier inversion. A theory of characters and a Brauer-Nesbitt theorem ensue.

Let S ⊆ K be a discrete valuation ring with field of fractions K, maximal ideal generated by π and residue field S̄ := S/πS of characteristic p ⩾ 0. The π-adic valuation of an element y ∈ K is denoted by v(y). Let T ⊆ L be the integral closure of S in L, and denote T̄ := T/πT. Choosing T ≀ G-lattices inside the simple L ≀ G-modules, we may restrict the Wedderburn isomorphism to a full embedding of T ≀ G into a tuple of matrix rings over certain extensions of S. We calculate the colength of this embedding, that is, the S-linear length of its cokernel.

Now suppose S̄ to be finite. The question for the number of indecomposable projective T ≀ G-modules leads to the question for the number of simple T̄ ≀ G-modules. If T̄ ≀ G/Jac(T̄ ≀ G) is a split semisimple S̄-algebra, then this number coincides with the S̄- linear dimension of Z(T̄ ≀ G/Jac(T̄ ≀ G)). This dimension in turn is given by a sum of certain vectorspace dimensions, indexed by the p′-classes of G. In the untwisted case, each of these dimensions equals 1. Thus we recover a theorem of Brauer.

0.2 Motivation

The untwisted group ring SG has the principal module S, endowed with the trivial Goperation. This module is projective if and only if the order of G is invertible in S.

The twisted group ring T ≀ G has the principal module T, endowed with the Galois operation of G. This module is projective if and only if the order of N is invertible in S and the extension T/S is tamely ramified (Speiser, E. Noether, cf. 2.4).

1This ring is also known as skew group ring of G with coefficients in L, see e.g. [12, p. 59]. Using the terminology twisted group ring of [9, §28], we should mention that there are more general twisted group rings still, involving 2-cocycles of G with coefficients in L^{∗}.

Therefore, wildness of T/S corresponds to a nontrivial homological behaviour of T over T ≀ G, which one is led to study. This seems to be more easily accessible than the related homological behaviour of T over SG. For instance, we have Ext_{T ≀G}^{∗}(T, T) ≃ Ext_{SG}^{∗}(S, T), and in particular cases it is possible to calculate the left hand side in order to obtain the right hand side. Here, we will restrict our considerations to some general properties of twisted group rings, however.

0.3 Known Results

Beautiful results on the classical case N = G abound. It might be worthwhile to attempt to extend certain of them to the twisted case. For example, so far there is no theory on induction and restriction that allows to independently vary the group and the ring (cf. [12, ch. 4], [9, §§ 10, 11, 19, 20]) (^{2}).

On the other hand, there are several results for the case N = 1. The theorem of Auslander-Goldman states that T ≀ G is isomorphic to S^{|G|×|G|} if and only if T/S is unramified and all occurring residue field extensions are separable [9, 28.5]. The theorem of Auslander-Rim states that T ≀ G is hereditary if and only if T/S is tamely ramified [9, 28.7].

If we admit wild ramification, or if nontrivial 2-cocycles enter, the situation gets more involved. Still, we assume N = 1.

• Wilson’s theorem calculates the locally free class group of O_{L} ≀ G to be the class group of a maximal O_{K}-order containing it, provided O_{L}/O_{K} is a finite extension of Dedekind domains giving rise to an extension L/K of global fields with Galois group G [10, 50.64, 49.33, 49.32].

• Williamson gives a criterion for π-principal heredity of T ≀_{f} G, where f is a 2cocycle of G with values in T^{∗} [28, 3.5]. In contrast to the above-mentioned theorems, it may occur that T ≀_{f} G is hereditary, and even maximal, whereas T/S is wildly ramified [27, 2.5].

• Benz and Zassenhaus showed that T ≀_{f} G is contained in a unique minimal hereditary overorder, S assumed to be complete [1].

• Cliff and Weiss showed how invariants of this hereditary overorder depend on the inertia and ramification indices of T/S, and the Schur index of L ≀_{f} G [8].

• If [L : K] = 2, Chalatsis and Theohari-Apostolidi classified the indecomposable O_{L} ≀ G-lattices [6, Th. 2].

• Weber described the image of the Wedderburn embedding for T = Z_{(p)}[ζ_{p^{m}}], G = C_{p} and S = Z_{(p)}[ζ_{p^{m−1}}] [25].

2For this purpose, it might be useful to consider a larger class of rings with group operation than merely Galois extensions of fields – cf. §1.3.

For N arbitrary, Nakayama and Shoda investigated twisted group rings L ≀ G over a field L such that the order of G is invertible in L, relating the representations of L ≀ G and KN [15]. Omitting this invertibility condition, Bonami investigated centers and radicals in [2].

Moreover, the ring K ≀_{f} G has been studied thoroughly, where f is a 2-cocycle with coefficients in the trivial G-module K^{∗}; one reason being its connection to Clifford theory (cf. [9, 11.19, 11.20]). See also [24, (10.5)].

0.4 Results

Let g := |G|, n := |N| and h := [L : K], so nh = g. Let {X_{i} | i ∈ [1, k]} be a complete set of nonisomorphic simple L ≀ G-right modules and let K_{i} := End_{L≀G}X_{i}. Let x_{i} denote the dimension of X_{i} as a right module over K_{i}, let r_{i} denote the degree [K_{i} : K] and let d_{i}^{2} denote the degree of K_{i} over its center. We have x_{i}d_{i}/h ∈ Z (1.14). Let V_{i} ⊆ X_{i} be a finitely generated T ≀ G-module such that KV_{i} = X_{i} and such that V_{i} is free as a module over S_{i} := End_{T ≀G}V_{i}.

The colength of an embedding of S-modules is defined to be the Jordan-Hölder multiplicity of S̄ in its quotient. If S̄ is finite, then colength = log_{|S̄|}(index).

Let S_{i}^{+} := {y ∈ K_{i} | Tr_{K_{i}/K}(yS_{i}) ⊆ S} and let δ_{S_{i}/S} be the colength of S_{i} ⊆ S_{i}^{+}.

Let T^{+} := {y ∈ L | Tr_{L/K}(yT) ⊆ S} and let δ_{T/S} be the colength of T ⊆ T^{+}.

Theorem (2.15). The colength of the Wedderburn embedding

T ≀ G [figure: ω] ∏_{i∈[1,k]}End_{S_{i}}V_{i}

is given by

\frac{1}{2} (g(δ_{T/S} + v(n)h) − ∑_{i∈[1,k]}x_{i}^{2}(δ_{S_{i}/S} + v(x_{i}d_{i}/h)r_{i})) .

In particular, this integer is positive or zero.

The essential ingredient for (2.15) is the Plancherel formula (1.16). Using an analogous central Plancherel formula (1.40), we obtain a central colength formula (2.25).

Schur relations allow to derive a Brauer-Nesbitt theorem. Let t be the maximal elementary divisor exponent of the Gram matrix of the trace bilinear form on T induced by Tr_{L/K}. Note that v(x_{i}) ⩽ v(g) + t − v(d_{i}) (2.21).

Theorem (3.1). Let i ∈ [1, k]. Suppose that K = K_{i} and that v(x_{i}) = v(g) + t. Then the reduction V_{i}/πV_{i} of the simple Λ-lattice V_{i} is a simple T̄ ≀ G-module.

Moreover, we shall count simple modules.

Theorem (1.29, 3.19).

(i) The K-linear dimension of the center Z(L ≀ G) equals the number of conjugacy classes of N. In particular, if L ≀ G is split semisimple over K, then this number equals the number of isoclasses of simple L ≀ G-modules.

(ii) Let T̄_{0} := T/Jac(T). Let

V_{σ} := S̄⟨(y^{σ} − y)z, y^{ρ} − y | y, z ∈ T̄_{0}, ρ ∈ C_{G}(σ)⟩ ⊆ T̄_{0} .

Let Cl(G, p′) be a set of representatives of p′-classes of G. We have

dimS̄ Z((T ≀ G)/Jac(T ≀ G)) = ∑_{σ∈Cl(G,p′)} dimS̄(T̄_{0}/V_{σ}) .

In particular, if T ≀ G/Jac(T ≀ G) is split semisimple over S̄, then this number equals the number of isoclasses of simple T ≀ G-modules, or, which is the same, the number of isoclasses of indecomposable projective T ≀ G-modules.

If T̄_{0} is a field, and if σ is not contained in the kernel N_{0} of the operation of G on T̄_{0}, then T̄_{0}/V_{σ} = 0. This indicates a similarity of (ii) to (i).

0.5 Acknowledgements

I thank G. Nebe for pointing out Plesken’s method to calculate colengths, and for several helpful discussions. I thank the referee for providing the isomorphism (1.25), yielding a splitting field for L ≀ G (1.27). Dedicated to K. W. Roggenkamp on the occasion of his 60th birthday.

0.6 Conventions

(i) Composition of maps is written on the right, ^{\overline{}^{a}} ^{\overline{}^{b}} = ^{\overline{}^{ab}} . Exception is made for ‘standard’ maps, such as traces, characters, Frobenius …

(ii) If A is an assertion, which might be true or false, we let {A} = 1 if A is true, and {A} = 0 if A is

false. (iii) For a, b ∈ Z, we denote by [a, b] := {c ∈ Z | a ⩽ c ⩽ b} the integral interval. (iv) Given a ∈ Z and a prime p, we denote by a[p] := p^{v_{p}(a)} the p-part of a.

(v) Given elements x, y of some set X, we let ∂_{x,y} = 1 in case x = y and ∂_{x,y} = 0 in case x ≠ y. (vi) Unless mentioned otherwise, modules are finitely generated right modules. (vii) Let A be a commutative ring. The trace of an element of an A-free A-algebra is the trace of the

A-linear map given by right multiplication with this element. (viii) Given a commutative ring A and x, y, z ∈ A, we write x ≡_{z} y if x − y ∈ Az. (ix) Given a ring A and m, n ⩾ 0, we denote by A^{m×n} the set of m × n-matrices over A.

(x) Given a ring A, we denote by Jac(A) ⊆ A its Jacobson radical. (xi) Given an element σ of a finite group G, we let o(σ) denote its order.

1 Rational

1.1 The Plancherel formula

Definition 1.1 Let C be a commutative ring, acted upon by a group G with fixed ring B := Fix_{G}C, an element τ ∈ G sending an element y ∈ C to y^{τ}. The twisted group ring C ≀ G is defined as follows. As a right C-module, it is free on the underlying set of the group G. The multiplication is defined by

(∑_{σ∈G}σy_{σ}) (∑_{τ∈G}τz_{τ}) := ∑_{ρ∈G}ρ (∑_{τ∈G}y_{ρτ^{−1}}^{τ}z_{τ}) ,

where y_{σ}, z_{τ} ∈ C for σ, τ ∈ G. Or, less formally, we extend (σy)(τz) = (στ)(y^{τ}z) B-linearly. Note that C ≀ G is a B-algebra.

Notation 1.2 Let G be a finite group acting on a field L with fixed field K := Fix_{G}L. Let H be the image of the operation map G \overline{}Aut_{fields} L, i.e. H = Gal(L/K), and let N denote the kernel of this operation map. The orders are denoted by

\begin{aligned}h & := & |H| = [L : K] \\ g & := & |G| \\ n & := & |N| , \end{aligned}

so g = hn. Let u ∈ L be such that Tr_{L/K}(u) = 1. Choose a K-basis (y_{l})_{l∈[1,h]} of L, and let (y_{l}^{∗})_{l∈[1,h]} be its dual basis with respect to Tr_{L/K}, i.e. Tr_{L/K}(y_{l}y_{m}^{∗}) = ∂_{l,m} for l, m ∈ [1, h]. Whenever useful, we assume y_{1} = 1.

Lemma 1.3 (Maschke, cf. [9, 28.7]) The K-algebra L ≀ G is semisimple if and only if n is invertible in K.

Suppose n to be invertible in K. Given an epimorphism of L ≀ G-modules X ^{\overline{}^{f}} X′′, we choose an L-linear coretraction X′′ ^{\overline{}^{i_{0}}} X and let

[figure: X′′; i; X; x′′; n^{−1}∑_{σ∈G}(x′′σ)i_{0} · uσ^{−1} ,]

which is L ≀ G-linear and satisfies if = 1_{X′′}.

Conversely, suppose L≀G to be semisimple. Since L≀G \overline{}L : σy \overline{}y is an epimorphism, it has a coretraction, which is necessarily of the form L \overline{}L≀G : 1 \overline{}z ∑_{σ∈G} σ for some z ∈ L. Composition yields n · Tr_{L/K}(z) = 1.

Henceforth, we assume n to be invertible in K. Now, alternatively, semisimplicity ensues from the following lemma [20, 7.18, 7.20] Lemma 1.4 The K-algebra L ≀ G is separable.

We have to show that the (L ≀ G)^{o} ⊗_{K} (L ≀ G)-linear epimorphism

[figure: (L ≀ G)^{o}; ⊗_{K}; (L ≀ G); L ≀ G; σu; ⊗; vρ; σuvρ]

is split, where σ, ρ ∈ G, u, v ∈ L. Let ∑_{i∈I} ξ_{i} ⊗ η_{i} ∈ L ⊗_{K} L ≃ ∏_{H} L be the element that satisfies ∑_{i∈I} ξ_{i}η_{i}^{λ} = ∂_{1,λ}u for λ ∈ H. Then

[figure: L ≀ G; (L ≀ G)^{o}; ⊗_{K}; (L ≀ G); 1; n^{−1} ∑_{κ∈G, i∈I}κξ_{i}; ⊗; η_{i}κ^{−1}]

is a coretraction as sought for.

Lemma 1.5 (Dedekind) Equipped with the operation y · (σz) := y^{σ}z, where y, z ∈ L, σ ∈ G, the field L becomes an absolutely simple module over L ≀ G, called the principal module. We have End_{L≀G}L = K. The module L is the simple module of the block L ≀ H of L ≀ G, unique up to isomorphism.

Consider the map

\begin{matrix}L ≀ H & ^{\overline{}^{ω_{H}}} & End_{K}L \\ σy & \overline{} & (z \overline{}z^{σ}y). \end{matrix}

Both source and target are of dimension h^{2} over K. Injectivity of ω_{H} is the content of Dedekind’s Lemma.

Definition 1.6 Let

(σy, τz) := ∂_{σ,τ^{−1}}Tr_{L/K}(y^{τ}z) = \frac{1}{n}Tr_{K}(u(−)σyτz)

define a K-bilinear form on L ≀ G, where σ, τ ∈ G, y, z ∈ L, and where ξ(−)ξ′ denotes the K-linear endomorphism of L≀G that sends η to ξηξ′, where ξ, ξ′ ∈ L≀G. This bilinear form is symmetric, associative and nondegenerate.

Notation 1.7 Let {X_{i} | i ∈ [1, k]} be a complete set of nonisomorphic simple L ≀ G-right modules and let K_{i} := End_{L≀G}X_{i} be the respective endomorphism skew field. Let x_{i} denote the dimension of X_{i} as a left module over K_{i}, let r_{i} denote the degree [K_{i} : K]. Choose X_{1} = L, whence K_{1} = K. Choose a maximal commutative subfield E_{i} of K_{i} that is separable over K, and let c_{i} := [Z(K_{i}) : K] and d_{i} := [E_{i} : Z(K_{i})] = [K_{i} : E_{i}] [20, 7.15]. In particular, r_{i} = c_{i}d_{i}^{2}. So altogether,

(K ⊆^{c_{i}} Z(K_{i}) ⊆^{d_{i}} E_{i} ⊆^{d_{i}} K_{i}) = (K ⊆^{r_{i}} K_{i}) .

Denote the operation of L ≀ G on X_{i} by

L ≀ G ^{\overline{}^{ω_{i}}} End_{K_{i}}X_{i} .

The Wedderburn isomorphism, of K-algebras, reads

[figure: L ≀ G; ω; ∼; ∏_{i∈[1,k]}End_{K_{i}}X_{i}; ξ; (ξω_{i})_{i∈[1,k]} .]

In particular, gh = ∑_{i∈[1,k]} r_{i}x_{i}^{2}. Note that we may interpret ω_{1}, up to isomorphism, as the surjective ring morphism L ≀ G \overline{}L ≀ H induced by G \overline{}H.

Let tr_{K_{i}/Z(K_{i})} : K_{i} \overline{}Z(K_{i}) denote the reduced trace [20, 9.6a, 9.3], and let

tr_{K_{i}/K} := Tr_{Z(K_{i})/K} ∘ tr_{K_{i}/Z(K_{i})} : K_{i} \overline{}K .

Remark 1.8 The next aim is to provide the Plancherel formula (1.16), which is needed in §2. Note that if h was assumed to be invertible in L, then the following considerations would simplify considerably, for then we would be able to write (σy, τz) = \frac{1}{g}Tr_{K}((−)σyτz). In our situation, however, we have to split up everything and to keep track of the various identifications.

Notation 1.9 (auxiliary) Let F be a finite galois extension of K that contains L and that contains E_{i} for all i ∈ [1, k]. Given i ∈ [1, k], we let Θ_{i} be the set of K-linear embeddings of Z(K_{i}) into F, so in particular |Θ_{i}| = c_{i}. For each ϑ ∈ Θ_{i}, we choose a prolongation ϑ̂: E_{i} \overline{}F of ϑ.

By choice of a K_{i}-linear basis, we identify X_{i} = K_{i}^{1×x_{i}}, considered as a row of End_{K_{i}}X_{i} = K_{i}^{x_{i}×x_{i}}.

Given μ ∈ H, we let e_{μ} ∈ F ⊗_{K}L ≃ ∏_{λ∈H} F be mapped to (∂_{μ,λ})_{λ} under this isomorphism.

Remark 1.10 Suppose given i ∈ [1, k]. We have an isomorphism of E_{i}-algebras

[figure: E_{i}; ⊗_{K}; K_{i}; ∼; End_{E_{i}}K_{i}; e; ⊗; w; e(−)w]

(cf. [20, pf. of 7.15, pf. of 7.11]). Therefore, we have an isomorphism of F -algebras

[figure: F; ⊗_{K}; K_{i}; ∼; ∏_{ϑ∈Θ_{i}} End_{F} (F _{ϑ̂}⊗K_{i}); f; ⊗; w; (f _{ϑ̂}(−)w)_{ϑ} ,]

where we denote F _{ϑ̂}⊗K_{i} := F ⊗_{E_{i}} K_{i} with respect to E_{i} [figure: ϑ̂] F, and where f _{ϑ̂}(−)w denotes the F -linear endomorphism of F _{ϑ̂}⊗K_{i} that sends f′ ⊗ w′ to ff′ ⊗ w′w (cf. [20, p. 100]).

Lemma 1.11 Given i ∈ [1, k], w ∈ K_{i} and ϑ ∈ Θ_{i}, we have

∑_{ϑ∈Θ_{i}} Tr_{F} (1_{ϑ̂}(−)w) = tr_{K_{i}/K}(w) ∈ K .

Using an E_{i}-linear basis of K_{i} as an F -linear basis of F _{ϑ̂}⊗_{K}K_{i}, we see, denoting by (−)w the E_{i}-linear multiplication endomorphism of K_{i}, that

\begin{aligned}Tr_{F} (1_{ϑ̂}(−)w) & = & (Tr_{E_{i}}((−)w))ϑ̂ \\ & = & (tr_{K_{i}/Z(K_{i})}(w))ϑ \end{aligned}

by definition of the reduced trace [20, 9.6a, 9.3, pf. of 7.15], cf. (1.10).

Lemma 1.12 Representatives for the isoclasses of simple F ⊗_{K} L ≀ G-modules are given by F _{ϑ̂}⊗X_{i}, where i ∈ [1, k], ϑ ∈ Θ_{i}. Note that X_{i} is a left E_{i}-module by restriction from K_{i}. Given f ∈ F and ξ ∈ L ≀ G, f ⊗ ξ acts on F _{ϑ̂}⊗X_{i} as

(f ⊗ ξ)ω_{i,ϑ} := f _{ϑ̂}⊗(ξ)ω_{i} ,

giving the component F ⊗_{K} L ≀ G ^{\overline{}^{ω_{i,ϑ}}} End_{F} F _{ϑ̂}⊗X_{i} of the Wedderburn isomorphism.

A block of (L ≀ G)ω splits under F ⊗_{K} − into

F ⊗_{K} End_{K_{i}}X_{i} = (F ⊗_{K} K_{i})^{x_{i}×x_{i}}
=^{(1.10)} ∏_{ϑ∈Θ_{i}}(End_{F} (F _{ϑ̂}⊗K_{i}))^{x_{i}×x_{i}}
= ∏_{ϑ∈Θ_{i}} End_{F} (F _{ϑ̂}⊗X_{i}) .

Now, f ⊗ φ corresponds to (f ⊗ φ_{a,b})_{a,b∈[1,x_{i}]}, then to ((f _{ϑ̂}(−)φ_{a,b})_{a,b∈[1,x_{i}]})_{ϑ}, hence finally to (f _{ϑ̂}⊗φ)_{ϑ}.

Lemma 1.13 Given i ∈ [1, k], ϑ ∈ Θ_{i} and φ ∈ End_{K_{i}}X_{i}, we have

∑_{ϑ∈Θ_{i}} Tr_{F} (1_{ϑ̂}⊗φ) = tr_{K_{i}/K}Tr_{K_{i}}φ .

In fact,

∑_{ϑ∈Θ_{i}} Tr_{F} (1_{ϑ̂}⊗φ) = ∑_{a∈[1,x_{i}]} ∑_{ϑ∈Θ_{i}} Tr_{F} (1_{ϑ̂}(−)φ_{a,a})
=^{(1.11)} ∑_{a∈[1,x_{i}]} tr_{K_{i}/K}φ_{a,a}
= tr_{K_{i}/K}Tr_{K_{i}}φ .

Lemma 1.14 Suppose given i ∈ [1, k] and ϑ ∈ Θ_{i}. We have x_{i}d_{i}/h ∈ Z. Moreover, given z ∈ Z(K_{i}) ⊆ End_{K_{i}}X_{i}, we have

Tr_{F} ((e_{μ})ω_{i,ϑ}) = x_{i}d_{i}/h .

We decompose F _{ϑ̂}⊗X_{i} = ⊕_{λ∈H}(F _{ϑ̂}⊗X_{i})e_{λ} as F -vectorspaces. Since (F _{ϑ̂}⊗X_{i})e_{λ} ≃ (F _{ϑ̂}⊗X_{i})e_{σ^{−1}λ} for σ ∈ G via multiplication with σ, using e_{μ}σ = σe_{σ^{−1}μ}, the F -linear dimension of (F _{ϑ̂}⊗X_{i})e_{μ} is given by x_{i}d_{i}/h. Now,

Tr_{F} ((e_{μ})ω_{i,ϑ}) = ∑_{λ∈H} Tr_{F} ((e_{μ})ω_{i,ϑ}|_{(F _{ϑ̂}⊗X_{i})e_{λ}})
= Tr_{F} ((e_{μ})ω_{i,ϑ}|_{(F _{ϑ̂}⊗X_{i})e_{μ}})
= x_{i}d_{i}/h .

Lemma 1.15 Given m ⩾ 1 and A, B ∈ F^{m×m}, we have Tr_{F} (A(−)B) = Tr_{F} (A)Tr_{F} (B), where A(−)B denotes the F -linear endomorphism of F^{m×m} that sends C ∈ F^{m×m} to ACB, and where Tr_{F} (A) denotes the trace of A as an element of F^{m×m}.

Proposition 1.16 (Plancherel formula) Given ξ, η ∈ L ≀ G, we have

(ξ, η) = ∑_{i∈[1,k]} \frac{x_{i}d_{i}}{g} · tr_{K_{i}/K}Tr_{K_{i}}((ξω_{i})(ηω_{i})) .

The right hand side is well defined by (1.14).

By associativity, we may assume η = 1. We extend the bilinear form F -linearly to F ⊗_{K} L ≀ G to obtain

(ξ, 1) =  (1 ⊗ ξ, 1 ⊗ 1)
=  n^{−1} Tr_{F} ((1 ⊗ u)(−)(1 ⊗ ξ))

independence of choice

= n^{−1} Tr_{F} (e_{1}(−)(1 ⊗ ξ))

algebra isomorphism

= n^{−1} ∑_{i∈[1,k]} ∑_{ϑ∈Θ_{i}} Tr_{F} (((e_{1})ω_{i,ϑ})(−)((1 ⊗ ξ)ω_{i,ϑ}))

=^{(1.15)}  n^{−1} ∑_{i∈[1,k]} ∑_{ϑ∈Θ_{i}} Tr_{F} ((e_{1})ω_{i,ϑ}) · Tr_{F} ((1 ⊗ ξ)ω_{i,ϑ})
=^{(1.14, 1.12)}  ∑_{i∈[1,k]} ∑_{ϑ∈Θ_{i}}(x_{i}d_{i}/g)Tr_{F} (1_{ϑ̂}⊗ξω_{i})
=^{(1.13)}  ∑_{i∈[1,k]}(x_{i}d_{i}/g) · tr_{K_{i}/K}Tr_{K_{i}}(ξω_{i}) .

1.2 Conclusions

Corollary 1.17 The quotient x_{i}d_{i}/h ∈ Z is invertible in K for any i ∈ [1, k].

If x_{i}d_{i}/h = 0 in K, then the bilinear form (−, =) on L ≀ G would be degenerate by (1.16). This is not the case, and so the result follows. Cf. (1.14).

Corollary 1.18 (Fourier inversion) The inverse of the Wedderburn isomorphism is given by

[figure: ∏_{i ∈ [1,k]}End_{K_{i}}X_{i}; ω^{−1}; ∼; L ≀ G; (φ_{i})_{i∈[1,k]}; \frac{1}{n} ∑_{σ ∈ G, l ∈ [1,h]}σy_{l} [ ∑_{i ∈ [1,k]} \frac{x_{i}d_{i}}{h} · tr_{K_{i}/K}Tr_{K_{i}} ((y_{l}^{∗}σ^{−1})ω_{i} · φ_{i})] .]

Given ρ ∈ G and m ∈ [1, h], we verify

(ρy_{m})ωω^{−1} = \frac{1}{n} ∑_{σ ∈ G, l ∈ [1,h]}σy_{l} [ ∑_{i ∈ [1,k]} \frac{x_{i}d_{i}}{h} · tr_{K_{i}/K}Tr_{K_{i}} ((y_{l}^{∗}σ^{−1})ω_{i} · (ρy_{m})ω_{i})]

=^{(1.16)} ∑_{σ ∈ G}σ [ ∑_{l ∈ [1,h]}y_{l}(y_{l}^{∗}σ^{−1}, ρy_{m})]

= ∑_{σ ∈ G}σ [∂_{ρ,σ} ∑_{l ∈ [1,h]}y_{l}Tr_{L/K}(y_{l}^{∗}y_{m})]
= ρy_{m} .

Suppose given i ∈ [1, k], y ∈ L, σ ∈ G. With respect to chosen K_{i}-linear bases of X_{i}, we write the endomorphism (σy)ω_{i} of X_{i} as a matrix ((σy)ω_{i;a,b})_{a,b∈[1,x_{i}]}.

Corollary 1.19 (Schur relations) Suppose given i, i′ ∈ [1, k], a, b ∈ [1, x_{i}], a′, b′ ∈ [1, x_{i′}]. Then

\frac{g}{x_{i}d_{i}} · ∂_{(i;a,b),(i′;a′,b′)} = ∑_{l∈[1,h], σ∈G}(σy_{l})ω_{i;a,b} · tr_{K_{i′}/K}((y_{l}^{∗}σ^{−1})ω_{i′;b′,a′}) .

By (1.18), we calculate the image of the endomorphism tuple given by the tuple of matrices ((∂_{(i′′;a′′,b′′),(i′;a′,b′)})_{a′′,b′′∈[1,x_{i′′}]})_{i′′∈[1,k]} under ω^{−1}ω to be

( ∑_{l∈[1,h], σ∈G}(σy_{l})ω_{j} \frac{x_{i′}d_{i′}}{g} tr_{K_{i′}/K}((y_{l}^{∗}σ^{−1})ω_{i′;b′,a′}))_{j∈[1,k]} ,

and it remains to compare entries at the position (i; a, b).

Notation 1.20 Given i ∈ [1, k], we let the character χ_{i} of X_{i} be defined by

[figure: L ≀ G; χ_{i}; K; ξ; χ_{i}(ξ) := tr_{K_{i}/K}Tr_{K_{i}}((ξ)ω_{i}) = ∑_{a∈[1,x_{i}]}tr_{K_{i}/K}((ξ)ω_{i;a,a}) .]

Corollary 1.21 (horizontal orthogonality relations) Suppose given i, i′ ∈ [1, k]. Then

gc_{i} ∂_{i,i′} = ∑_{l∈[1,h], σ∈G}χ_{i}(σy_{l})χ_{i′}(y_{l}^{∗}σ^{−1}) .

Corollary 1.22 (to 1.18) The primitive central idempotent ε_{i} of L ≀ G that belongs to X_{i}, i ∈ [1, k], is given by

ε_{i} = \frac{x_{i}d_{i}}{g} ∑_{σ ∈ G, l ∈ [1,h]}σy_{l}^{∗} · χ_{i}(y_{l}σ^{−1}) .

In particular, ε_{1} = n^{−1}∑_{ν∈N}ν.

To evaluate ε_{1}, it suffices to remark that for y ∈ L and σ ∈ G, the K-linear trace of (yσ)ω_{1} is given by {σ ∈ N}Tr_{L/K}(y), as to be calculated as L-linear trace on L ⊗_{K} L ≃ ∏_{λ∈H} L using the primitive idempotent basis.

Remark 1.23 For the principal module, there is an inversion formula due to E. Noether [18, §1]. Write φ ∈ End_{K}L as y_{l}φ =: ∑_{m∈[1,h]} φ_{l,m}y_{m} with φ_{l,m} ∈ K. Then

[figure: End_{K}L; ω_{H}^{−1}; ∼; L ≀ H ( ≃ L ≀ Gε_{1} ⊆ L ≀ G ); φ; ∑_{l,m∈[1,h]} φ_{l,m}y_{l}^{∗}(∑_{λ∈H} λ)y_{m} .]

To see this, we let φ_{1} = (ρz)ω_{1} with ρ ∈ G and z ∈ L and apply (1.18) to

(φ_{1}, 0, …, 0) ^{\overline{}^{ω^{−1}}} n^{−1} ∑_{σ∈G} σ ∑_{l∈[1,h]} y_{l}Tr_{K}((y_{l}^{∗}σ^{−1}ρz)ω_{1})
= n^{−1} ∑_{σ∈G} σ{σ^{−1}ρ ∈ N} ∑_{l∈[1,h]} y_{l}Tr_{L/K}(y_{l}^{∗}z)
= ρzε_{1} ,

whereas Noether’s formula yields

\begin{aligned}φ_{1} & \overline{} & ∑_{l∈[1,h]} y_{l}^{∗}(∑_{λ∈H} λ)(y_{l}φ_{1}) \\ & = & (∑_{λ∈H} λ ∑_{l∈[1,h]} y_{l}^{∗,λ}y_{l}) ρz , \end{aligned}

and it remains to be shown that ∑_{l∈[1,h]} y_{l}^{∗,λ}y_{l} = ∂_{λ,1}. Since the left hand side is independent of the choice of the basis, we may tensor with L⊗_{K} − and use the primitive idempotent basis of L ⊗_{K} L ≃ ∏_{μ∈H} L, which is self dual. Orthogonality of these idempotents yields the result, for λ ≠ 1 permutes them without fixed points.

Question 1.24 Suppose N = 1. Given f ∈ H^{2}(G, L^{∗}), there is a central simple K-algebra L ≀_{f} G (denoted (L/K, f) in [20, 29.6]). If f is trivial, (1.18) or (1.23) allow to deduce an orthogonal primitive idempotent decomposition. I do not know such a decomposition in the general case, nor a workable description of the simple module (as an L-vectorspace with compatible G-operation, say). Cf. [20, 29.22, 32.19], [26, 2.4].

1.3 A Morita equivalence

Lemma 1.25 Considering L ≀ G as a left module over LN, we have an isomorphism

\begin{matrix}L ⊗_{K} L ≀ G & ^{\overline{}^{∼}} & End_{LN}(L ≀ G) ≃ (LN)^{h×h} \\ y ⊗ zσ & \overline{} & (wρ \overline{}y · wρ · zσ) \end{matrix}

of algebras over L.

Since both source and target are of dimension gh over L, it suffices to prove injectivity. Suppose given an element ∑_{σ∈G, i∈I_{σ}} y_{σ,i} ⊗ z_{σ,i}σ that is mapped to zero. From ∑_{σ∈G, i∈I_{σ}} y_{σ,i}ρz_{σ,i}σ = 0 for all ρ ∈ G we conclude that ∑_{i∈I_{σ}} y_{σ,i}z_{σ,i}^{ρ^{−1}} = 0 for all σ, ρ ∈ G. The isomorphism L ⊗_{K} L ^{\overline{}^{∼}} ∏_{λ∈H} L now shows that ∑_{i∈I_{σ}} y_{σ,i} ⊗ z_{σ,i} = 0 for all σ ∈ G, as was to be shown.

Corollary 1.26 L ⊗_{K} L ≀ G is Morita equivalent to LN.

Concerning the situation over K, cf. (1.34, 1.36, 1.37).

Corollary 1.27 The field L(ζ_{exp(N)}) is a splitting field for L ≀ G over K.

By (1.25), we have L(ζ_{exp(N)}) ⊗_{K} (L ≀ G) ≃ (L(ζ_{exp(N)})N)^{h×h}, and L(ζ_{exp(N)}) is a splitting field for N by [22, 12.3, cor. to th. 24]. Cf. also (1.36).

1.4 The center of L ≀ G

Let Cl_{N}^{G} ⊆ N denote a set of representatives of the G-orbits of N by conjugation. Given σ ∈ N, we let h_{σ} := h/|C_{G}(σ)N/N| = g/|C_{G}(σ)N|. Let (y_{σ,i})_{i∈[1,h_{σ}]} be a K-linear basis of L_{σ} := Fix_{C_{G}(σ)}L ⊆ L, where σ ∈ Cl_{N}^{G}.

Lemma 1.28 ([15, p. 113]) The center Z(L ≀ G) of L ≀ G has the K-linear basis

( ∑_{ρ∈C_{G}(σ)\G}σ^{ρ}y_{σ,i}^{ρ})_{σ∈Cl_{N}^{G} , i∈[1,h_{σ}]} .

Suppose given ∑_{σ∈G} σt_{σ} ∈ Z(L≀G). For any y ∈ L, we have y(∑_{σ∈G} σt_{σ})−(∑_{σ∈G} σt_{σ})y = 0, whence ∑_{σ∈G} σt_{σ} = ∑_{σ∈N} σt_{σ}. Suppose given ρ ∈ G. From

∑_{σ∈N} σt_{σ} = (∑_{σ∈N} σt_{σ})^{ρ} = ∑_{σ∈N} σt_{σ^{ρ^{−1}}}^{ρ},

we conclude that t_{σ^{ρ}} = t_{σ}^{ρ} for all σ ∈ N. Letting ρ ∈ C_{G}(σ), we conclude that t_{σ} ∈ L_{σ} for all σ ∈ Cl_{N}^{G}.

Corollary 1.29 ([15, Satz 2]) The K-linear dimension of the center Z(L ≀ G), i.e. ∑_{l∈[1,k]} dim_{K} Z(K_{l}), equals the number of conjugacy classes of N.

First proof. We let G act on the set of conjugacy classes of N, the G-orbits being represented by the N-orbits of the elements of Cl_{N}^{G}. The stabilizer of the N-orbit of σ ∈ Cl_{N}^{G} in G is given by C_{G}(σ)N. But the dimension of Z(L ≀ G) over K is given by

∑_{σ∈Cl_{N}^{G}}h_{σ} = ∑_{σ∈Cl_{N}^{G}}g/|C_{G}(σ)N|,

which is the sum of the orbit lengths of this operation, indexed by representatives of orbit representatives.

Second proof. We have dim_{K} Z(L ≀ G) = dim_{L} Z(L ⊗_{K} L ≀ G) =^{(1.26)} dim_{L} Z(LN).

Corollary 1.30 If K = K_{i} for all i ∈ [1, k], the number of irreducible modules of L ≀ G is given by number of conjugacy classes of N.

Corollary 1.31 Given i ∈ [1, k], we have χ_{i}(σy) = 0 if σ ∉ N. If σ ∈ N, then

∑_{l∈[1,h]} y_{l}^{∗,ρ}χ_{i}(y_{l}σ) = ∑_{l∈[1,h]} y_{l}^{∗}χ_{i}(y_{l}^{ρ^{−1}}σ) .

for ρ ∈ G.

The primitive central idempotent ε_{i} described in (1.22) is contained in Z(L ≀ G), so that the result follows by (1.28).

Assumption 1.32 For all ν ∈ N, we have C_{G}(ν) · N = G.

This assumption holds if N = 1, if N = G, or if N ⩽ Z(G).

Lemma 1.33 (cf. [2, p. 21]) If (1.32) holds, then Z(KN) = Z(L ≀ G).

Consider a basis element ∑_{ρ∈C_{G}(σ)\G} σ^{ρ}y_{σ,i}^{ρ} of L ≀ G as in (1.28). Since L_{σ} = K by (1.32), we may assume y_{σ,i} = 1. By (1.32), the G-orbits and the N-orbits on N coincide. Thus our basis element equals the class sum ∑_{ρ∈C_{N} (σ)\N} σ^{ρ} ∈ Z(KN).

Corollary 1.34 If (1.32) holds, and if moreover all endomorphism rings of simple modules of KN and of L ≀ G are commutative, then KN and L ≀ G are Morita equivalent.

Concerning the situation after scalar extension over K to L, cf. (1.26). Cf. also (1.36, 1.37).

1.5 (Counter)examples

Example 1.35 Suppose given a prime p ⩾ 3, and let ζ_{p^{2}} be a primitive p^{2}th root of unity over Q. Let π_{p^{2}} ∈ Q(ζ_{p^{2}}) be defined by π_{p^{2}} := ∏_{j∈[1,p−1]}(ζ_{p^{2}}^{j^{p}} − 1). Let ζ_{p} := ζ_{p^{2}}^{p}.

Consider the operation of C_{p^{2}} on Q(π_{p^{2}}) given by composition of the surjection C_{p^{2}} \overline{}C_{p} with the operation of the Galois group over Q. Let σ be a generator of C_{p^{2}}, let z ∈ Q(π_{p^{2}}). The principal module is given by

\begin{matrix}Q(π_{p^{2}}) ≀ C_{p^{2}} & \overline{} & End_{Q}Q(π_{p^{2}}) \\ σ & \overline{} & (y \overline{}y^{σ}) \\ z & \overline{} & (y \overline{}yz) . \end{matrix}

Moreover, Q(ζ_{p^{2}}) becomes a module by means of

\begin{matrix}Q(π_{p^{2}}) ≀ C_{p^{2}} & \overline{} & End_{Q(ζ_{p})}Q(ζ_{p^{2}}) \\ σ & \overline{} & (y \overline{}y^{σ}ζ_{p^{2}}) \\ z & \overline{} & (y \overline{}yz), \end{matrix}

where C_{p^{2}} operates on Q(ζ_{p^{2}}) via composition of the surjection C_{p^{2}} \overline{}C_{p} with the operation of the Galois group C_{p} of Q(ζ_{p^{2}}) over Q(ζ_{p}), so that σ maps ζ_{p^{2}} to ζ_{p^{2}}^{1+p}. The element σ^{p} is mapped to the multiplication with

N_{Q(ζ_{p^{2}} )/Q(ζ_{p})}(ζ_{p^{2}}) = ζ_{p^{2}}^{p+( \begin{matrix}p \\ 2 \end{matrix} )p} = ζ_{p^{2}}^{p} .

In particular, σ^{p^{2}} is mapped to the identity.

We claim that Q(ζ_{p^{2}}) is a simple module not isomorphic to Q(π_{p^{2}}), more precisely, we claim its endomorphism ring to be equal to Q(ζ_{p}). Since such an endomorphism is required to commute with Q(π_{p^{2}}) and with σ^{p}, it is determined by the image y of 1. Then commutativity with σ yields y = y^{σ}. As Wedderburn isomorphism, we obtain

Q(π_{p^{2}}) ≀ C_{p^{2}} \overline{}^{ω}_{∼} Q^{p×p} × Q(ζ_{p})^{p×p},

since this surjection is actually an isomorphism for dimension reasons. So in this case QC_{p} and Q(π_{p^{2}})≀C_{p^{2}} are in fact Morita equivalent, as shown by construction of the progenerator Q(π_{p^{2}}) ⊕ Q(ζ_{p^{2}}) (cf. 1.34).

Example 1.36 Let G = S_{3}, N = C_{3}, L = Q(i), K = Q, where i^{(1,2)} = −i. We note that (1.32) fails, since C_{G}((1, 2, 3)) = N. According to (1.28), the center of Q(i) ≀ S_{3} has the basis (1, (1, 2, 3)+(1, 3, 2), i(1, 2, 3)−i(1, 3, 2)). Thus we have an isomorphism of Q-algebras

[figure: Q; ×; Q(\sqrt{3}); ∼; Z(Q(i) ≀ S_{3}); 1; ×; 0; \frac{1}{3}(1 + (1, 2, 3) + (1, 3, 2)); 0; ×; 1; \frac{1}{3}(2 − (1, 2, 3) − (1, 3, 2)); 0; ×; \sqrt{3}; i(1, 2, 3) − i(1, 3, 2) .]

In particular, Z(L ≀ G) is not isomorphic to Z(KN) (cf. 1.33 and 1.34). Moreover, K(ζ_{|G|}) is not a splitting field for the K-algebra L ≀ G (but cf. 1.27).

Example 1.37 Let G = Q_{8} = ⟨i, j | i^{4}, j^{4}, i^{2} = j^{2}, ij = ji^{3}⟩ be the quaternion group with 8 elements, N = ⟨i^{2}⟩ ⩽ Z(G), L = Q(ζ_{8}), K = Q, ζ_{8}^{i} = ζ_{8}^{−1}, ζ_{8}^{j} = ζ_{8}^{3}. We remark that (1.32) is fulfilled. We have an idempotent

e := \frac{1 − i^{2}}{2} · \frac{1 + ζ_{8}j}{2} ∈ Q(ζ_{8}) ≀ Q_{8}

and an isomorphism

[figure: H_{Q}; ∼; e(Q(ζ_{8}) ≀ Q_{8})e; I; eζ_{8}ie; J; e(ζ_{8} + i)e ,]

where H_{Q} = Q⟨I, J⟩/(I^{2} + 1, J^{2} + 1, IJ + JI) is a rational quaternion skewfield. Now, Z(KN) = Z(L ≀ G) by (1.33), but KN and L ≀ G are not Morita equivalent (cf. 1.34).

1.6 The central Plancherel formula

We have an isomorphism

[figure: Z(L ≀ G); ω^{Z}; ∼; ∏_{i∈[1,k]}Z(K_{i}); ξ; ((ξ)ω_{i}^{Z})_{i} := ((ξ)ω_{i})_{i} ,]

where (ξ)ω_{i} ∈ End_{K_{i}}X_{i} is actually contained in Z(K_{i}) ⊆ End_{K_{i}}X_{i}. Recall that (ε_{j})ω_{i}^{Z} = ∂_{j,i} for i, j ∈ [1, k] (1.22).

Given ∑_{ρ∈C_{G}(ν)\G}(νy)^{ρ}, ∑_{ρ′∈C_{G}(ν′)\G}(ν′y′)^{ρ′} ∈ Z(L ≀ G), where ν, ν′ ∈ N, y ∈ L_{ν} and y′ ∈ L_{ν′} , we let

( ∑_{ρ∈C_{G}(ν)\G}(νy)^{ρ}, ∑_{ρ′∈C_{G}(ν′)\G}(ν′y′)^{ρ′})^{Z}

:= \{ \begin{matrix}[N : C_{N}(ν)] · Tr_{L_{ν}/K}(yy^{′τ}) & if ν = (ν^{′−1})^{τ} with τ ∈ G \\ 0 & if ν and ν^{′−1} are not conjugate in G \end{matrix}

define a K-bilinear form on Z(L ≀ G) (cf. 1.28). Note that yy^{′τ} ∈ L_{ν}. This bilinear form is symmetric and nondegenerate.

Remark 1.38 We have h · (ξ, ξ′)^{Z} = (ξ, ξ′) .

Let ν ∈ Cl_{N}^{G} and ν′ ∈ (Cl_{N}^{G})^{−1}, y ∈ L_{ν} and y′ ∈ L_{ν′} . Then

( ∑_{ρ∈C_{G}(ν)\G}(νy)^{ρ}, ∑_{ρ′∈C_{G}(ν′)\G}(ν′y′)^{ρ′}) = ∑_{ρ∈C_{G}(ν)\G} ∂_{ν,ν^{′−1}}Tr_{L/K}(y^{ρ}y^{′ρ})
= [G : C_{G}(ν)] · ∂_{ν,ν^{′−1}}Tr_{L/K}(yy′)
= [G : C_{N}(ν)] · ∂_{ν,ν^{′−1}}Tr_{L_{ν}/K}(yy′) .

Thus, if h is invertible in K, the following discussion simplifies considerably.

Lemma 1.39 The bilinear form (−, =)^{Z} is associative.

Let ν ∈ Cl_{N}^{G} and ν′ ∈ (Cl_{N}^{G})^{−1}, y ∈ L_{ν} and y′ ∈ L_{ν′} . Then

((∑_{ρ∈C_{G}(ν)\G}(νy)^{ρ}) (∑_{ρ′∈C_{G}(ν′)\G}(ν′y′)^{ρ′}) , 1)_{Z}

= ∂_{ν,ν^{′−1}} ∑_{ρ∈C_{G}(ν)\G} y^{ρ}y^{′ρ}
= [N : C_{N}(ν)] · ∂_{ν,ν^{′−1}}Tr_{L_{ν}/K}(yy′) .

Proposition 1.40 (central Plancherel formula) Given ξ, ξ′ ∈ Z(L ≀ G), we have

(ξ, ξ′)^{Z} = ∑_{i∈[1,k]} \frac{1}{n} (\frac{x_{i}d_{i}}{h} )^{2} Tr_{Z(K_{i})/K} ((ξ)ω_{i}^{Z}(ξ′)ω_{i}^{Z}) .

By associativity, we may assume ξ′ = 1 (1.39). Let ξ =: (ζ)(ω^{Z})^{−1}, where ζ =: (z_{i})_{i∈[1,k]} ∈ ∏_{i∈[1,k]} Z(K_{i}) ⊆ ∏_{i∈[1,k]} End_{K_{i}}X_{i}. Using the auxiliary notation (1.9), and viewing L ⊆ F ⊗_{K} L, we obtain

((ζ)(ω^{Z})^{−1}, 1)^{Z}
=^{(1.18)}  (∑_{σ∈G, l∈[1,h]} σy_{l} [∑_{i∈[1,k]} \frac{x_{i}d_{i}}{g} · tr_{K_{i}/K}Tr_{K_{i}}((y_{l}^{∗}σ^{−1})ω_{i} · z_{i})] , 1)_{Z}
=^{definition} ∑_{i∈[1,k]} ∑_{l∈[1,h]} y_{l} · \frac{x_{i}d_{i}}{g} · tr_{K_{i}/K}Tr_{K_{i}}((y_{l}^{∗})ω_{i} · z_{i})
=^{(1.13, 1.12)} ∑_{i∈[1,k]} ∑_{l∈[1,h]} ∑_{ϑ∈Θ_{i}}(1 ⊗ y_{l}) · \frac{x_{i}d_{i}}{g} · Tr_{F} ((1 ⊗ y_{l}^{∗})ω_{i,ϑ} · (1_{ϑ̂}⊗z_{i}))

independence

of choice = ∑_{i∈[1,k]} ∑_{μ∈H} ∑_{ϑ∈Θ_{i}} e_{μ} · \frac{x_{i}d_{i}}{g} · Tr_{F} ((e_{μ})ω_{i,ϑ} · (z_{i}ϑ_{ϑ̂}⊗1))

=^{(1.14)}  ∑_{i∈[1,k]} ∑_{μ∈H} ∑_{ϑ∈Θ_{i}} e_{μ} · \frac{x_{i}d_{i}}{g} · \frac{x_{i}d_{i}}{h} · z_{i}ϑ
=  ∑_{i∈[1,k]} \frac{1}{n} (\frac{x_{i}d_{i}}{h} )^{2} Tr_{Z(K_{i})/K}(z_{i}) .

2 Locally integral

Notation 2.1 Let S ⊆ K be a discrete valuation ring with field of fractions K, maximal ideal generated by π and residue field S̄ := S/πS of characteristic p ⩾ 0. The π-adic valuation of an element y ∈ K is denoted by v(y). Let T ⊆ L be the integral closure of S in L. Note that T is a principal ideal domain. Let π^{s}S := Tr_{L/K}(T). We have πT = ∏_{i∈[1,d]} 𝔮_{i}^{e}, where the 𝔮_{i} ⊆ T denote the maximal ideals, forming a single Galois orbit, and where e is the ramification index of π in T [16, I.§9]. Recall that π is said to be tamely ramified in T if T/𝔮_{i} is separable over S̄ for some (hence for all) i ∈ [1, d], and if, in addition, the ramification index e of π in T is not divisible by p. Otherwise, it is said to be wildly ramified.

2.1 Projectivity

Lemma 2.2 (Maschke locally) For S-free T ≀ G-modules X and Y, we have π^{s}n · Ext_{T ≀G}^{1}(Y, X) = 0. In particular, if ε is a central idempotent of L ≀ G, we have π^{s}nε ∈ T ≀ G.

Let u_{0} ∈ T be such that Tr_{L/K}(u_{0}) = π^{s}. Consider an extension

0 \overline{}X \overline{}E ^{\overline{}^{f}} Y \overline{}0

of T ≀ G-modules. We choose a T -linear coretraction Y ^{\overline{}^{i_{0}}} E to E ^{\overline{}^{f}} Y and let

[figure: Y; i; E; y; ∑_{σ∈G}(yσ)i_{0} · u_{0}σ^{−1} ,]

which is T ≀ G-linear and satisfies if = π^{s}n · 1_{Y}.

Applying this construction to the epimorphism T ≀G ^{\overline{}^{f}} εT ≀G given by left multiplication with ε, the map i sends ε to an element of the form ξ = ξε ∈ T ≀ G. Composition with f

yields π^{s}nε = εξε =^{ε central} ξε ∈ T ≀ G.

Corollary 2.3 Given i ∈ [1, k], σ ∈ G and l ∈ [1, h], we have π^{s} \frac{x_{i}d_{i}}{h} y_{l}^{∗}χ_{i}(y_{l}σ) ∈ T.

Apply (2.2) to (1.22).

Proposition 2.4 (Speiser, E. Noether, see [11, I.§3, Th. 3]) The following assertions are equivalent.

(i) As a module over T ≀ G, T is projective.

(ii) As a module over SG, T is projective.

(iii) The prime p does not divide n, and the trace Tr_{L/K} maps T onto S.

(iv) The prime p does not divide n, and π is tamely ramified in T.

Assertion (i) implies (ii) since T ≀ G is free over SG. We claim that (ii) implies (i). Given a split SG-linear epimorphism (SG)^{⊕i} \overline{}T, we obtain a split T ≀ G-linear epimorphism (T ≀ G)^{⊕i} \overline{}T ⊗_{SG} T ≀ G when tensoring with T ≀ G over SG. But the T ≀ G-linear epimorphism (the counit of the tensor adjunction)

[figure: T; ⊗_{SG}; T ≀ G; T; y; ⊗; σz; y^{σ}z; 1; ⊗; z; z]

is T ≀ G-linearly split as indicated.

We claim equivalence of (i) and (iii). As a module over T ≀ G, T is projective if and only if the T ≀ G-linear epimorphism

[figure: T ≀ G; η; T; 1; 1]

is split by some coretraction T ≀G^{\overline{}^{φ}} T. Amongst the T -linear maps from T to T ≀G, those are T ≀ G-linear that send 1 to y ∑_{σ∈G} σ for some y ∈ T. But this map is a coretraction to η if and only if n · Tr_{L/K}y = 1.

We claim equivalence of (iii) and (iv). Reducing modulo π, the surjectivity of the trace in question is equivalent to the nonvanishing of the S̄-linear trace on some factor T/𝔮_{i}^{e} of T̄. This trace in turn equals e times the trace on T/𝔮_{i}, by a filtration argument. Finally, the trace on T/𝔮_{i} does not vanish if and only if T/𝔮_{i} is separable over S̄.

Remark 2.5

(i) If the equivalent conditions of (2.4) are fulfilled, then T ≃_{SG} SH [9, 32.1].

(ii) If N = 1, another proof of (2.4), (iii ⇒ i) can be obtained by [9, 28.7], and by noting that T ≃ (∑_{σ∈G} σ)T ≀ G ⊆ T ≀ G is a right ideal.

2.2 The Wedderburn colength

We give a formula for the colength of the Wedderburn embedding of twisted group rings (2.15), following basically the method of Plesken [19] (that G. Nebe taught me).

Notation 2.6 For each i ∈ [1, k], we choose a finitely generated T ≀ G-module V_{i} ⊆ X_{i} such that KV_{i} = X_{i} and such that S_{i} := End_{T ≀G}V_{i} is a maximal order in K_{i} (cf. 2.7). In particular, we choose V_{1} = T ⊆ L = X_{1}, so that S_{1} = S. Note that S_{i} is finitely generated free as a module over S, and that KS_{i} = K_{i}, since S_{i} = End_{S}V_{i} ∩ K_{i} ⊆ End_{K}X_{i}. Note that V_{i} is free as a module over S_{i} [20, 18.7 (i)], of rank x_{i}. We fix S_{i}-linear bases for V_{i}, i ∈ [1, k], and write S_{i}- (resp. K_{i}-) linear endomorphisms of V_{i} (resp. of X_{i}) as matrices over S_{i} (resp. over K_{i}), i.e. as elements of S_{i}^{x_{i}×x_{i}} (resp. of K_{i}^{x_{i}×x_{i}}). In particular, we identify End_{S_{i}}V_{i} = S_{i}^{x_{i}×x_{i}}.

Remark 2.7 The required choice of V_{i} ⊆ X_{i} can be achieved as follows. Let S_{i} be a maximal order in K_{i} (1.4), [20, 7.18 (ii), 10.4]. The S-subalgebra Ξ of End_{K}X_{i} generated by S_{i} and the image of T ≀ G therein, is finitely generated as a module over S. Let V_{i} be a finitely generated Ξ-submodule of X_{i} such that KV_{i} = X_{i}. Since S_{i} ⊆ End_{T ≀G}V_{i} ⊆ K_{i}, maximality of S_{i} yields S_{i} = End_{T ≀G}V_{i}.

Notation 2.8 Given a finitely generated torsion S-module M, we denote by l(M) the length of M in the sense of Jordan-Hölder. Given an inclusion of S-modules N ⊆ M such that M/N is a finitely generated torsion S-module, the colength of N in M is defined to be the length l(M/N), i.e. the multiplicity of S̄ in M/N.

We write shorthand

\begin{aligned}A & := & L ≀ G \\ Λ & := & T ≀ G . \end{aligned}

The K-bilinear form on A defined in (1.6) restricts to an S-bilinear form on Λ. For an S-submodule U ⊆ A, we let

U^{♯} := {y ∈ A | (y, U) ⊆ S} ⊆ A

be its dual S-submodule.

Lemma 2.9 For a finitely generated S-submodule U of A such that KU = A, the module U^{♯} is also a finitely generated S-submodule of A such that KU^{♯} = A. In this case, we obtain

U = U^{♯♯} .

Therefore U \overline{}U^{♯} is an antiinvolution of the lattice U^{f} of finitely generated S-submodules of A such that KU = A. In particular, given U, V ∈ U^{f}, U ⊆ V, we have l(V/U) = l(U^{♯}/V^{♯}).

We choose two S-linear bases (u_{i})_{i∈[1,gh]} and (u′_{i})_{i∈[1,gh]} of U such that (u_{i}, u′_{j}) = ∂_{i,j}z_{i} for some z_{i} ∈ K^{∗}, i, j ∈ [1, gh]. We obtain S-linear bases (z_{i}^{−1}u_{i})_{i∈[1,gh]} and (z_{i}^{−1}u′_{i})_{i∈[1,gh]} of U^{♯}. With respect to these bases, we have (z_{i}^{−1}u_{i}, z_{j}^{−1}u′_{j}) = ∂_{i,j}z_{i}^{−1}, i, j ∈ [1, gh]. Hence by the same argument, applied to this situation, we obtain U^{♯♯} = U.

Notation 2.10 Suppose given i ∈ [1, k]. Let

S_{i}^{+} := {y ∈ K_{i} | tr_{K}(yS_{i}) ⊆ S} .

The colength of S_{i} in S_{i}^{+} is denoted by

δ_{S_{i}/S} := l(S_{i}^{+}/S_{i}) .

Furthermore, let

\begin{aligned}T^{+} & := & \{y ∈ L | Tr_{L/K}(yT) ⊆ S\} \\ δ_{T/S} & := & l(T^{+}/T) \\ Λ^{+} & := & \{∑_{σ∈G}σy_{σ} ∈ A | y_{σ} ∈ T^{+}\} . \end{aligned}

Cf. [16, III.2.1]. Remark 2.11 We have

l(Λ^{+}/Λ) = gδ_{T/S} .

Lemma 2.12 We have

Λ^{♯} = Λ^{+} .

Given ∑_{σ∈G}σy_{σ} ∈ A, y_{σ} ∈ L, we obtain

(∑_{σ∈G}σy_{σ}, ρ^{−1}z) = Tr_{L/K}(y_{ρ}^{ρ^{−1}}z) ,

which is in S for all ρ ∈ G and for all z ∈ T if and only if ∑_{σ∈G} σy_{σ} is in Λ^{♯}, but also if and only if it is in Λ^{+}. Notation 2.13 Let

\begin{aligned}Γ & := & ( ∏_{i∈[1,k]}S_{i}^{x_{i}×x_{i}} ) ω^{−1} & ⊆ & A \\ Γ^{+} & := & ( ∏_{i∈[1,k]} \frac{h}{x_{i}d_{i}} (S_{i}^{+})^{x_{i}×x_{i}} ) ω^{−1} & ⊆ & A , \end{aligned}

where (S_{i}^{+})^{x_{i}×x_{i}} is the S-submodule of K_{i}^{x_{i}×x_{i}} consisting of matrices entrywise in S_{i}^{+}. Cf. (1.14, 1.17). Lemma 2.14 We have

Γ^{♯} = ( ∏_{i∈[1,k]} \frac{g}{x_{i}d_{i}} (S_{i}^{+})^{x_{i}×x_{i}} ) ω^{−1} .

Hence

\begin{aligned}l(Γ^{+}/Γ) & = & ∑_{i∈[1,k]}x_{i}^{2} (δ_{S_{i}/S} + v(x_{i}d_{i}/h)r_{i}) \\ l(Γ^{+}/Γ^{♯}) & = & v(n)gh . \end{aligned}

We use (1.16) to apply the bilinear form on the right hand side of the Wedderburn isomorphism to a tuple of matrices with only one nonzero matrix which in turn has only one nonzero entry, and an arbitrary tuple.

Theorem 2.15 The colength of the Wedderburn embedding

T ≀ G [figure: ω] ∏_{i∈[1,k]}End_{S_{i}}V_{i}

is given by

\frac{1}{2} (g(δ_{T/S} + v(n)h) − ∑_{i∈[1,k]}x_{i}^{2}(δ_{S_{i}/S} + v(x_{i}d_{i}/h)r_{i})) .

Concerning notation, cf. (1.2, 1.7, 2.1, 2.6, 2.8, 2.13).

Using (2.12), the diagram

[figure: Γ^{♯}; Λ^{♯}; =; Λ^{+}; Λ; Γ; Γ^{+}]

of S-submodules of A shows that

l(Λ^{♯}/Γ^{♯}) − l(Λ^{+}/Λ) + l(Γ/Λ) + l(Γ^{+}/Γ) − l(Γ^{+}/Γ^{♯}) = 0 .

The result follows by (2.9, 2.11, 2.14).

Question 2.16 Given i ∈ [1, k], it would be desirable to know the colength of the embedding of the quasiblock T ≀ Gε_{i} into End_{S_{i}}V_{i} via ω_{i}.

Corollary 2.17 If G acts faithfully on L, the colength of the Wedderburn embedding T ≀ G [figure: ω] End_{S}T is given by

\frac{1}{2} g δ_{T/S} .

Corollary 2.18 (cf. [14, 1.1.5]) If G acts trivially on L, the colength of the Wedderburn embedding SG [figure: ω] ∏_{i∈[1,k]}End_{S_{i}}V_{i} is given by

\frac{1}{2} (gv(g) − ∑_{i∈[1,k]}x_{i}^{2}(δ_{S_{i}/S} + v(x_{i}d_{i})r_{i})) .

2.3 Some estimates

Notation 2.19 Suppose the basis (y_{i})_{i∈[1,h]} chosen in (1.2) to be an S-linear basis of T. Let

Sπ^{t} := {z ∈ S | zy_{i}^{∗} ∈ T for all i, j ∈ [1, h]} .

So t is the maximal elementary divisor exponent of the Gram matrix of the trace bilinear form on T induced by Tr_{L/K}. In particular, s ⩽ t

Lemma 2.20 The cokernel of the Wedderburn embedding

T ≀ G [figure: ω] ∏_{i∈[1,k]} End_{S_{i}}V_{i}

is annihilated by nπ^{t}.

This follows by Fourier inversion (1.18) and by (1.14).

Lemma 2.21 Given i ∈ [1, k], we have v(x_{i}) + v(d_{i}) ⩽ v(g) + t. Cf. (1.17).

The multiple

\frac{π^{t}g}{x_{i}d_{i}} · ε_{i} = π^{t} ∑_{σ ∈ G, l ∈ [1,h]}σy_{l}^{∗} · χ_{i}(y_{l}σ^{−1}) ∈ T ≀ G

of the central primitive idempotent ε_{i} is mapped under ω_{i} to (π^{t}g)/(x_{i}d_{i}) times the identity on V_{i} (1.22).

2.4 The central colength

Notation 2.22 Given ν ∈ N, we let T_{ν} := Fix_{C_{G}(ν)}T = T ∩ L_{ν}. Note that T_{ν} is the integral closure of S in L_{ν}. Let T_{ν}^{+} := {y ∈ L_{ν} | Tr_{L_{ν}/K}(yT_{ν}) ⊆ S}, let δ_{T_{ν}/S} := l(T_{ν}^{+}/T_{ν}). Let Z(S_{i})^{+} := {z ∈ Z(K_{i}) | Tr_{Z(K_{i})/K}(zZ(S_{i})) ⊆ S}, let δ_{Z(S_{i})/S} := l(Z(S_{i})^{+}/Z(S_{i})).

Lemma 2.23 With respect to the bilinear form (−, =)^{Z} on Z(A) (§1.6), we obtain

\begin{aligned}Z(Λ)^{♯} & := & \{ζ ∈ Z(A) | (ζ, Z(Λ))^{Z} ⊆ S\} \\ & = & \{ ∑_{σ∈Cl_{N}^{G}} ∑_{ρ∈C_{G}(σ)\G}(σy_{σ})^{ρ} | y_{σ} ∈ [N : C_{N}(σ)]^{−1}T_{σ}^{+}\} . \end{aligned}

In particular, the S-linear colength of Z(Λ) in Z(Λ)^{♮} is given by

∑_{ν∈Cl_{N}^{G}} δ_{T_{ν}/S} + v([N : C_{N}(ν)])h_{ν} .

Suppose given ζ = ∑_{σ∈Cl_{N}^{G}} ∑_{ρ∈C_{G}(σ)\G}(σy_{σ})^{ρ} ∈ Z(A), where y_{σ} ∈ L_{σ} (1.28), and suppose given ∑_{ρ′∈C_{G}(ν)\G}(ν^{−1}z)^{ρ′} ∈ Z(Λ), where ν ∈ Cl_{N}^{G} and z ∈ T_{ν}. We obtain

( ∑_{σ∈Cl_{N}^{G}} ∑_{ρ∈C_{G}(σ)\G}(σy_{σ})^{ρ} , ∑_{ρ′∈C_{G}(ν)\G}(ν^{−1}z)^{ρ′} )^{Z} = [N : C_{N}(ν)] · Tr_{L_{ν}/K}(y_{ν}z) .

Lemma 2.24 With respect to the bilinear form (−, =)^{Z} on Z(A) (cf. §1.6), we obtain

\begin{aligned}Z(Γ)^{♯} & := & \{ζ ∈ Z(A) | (ζ, Z(Γ))^{Z} ⊆ S\} \\ & = & \{((z_{i})_{i∈[1,k]})(ω^{Z})^{−1} | z_{i} ∈ \frac{gh}{x_{i}^{2}d_{i}^{2}} Z(S_{i})^{+}\} . \end{aligned}

This follows by (1.40). Proposition 2.25 The colength of the central Wedderburn embedding

Z(T ≀ G) [figure: ω^{Z}] ∏_{i∈[1,k]}Z(S_{i})

is given by

\frac{1}{2} (( ∑_{ν∈Cl_{N}^{G}} δ_{T_{ν}/S} + v([N : C_{N}(ν)])h_{ν}) − ( ∑_{i∈[1,k]} δ_{Z(S_{i})/S} + v(\frac{x_{i}^{2}d_{i}^{2}}{gh} ) c_{i})) .

This follows by (2.23, 2.24) by the argument of (2.15). Corollary 2.26 (cf. [7, 4.1]) If G acts trivially on L, the colength of the central Wedderburn embedding Z(T ≀ G) [figure: ω^{Z}] ∏_{i∈[1,k]}Z(S_{i}) is given by

\frac{1}{2} (( ∑_{σ∈Cl_{G}^{G}} v([G : C_{G}(σ)])) − ( ∑_{i∈[1,k]} δ_{Z(S_{i})/S} + v(\frac{x_{i}^{2}d_{i}^{2}}{g} ) c_{i})) .

3 Modular

3.1 Brauer-Nesbitt

Corollary 3.1 (to 1.19) Let i ∈ [1, k]. Suppose that K = K_{i} and that v(x_{i}) = v(g) + t (cf. 2.19, 2.21). Then the reduction V_{i}/πV_{i} is a simple T̄ ≀ G-module.

Assume V_{i}/πV_{i} to have a nontrivial submodule. Choose an S-linear basis of V_{i} such that v ((σy_{l})ω_{i; x_{i},1}) > 0 for all σ ∈ G, l ∈ [1, h]. Then

\frac{g}{x_{i}} =^{(1.19)} ∑_{l∈[1,h], σ∈G}(σy_{l})ω_{i; x_{i},1} · (y_{l}^{∗}σ^{−1})ω_{i; 1,x_{i}} .

Now the left hand side has valuation −t, whereas the right hand side has valuation > −t. This contradicts our assumption, and therefore V_{i}/πV_{i} is simple.

Remark 3.2 The S̄-algebra T̄ ≀ G is semisimple if and only if e = 1, T/𝔮_{1} is separable over S̄ and p does not divide n. We have e = 1 if and only if t = 0.

For the necessity of these three conditions, we consider the split epimorphisms T̄ \overline{}T̄_{0}, which implies that e = 1, and T̄ ≀ G \overline{}T̄, which then implies that T/𝔮_{1} is separable over S̄ and that p does not divide n (cf. 2.4).

First of all, if e = 1 and T/𝔮_{1} is separable over S̄, then Tr_{T̄/S̄}(T̄) = S̄ (cf. 2.4). Sufficiency now follows as in (2.2), using an element ū_{0} ∈ T̄ such that Tr_{T̄/S̄}(ū_{0}) = 1.

We have t = 0 if and only if T^{+} = T, i.e. if and only if the discriminant of T/S is invertible in S. By [16, III.2.12], this holds if and only if e = 1.

Remark 3.3 Let i ∈ [1, k]. If T̄ ≀ G is semisimple and if K = K_{i}, then the reduction V_{i}/πV_{i} is a simple T̄ ≀ G-module.

By (3.1), we have to show that v(x_{i}) = v(h) + v(n) + t. Now t = 0 and v(n) = 0 by (3.2), v(x_{i}) ⩾ v(h) by (1.14) and v(x_{i}) ⩽ v(h) by (2.21).

Question 3.4 Suppose T̄_{0} ≀ G to be semisimple. For instance, this is the case if T̄ ≀ G is semisimple (3.2). Then Jac(T̄ ≀ G) = Jac(T)(T̄ ≀ G), being a nilpotent ideal with semisimple quotient. Are the isoclasses of simple T̄ ≀ G-modules represented by V_{i}/Jac(T)V_{i} for i ∈ [1, k] ? Cf. [9, §28, ex. 5].

3.2 Counting simple modules

The arguments in this section are an adaption of Brauer’s proof of the classical case [3, 3B]. For sake of self-containedness, and without claiming originality, we repeat some lemmas from [13].

Notation 3.5 Recall from (2.1) that we write S̄ = S/πS.

Assume S̄ to be finite, and let S̄ =: p^{a} =: q.

On the other hand, in this section we may admit n · 1_{K} = 0.

Denote T̄ := T/πT and 𝔮̄_{i} := 𝔮_{i}/πT. Note that T̄ ≃ ∏_{i∈[1,d]} T̄_{𝔮_{i}} ≃ ∏_{i∈[1,d]} T/𝔮_{i}^{e}.

Let Λ̄ := T̄ ≀ G, so A ⊇ Λ \overline{}Λ̄. Let [Λ̄, Λ̄] be the S̄-subspace of Λ̄ generated by the elements ab − ba, for a, b ∈ Λ̄. Let

[Λ̄, Λ̄]^{p^{−ı}} := {ξ ∈ Λ̄ | ξ^{p^{m}} ∈ [Λ̄, Λ̄] for some m ⩾ 0} .

Lemma 3.6 Given ξ, η ∈ Λ̄ and m ⩾ 0, we have (ξ + η)^{p^{m}} ≡_{[Λ̄,Λ̄]} ξ^{p^{m}} + η^{p^{m}}.

Note that C_{p^{m}} acts on {ξ, η}^{[1,p^{m}]}. Let {ξ, η}^{[1,p^{m}]}/C_{p^{m}} denote a set of representatives for the orbits. If ξ ≠ η then

(ξ + η)^{p^{m}} =  ∑_{f∈{ξ,η}^{[1,p^{m}]}} ∏_{j∈[1,p^{m}]} f(j)
≡_{[Λ̄,Λ̄]} ∑_{f∈{ξ,η}^{[1,p^{m}]}/C_{p^{m}}} |fC_{p^{m}}| ∏_{j∈[1,p^{m}]} f(j)
=  ξ^{p^{m}} + η^{p^{m}} .

Lemma 3.7 We have [Λ̄, Λ̄]^{p} ⊆ [Λ̄, Λ̄].

By (3.6), it suffices to show that (ξη − ηξ)^{p} ∈ [Λ̄, Λ̄] for ξ, η ∈ Λ̄. In fact,

(ξη − ηξ)^{p} ≡^{(3.6)}_{[Λ̄,Λ̄]} (ξη)^{p} − (ηξ)^{p}
=  ξ((ηξ)^{p−1}η) − ((ηξ)^{p−1}η)ξ
≡_{[Λ̄,Λ̄]} 0 .

Lemma 3.8 The set [Λ̄, Λ̄]^{p^{−ı}} is an S̄-linear subspace of Λ̄.

Suppose given ξ, η ∈ [Λ̄, Λ̄]^{p^{−ı}}. Let m ⩾ 0 be such that ξ^{p^{m}}, η^{p^{m}} ∈ [Λ̄, Λ̄] (3.7). Given α, β ∈ S̄, we obtain

(αξ + βη)^{p^{m}} ≡^{(3.6)}_{[Λ̄,Λ̄]} α^{p^{m}}ξ^{p^{m}} + β^{p^{m}}η^{p^{m}} ≡_{[Λ̄,Λ̄]} 0 .

Lemma 3.9 There exists M ⩾ 0 such that for any m ⩾ M

[Λ̄, Λ̄]^{p^{−ı}} = {ξ ∈ Λ̄ | ξ^{p^{m}} ∈ [Λ̄, Λ̄]} =: [Λ̄, Λ̄]^{p^{−m}} .

We choose finitely many S̄-linear generators of [Λ̄, Λ̄]^{p^{−ı}} (3.8) and choose M ⩾ 0 such that (−)^{p^{M}} maps them into [Λ̄, Λ̄], thus likewise for (−)^{p^{m}} (3.7). By the argument of (3.8), extended to arbitrary linear combinations, we obtain ([Λ̄, Λ̄]^{p^{−ı}})^{p^{m}} ⊆ [Λ̄, Λ̄].

Corollary 3.10 Let μ ⩾ 0 be such that aμ ⩾ M and such that aμ ⩾ v_{p}(o(σ)) for all σ ∈ G. We have an injective S̄-linear map

Λ̄/[Λ̄, Λ̄]^{p^{−ı}} = Λ̄/[Λ̄, Λ̄]^{q^{−μ}}  [figure: ι]  Λ̄/[Λ̄, Λ̄]
ξ \overline{}ξ^{q^{μ}} .

The S̄-linearity follows by (3.6) and by the choice of the exponent to be a power of q.

Lemma 3.11 We have

dimS̄ Z(Λ̄/Jac(Λ̄)) = dimS̄(Λ̄/[Λ̄, Λ̄]^{p^{−ı}}) .

We write X_{0} for the image of a subset X ⊆ Λ̄ modulo Jac(Λ̄). In particular, Λ̄_{0} = Λ̄/Jac(Λ̄). We claim that

[Λ̄, Λ̄]_{0} = ([Λ̄, Λ̄]_{0})^{p^{−∞}} = ([Λ̄, Λ̄]^{p^{−ı}})_{0} ,

where ([Λ̄, Λ̄]_{0})^{p^{−∞}} := {ξ_{0} ∈ Λ̄_{0} | ξ_{0}^{p^{m}} ∈ [Λ̄, Λ̄]_{0} for some m ⩾ 0}. Since each element of Jac(Λ̄) is nilpotent, we have Jac(Λ̄) ⊆ [Λ̄, Λ̄]^{p^{−ı}}, and so this claim implies dimS̄(Λ̄/[Λ̄, Λ̄]^{p^{−ı}}) = dimS̄(Λ̄_{0}/[Λ̄, Λ̄]_{0}).

Concerning the second equality, we need to show ⊆. Suppose given ξ ∈ Λ̄ such that ξ_{0}^{p^{m}} ∈ [Λ̄, Λ̄]_{0}, i.e. such that ξ^{p^{m}} = c + r, where c ∈ [Λ̄, Λ̄] and r ∈ Jac(Λ̄). Choose m′ ⩾ 0 with r^{p^{m′}} = 0. Then

ξ^{p^{m+m′}} =  (c + r)^{p^{m′}}
≡^{(3.6)}_{[Λ̄,Λ̄]} c^{p^{m′}} + r^{p^{m′}}
=  c^{p^{m′}}
≡^{(3.7)}_{[Λ̄,Λ̄]} 0 .

Concerning the first equality, we need to show ⊇. We may consider the blocks of Λ̄_{0} separately and assume that Λ̄_{0} = S̃^{l×l} for some finite field extension S̃/S̄. By assumption, S̃ is commutative. The S̃-linear trace S̃^{l×l} \overline{}S̃ factors over S̃^{l×l}/[S̃^{l×l}, S̃^{l×l}] ^{\overline{}^{∼}} S̃. Now the first equality follows by TrS̃(ξ_{0}^{p}) = TrS̃(ξ_{0})^{p}, as to be seen using the Jordan normal form.

Finally, dimS̄(Λ̄_{0}/[Λ̄, Λ̄]_{0}) = dimS̄ Z(Λ̄_{0}).

Notation 3.12 Let T̄_{0} := T̄/Jac(T̄) = T/Jac(T) ≃ ∏_{i∈[1,d]} T̄_{0,𝔮_{i}} ≃ ∏_{i∈[1,d]} T/𝔮_{i}. Deviating from former notation, we let Cl(G) be a system of representatives of conjugacy classes of G, and let Cl(G, p′) := {σ ∈ Cl(G) | o(σ) ≢_{p} 0} ⊆ Cl(G) be the subset of p′-representatives.

Given σ ∈ Cl(G), we define the subspace

V_{σ} := S̄⟨(y^{σ} − y)z, y^{ρ} − y | y, z ∈ T̄_{0}, ρ ∈ C_{G}(σ)⟩ ⊆ T̄_{0} .

Lemma 3.13 There is an S̄-linear map

[figure: Λ̄/[Λ̄, Λ̄]; φ; ⨁_{κ∈Cl(G)}T̄_{0}/V_{κ}; σ^{ρ}y; (∂_{σ,κ}y^{ρ^{−1}})_{κ} ,]

where y ∈ T̄, σ ∈ Cl(G) and ρ ∈ G.

We need to show the independence of the choice of ρ. But if τ ∈ C_{G}(σ), then y^{ρ^{−1}} − y^{ρ^{−1}τ^{−1}} ∈ V_{σ}.

We need to show that given τ, τ̃ ∈ G, y, ỹ ∈ T̄, the element τyτ̃ỹ − τ̃ỹτy is mapped to zero. Writing ττ̃ = σ^{ρ} with σ ∈ Cl(G) and ρ ∈ G, we get

(τyτ̃ỹ − τ̃ỹτy)φ = (σ^{ρ}yτ̃ỹ − σ^{ρτ}ỹ^{τ}y)φ
= (∂_{σ,κ}y^{τ̃ρ^{−1}}ỹ^{ρ^{−1}} − ỹ^{ρ^{−1}}y^{τ^{−1}ρ^{−1}})_{κ}
= (∂_{σ,κ}(y^{τ̃ρ^{−1}} − y^{τ̃ρ^{−1}σ^{−1}})ỹ^{ρ^{−1}})_{κ}
∈ V_{σ} .

Lemma 3.14 There is a commutative diagram of S̄-linear maps

[figure: Λ̄/[Λ̄, Λ̄]^{p^{−ı}}; φ_{0}; ⨁_{κ∈Cl(G,p′)}T̄_{0}/V_{κ}; ι; Λ̄/[Λ̄, Λ̄]; φ; ⨁_{κ∈Cl(G)}T̄_{0}/V_{κ} ,]

thus defining φ_{0}.

Given τ ∈ G and y ∈ T̄, we have (τy)^{q^{μ}} = τ^{q^{μ}}y′ for some y′ ∈ T̄. Now τ^{q^{μ}} is a p′-element by choice of μ (3.10).

Lemma 3.15 There is an S̄-linear map

⨁_{κ∈Cl(G,p′)}T̄_{0}/V_{κ}  ^{\overline{}^{ψ}}  Λ̄/[Λ̄, Λ̄]^{p^{−ı}}
(y_{κ})_{κ}  \overline{}∑_{κ∈Cl(G,p′)} κy_{κ} .

Given y ∈ Jac(T̄) and σ ∈ Cl(G, p′), we have to show that the image of (∂_{σ,κ}y)_{κ} vanishes. We have (σy)^{e·o(σ)} = (∏_{i∈[0,o(σ)−1]} y^{σ^{i}})^{e} = 0, whence σy ∈ [Λ̄, Λ̄]^{p^{−ı}}.

Now, given ρ ∈ C_{G}(σ), y, z ∈ T̄_{0}, we have to show that the images of (∂_{σ,κ}(z^{ρ} − z))_{κ} and of (∂_{σ,κ}(y^{σ} − y)z)_{κ} vanish. Note that

σy^{ρ^{−1}σ}z = ρyσz^{ρ}ρ^{−1} ≡_{[Λ̄,Λ̄]} σyz^{ρ} .

Putting ρ = 1, we see σy^{σ}z ≡_{[Λ̄,Λ̄]} σyz. Putting y = 1, we see σz ≡_{[Λ̄,Λ̄]} σz^{ρ}.

Lemma 3.16 Let f ⩾ 1. Given α ∈ Gal(F_{q^{f}} /F_{q}) ≃ C_{f} of order o(α) prime to p, the map

[figure: F_{q^{f}}; ^{N^{α,q^{μ}}}; F_{q^{f}}; y; ∏_{l∈[0,q^{μ}−1]} y^{α^{l}}]

is surjective.

Let α = F^{s}, where F(y) = y^{q}. Thus N_{α,q^{μ}}(y) = y^{\frac{q^{sq^{μ}} −1}{q^{s}−1}} , and so we have to show that \frac{q^{sq^{μ}}−1}{q^{s}−1} is prime to q^{f} − 1. Since o(α) is assumed to be prime to p, we have f[p] = s[p]. Writing f′ = f/f[p], s′ = s/s[p] and r = q^{f[p]} = q^{s[p]}, we have to show that

gcd (r^{f′} − 1, \frac{r^{s′q^{μ}} − 1}{r^{s′} − 1}) = 1 .

We remark that given u, v, w ∈ Z_{>0} with w | v and gcd(w, v/w) = 1, we have gcd(u, v/w) = gcd(u, v)/ gcd(u, w). Moreover, we remark that given u, v ∈ Z_{>0}, we have

gcd(r^{u} − 1, r^{v} − 1)  =  (r − 1) · gcd (\frac{r^{u}−1}{r−1}, \frac{r^{v}−1}{r−1})

Euclid’s algorithm

= (r − 1) · \frac{r^{gcd(u,v)}−1}{r−1}

=  r^{gcd(u,v)} − 1 .

Now, since gcd(r^{s′} − 1, \frac{r^{s′q^{μ}}−1}{r^{s′}−1}) = gcd(r^{s′} − 1, q^{μ}) = 1, the claim follows from

gcd(r^{f′} − 1, r^{s′q^{μ}} − 1) = r^{gcd(f′,s′q^{μ})} − 1
= r^{gcd(f′,s′)} − 1
= gcd(r^{f′} − 1, r^{s′} − 1) .

Lemma 3.17 The map ψ is surjective.

Suppose given τ ∈ G and y ∈ T̄. We claim that τy ∈ Λ̄/[Λ̄, Λ̄]^{p^{−ı}} is contained in the image of ψ. By additivity, we may assume that y corresponds to (∂_{j,i}y)_{i} under T̄ ^{\overline{}^{∼}} ∏_{i∈[1,d]} T/𝔮_{i}^{e} for some j ∈ [1, d]. Let e_{j} be the primitive idempotent corresponding to (∂_{j,i})_{i}, so y = ye_{j}.

Case e_{j}^{τ} ≠ e_{j}. Then e_{j}^{τ} = e_{j′} for some j′ ≠ j since τ preserves primitive idempotents. Therefore, τy = τye_{j} ≡_{[Λ̄,Λ̄]} e_{j}τy = τye_{j}^{τ} = 0.

Case e_{j}^{τ} = e_{j}. From y = ye_{j}, we infer y^{τ^{l}} = y^{τ^{l}}e_{j}^{τ^{l}} = y^{τ^{l}}e_{j} for l ∈ Z.

We write τ = τ_{s}τ_{u} = τ_{u}τ_{s} with o(τ_{u}) a power of p and o(τ_{s}) prime to p. It suffices to show that there exists a ỹ ∈ T̄ such that τy ≡_{[Λ̄,Λ̄]^{p^{−ı}}} τ_{s}ỹ. With the ansatz that ỹ should correspond to (∂_{j,i}ỹ)_{i} under T̄ ^{\overline{}^{∼}} ∏_{i∈[1,d]} T/𝔮_{i}^{e}, we obtain (τ_{s}ỹ)^{q^{μ}} = τ^{q^{μ}} ∏_{l∈[0,q^{μ}−1]} ỹ^{τ_{s}^{l}}, where ∏_{l∈[0,q^{μ}−1]} ỹ^{τ_{s}^{l}} corresponds to

(∂_{j,i} ∏_{l∈[0,q^{μ}−1]} ỹ^{α^{l}} )_{i∈[1,d]}

for some automorphism α of T/𝔮_{j}^{e} that fixes S̄, of order o(α) prime to p. Now (τy)^{q^{μ}} = τ^{q^{μ}}y′, where y′ corresponds to (∂_{j,i}y′)_{i}. By (3.16), applied to T/𝔮_{j} ≃ F_{q^{f}}, the map

T/𝔮_{j}^{\overline{}^{N^{α,q^{μ}}}}T/𝔮_{j} is surjective, and so we may find ỹ ∈ T̄ as in the ansatz such that (τy)^{q^{μ}} − (τ_{s}ỹ)^{q^{μ}} is nilpotent; choose μ′ ⩾ 0 such that the exponent q^{μ′} annihilates it. Now since Λ̄/[Λ̄, Λ̄]^{p^{−ı}} \overline{}Λ̄/[Λ̄, Λ̄], ξ \overline{}ξ^{q^{μ+μ′}} is likewise injective (cf. 3.9, 3.10), we may conclude from

(τy)^{q^{μ+μ′}} − (τ_{s}ỹ)^{q^{μ+μ′}} ≡^{(3.6)}_{[Λ̄,Λ̄]} ((τy)^{q^{μ}} − (τ_{s}ỹ)^{q^{μ}})^{q^{μ′}} = 0

that τy ≡_{[Λ̄,Λ̄]^{p^{−ı}}} τ_{s}ỹ.

Lemma 3.18 The endomorphism ψφ_{0} is bijective.

Suppose given an element (∂_{σ,κ}y) in ⨁_{κ∈Cl(G,p′)} T̄_{0}/V_{κ}. It is mapped to σy under ψ, and then to (σy)^{q^{μ}} via ι. Now let σ̃^{ρ} = σ^{q^{μ}}, where σ̃ ∈ Cl(G, p′), ρ ∈ G. Then (σy)^{q^{μ}} is mapped

via φ to the tuple that has entry (∏_{i∈[0,q^{μ}−1]} y^{σ^{i}})^{ρ^{−1}} ∈ T̄_{0}/Vσ̃ at σ̃, and 0 elsewhere. The corresponding subtuple in ⨁_{κ∈Cl(G,p′)} T̄_{0}/V_{κ} is the image of (∂_{σ,κ}y) under ψφ_{0}.

Since σ \overline{}σ^{q^{μ}} induces a permutation of Cl(G, p′) and since

[figure: T̄_{0}/Vσ̃; T̄_{0}/V_{}_{σ̃^{ρ}} = T̄_{0}/V_{σ^{q^{μ}}}; y; y^{ρ}]

is bijective, it remains to be shown that

[figure: T̄_{0}/V_{σ}; T̄_{0}/V_{σ^{q^{μ}}}; y; ∏_{i∈[0,q^{μ}−1]} y^{σ^{i}}]

is bijective. Since σ is a p′-element of G, the subspace V_{σ^{q^{μ}}} contains the ideal generated by the elements of the form z^{σ} − z, where z ∈ T̄_{0}. Thus ∏_{i∈[0,q^{μ}−1]} y^{σ^{i}} ≡_{V_{σ^{q^{μ}}}} y^{q^{μ}}. Moreover, note that C_{G}(σ^{q^{μ}}) = C_{G}(σ), and so V_{σ^{q^{μ}}} = V_{σ}. Since z \overline{}z^{q^{μ}} is an automorphism on T̄_{0}, the endomorphism it induces on T̄_{0}/V_{σ} is an automorphism, too.

Theorem 3.19 We have

dimS̄ Z(Λ/Jac(Λ)) = dimS̄ Z(Λ̄/Jac(Λ̄)) = ∑_{σ∈Cl(G,p′)} dimS̄(T̄_{0}/V_{σ}) =: z(Λ) .

Concerning notation, cf. (3.5, 3.12).

Consider the following diagram of S̄-linear maps (cf. 3.13, 3.14, 3.15).

[figure: ⨁_{κ∈Cl(G,p′)}T̄_{0}/V_{κ}; ψ; Λ̄/[Λ̄, Λ̄]^{p^{−ı}}; φ_{0}; ⨁_{κ∈Cl(G,p′)}T̄_{0}/V_{κ}]

Since ψ is surjective (3.17) and ψφ_{0} is bijective (3.18), ψ is an S̄-linear isomorphism. Thus

∑_{κ∈Cl(G,p′)} dimS̄(T̄_{0}/V_{κ}) = dimS̄(Λ̄/[Λ̄, Λ̄]^{p^{−ı}}) =^{(3.11)} dimS̄ Z(Λ̄/Jac(Λ̄)) .

Corollary 3.20 The number of isoclasses of simple Λ̄-modules is less than or equal to z(Λ).

Corollary 3.21 Suppose Λ̄/Jac(Λ̄) to be split semisimple over S̄, i.e. suppose all simple Λ̄-modules to be absolutely simple. Then the number of isoclasses of simple Λ̄-modules is given by z(Λ).

We may reformulate (3.21) to an assertion on T ≀ G.

Corollary 3.22 Suppose all simple T ≀ G-modules to be absolutely simple. Then the number of isoclasses of indecomposable projective T ≀ G-modules is given by z(T ≀ G).

Remark 3.23 (particular cases) We drop the finiteness assumption on S̄.

(i) The T̄ ≀ G-module T̄_{0} is simple, with endomorphism ring End_{T̄≀G}T̄_{0} ≃ Fix_{C_{H}(𝔮_{1})}T/𝔮_{1}. If T/𝔮_{1} is separable over S̄, then T̄_{0} is absolutely simple, i.e. End_{T̄≀G}T̄_{0} = S̄. Cf. (3.26).

(ii) If G is a p-group, then T̄_{0} is the only simple Λ̄-module, up to isomorphism. In particular, if S̄ is finite, (3.19) yields dimS̄ T̄_{0}/V_{1} = dimS̄ Fix_{G}T̄_{0}.

(iii) If S̄ is finite, and if G acts trivially on T̄_{0}, then z(Λ) = |Cl(G, p′)| · dimS̄ T̄_{0} (3.19).

(iv) Suppose d = 1. Write N_{0} for the kernel of the operation of G on T̄_{0}. If σ ∈ G is not contained in N_{0}, then T̄_{0}/V_{σ} = 0. Cf. (1.29).

(v) Suppose G to be a p′-group. Then T̄_{0} ≀ G is semisimple. In particular, if in addition T/S is unramified, then Λ̄ = T̄ ≀ G is semisimple.

(vi) The isoclasses of simple modules of Λ̄ = T̄ ≀G correspond bijectively to the isoclasses of simple modules of T̄_{0} ≀ G via composition of the operation with T̄ ≀ G \overline{}T̄_{0} ≀ G. Moreover, we have an isomorphism (T̄ ≀ G)/Jac(T̄ ≀ G) ^{\overline{}^{∼}} (T̄_{0} ≀ G)/Jac(T̄_{0} ≀ G).

(vii) If e = 1 and if T/𝔮_{1} is separable over S̄, then V_{1} has codimension 1 in T̄_{0}.

(viii) The S̄-linear dimension of a T̄ ≀ G-module is divisible by dimS̄ T̄_{0} = h/e.

Ad (i). Firstly, End_{T̄≀G}T̄_{0} ≃ Fix_{G}T̄_{0}. Now the projection of Fix_{G}T̄_{0} to T/𝔮_{1} is injective and surjects onto Fix_{C_{H}(𝔮_{1})}T/𝔮_{1}. If the extension T/𝔮_{1} over S̄ is separable, we obtain a surjection of C_{H}(𝔮_{1}) onto Gal((T/𝔮_{1})/S̄) [16, I.9.4].

Ad (ii). Any Λ̄-module M contains a fixed point different from 0. In fact, passing to a cyclic module, we may assume it to be finite, of cardinality divisible by p. Since p divides the length of a nontrivial G-orbit, the claim follows. Cf. [22, prop. 26].

Let m ∈ M be a nonzero fixed point. Then m · T̄ is a submodule of M. Hence, if M is simple, M is isomorphic to a quotient of T̄. But the only simple quotient of T̄ is T̄_{0}.

Ad (iv). By assumption, there exists a y ∈ T̄_{0} such that y^{σ} − y ≠ 0. The subspace V_{σ} of T̄_{0} contains the ideal generated by y^{σ} − y. Since now T̄_{0} is a field, this implies V_{σ} = T̄_{0}.

Ad (v). Given an epimorphism of Λ̄-modules, we may choose a T̄_{0}-linear coretraction, for T̄_{0} is semisimple. The sum over the G-conjugates divided by |G| yields a Λ̄-linear coretraction.

Ad (vi). The kernel of T̄ ≀ G \overline{}T̄_{0} ≀ G is a nilpotent ideal, thus contained in Jac(T̄ ≀ G).

Ad (vii). By tameness of T/S, (2.4) and [9, 32.1] show that T̄_{0}|_{S̄H} ≃ S̄H. Under such an isomorphism, V_{1} corresponds to the augmentation ideal.

Ad (viii). We are reduced to consider a simple T̄≀G-module X. Let e_{j} correspond to (∂_{j,i})_{i} under T̄_{0} ^{\overline{}^{∼}} ∏_{i∈[1,d]} T/𝔮_{i}. We may decompose X as a T̄_{0}-module into X = ⨁_{j∈[1,d]} Xe_{j}. Since G acts transitively on {e_{j} | j ∈ [1, d]}, we have for each j ∈ [1, d] an element σ ∈ G that induces Xe_{1} ^{\overline{}^{∼}} Xe_{j} by multiplication. Since Xe_{1} is a module over T/𝔮_{1}, d · dimS̄ T/𝔮_{1} = dimS̄_{0} T̄_{0} divides dimS̄ X.

3.3 Examples

When we count simple modules, we shall tacitly count them up to isomorphism. We use Maple, Magma and the Meat Axe version of M. Ringe [21].

Example 3.24 We continue (1.36).

p = 2. Let S = Z_{(2)} and T = Z_{(2)}[i]. Since T̄_{0} ≃ F_{2}, the simple T̄ ≀ G-modules are given by the two simple F_{2}S_{3}-modules (3.23 iii).

p = 3. Let S = Z_{(3)} and T = Z_{(3)}[i]. Now T̄ = T̄_{0} = F_{3}[i] ≃ F_{9}. We obtain V_{1} = F_{3}⟨i⟩ and V_{(1,2)} = F_{3}[i]. Therefore z(Z_{(3)}[i] ≀ S_{3}) = 1, and thus T̄_{0} is the unique (absolutely) simple T̄ ≀ G-module (3.23 i).

Example 3.25 Let K = Q. Let γ be a root of the irreducible polynomial

μ(X) = X^{6} − 3X^{5} + 7X^{4} − 9X^{3} + 7X^{2} − 3X + 1 ∈ Q[X] ,

and let L = Q(γ). Then Z[γ] is the ring of algebraic integers in the Galois extension L/K. We have Gal(L/K) ≃ S_{3}. Thus there is an operation of S_{4} on L, where

\begin{aligned}γ^{(1,2)} & = & 1 − γ \\ γ^{(1,2,3,4)} & = & 1/γ . \end{aligned}

p = 2. Let S = Z_{(2)} and T = Z_{(2)}[γ]. We have μ(X) ≡_{2} (X^{3} + X + 1)(X^{3} + X^{2} + 1), hence T̄ = T̄_{0} ≃ F_{8} × F_{8}. Let Fx = x^{2} for x ∈ F_{8}. The operation of S_{4} on T̄_{0} is given by

[figure: F_{8}; ×; F_{8}; F_{8}; ×; F_{8}; x; ×; y; (1,2); y; ×; x; x; ×; y; (1,2,3,4); Fy; ×; F^{2}x .]

Let α ∈ F_{8} with α^{3} + α + 1 = 0. Then V_{1} = F_{2}⟨1 × 1, α × 0, α^{2} × 0, 0 × α, 0 × α^{2}⟩ (cf. 3.23 vii), and V_{(1,2,3)} = T̄_{0}. Altogether, z(T ≀ G) = 1. Therefore, T̄_{0} = T̄ is the only (absolutely) simple T̄ ≀ G-module (3.19, 3.23 i).

p = 3. Let S = Z_{(3)} and T = Z_{(3)}[γ]. We have μ(X) ≡_{3} (X^{2} + 1)(X^{2} + X − 1)(X^{2} − X − 1), hence T̄ = T̄_{0} ≃ F_{9} × F_{9} × F_{9}. Let Fx = x^{3} for x ∈ F_{9}. The operation of S_{4} on T̄_{0} is given by

F_{9} × F_{9} × F_{9}  \overline{}F_{9} × F_{9} × F_{9}
x × y × z ^{\overline{}^{(1,2)}}  y × x × Fz
x × y × z ^{\overline{}^{(1,2,3,4)}} Fx × z × y .

Choose ι ∈ F_{9} with ι^{2} + 1 = 0. Then V_{1} = F_{3}⟨1 × −1 × 0, 0 × 1 × −1, ι × 0 × 0, 0 × ι × 0, 0 × 0 × ι⟩ (cf. 3.23 vii). Moreover, V_{(1,2)(3,4)} = F_{3}⟨ι × 0 × 0, 0 × 1 × −1, 0 × ι × −ι⟩. Finally, V_{(1,2)} = V_{(1,2,3,4)} = T̄_{0}. Altogether, z(T ≀ G) = 4. It turns out that there are three simple T̄ ≀G-modules, two of dimension 6 and endomorphism ring F_{3}, and one of dimension 12 and endomorphism ring F_{9}. In particular, the center of T̄ ≀ G/Jac(T̄ ≀ G) is isomorphic to F_{3} × F_{3} × F_{9}. It is of dimension 4 over F_{3}, as predicted by (3.19).

p = 31. Let S = Z_{(31)} and T = Z_{(31)}[γ]. We have μ(X) ≡_{31} (X −2)^{2}(X +15)^{2}(X +1)^{2}, hence T̄ ≃ F_{31}[ε] × F_{31}[ε] × F_{31}[ε], where ε^{2} = 0. In particular, T̄_{0} ≃ F_{31} × F_{31} × F_{31}. The operation of S_{4} on T̄_{0} is given by

[figure: F_{31}; ×; F_{31}; ×; F_{31}; F_{31}; ×; F_{31}; ×; F_{31}; x; ×; y; ×; z; (1,2); z; ×; y; ×; x; x; ×; y; ×; z; (1,2,3,4); y; ×; x; ×; z .]

We obtain V_{1} = F_{31}⟨1 × −1 × 0, 1 × 0 × −1⟩, V_{(1,2)} = F_{31} × 0 × F_{31}, V_{(1,2)(3,4)} = F_{31}⟨1 × 0 × −1⟩, V_{(1,2,3)} = T̄_{0} and V_{(1,2,3,4)} = F_{31} × F_{31} × 0. Altogether, z(T ≀ G) = 5. In fact, there are 5 absolutely simple T̄ ≀ G-modules, four of dimension 3 and one of dimension 6.

Example 3.26 Let G = H = C_{2} = ⟨c⟩, let L = Q(X), let X^{c} = −X, so that K = Q(X^{2}). Let 𝔭 := 2Z[X^{2}] ⊆ Z[X^{2}], let S = Z[X^{2}]_{𝔭}, hence T = Z[X]_{𝔭}. We obtain S̄ = F_{2}(X^{2}) and T̄ = T̄_{0} = F_{2}(X). In particular, c acts trivially on T̄_{0}, so End_{T̄≀G}T̄_{0} = F_{2}(X). Therefore, T̄_{0} is simple, but not absolutely simple over the S̄-algebra T̄ ≀ G.

References

[1] Benz, H., Zassenhaus, H., Über verschränkte Produktordnungen, J. Num. Th. 20, p. 282-298, 1985.

[2] Bonami, L., On the structure of skew group rings, Algebra-Berichte 48, Fischer, 1984.

[3] Brauer, R., Zur Darstellungstheorie der Gruppen endlicher Ordnung, Math. Z. 63, p. 406-444, 1956.

[4] Brauer, R.; Nesbitt, C., On the modular characters of groups, Ann. Math. (2) 42, p. 556-590, 1941. 556–590.

[5] le Bruyn, L., Van den Bergh, M., Van Oystaeyen, F., Graded Orders, Birkhäuser, 1988.

[6] Chalatsis, A., Theohari-Apostolidi, Th., Integral Representations of a Certain Twisted Group Ring, J. Alg. 124, p. 1-8, 1986.

[7] Cliff, G.H., Plesken, W., Weiss, A., Order-Theoretic Properties of the Center of a Block, Proc. Symp. Pure Math. 47, p. 413-420, 1987.

[8] Cliff, G.H., Weiss, A. R., Crossed product orders and wild ramification, in Orders and their applications, SLN 1142, p. 96-104, 1984.

[9] Curtis, C. W.; Reiner, I., Methods of representation theory I, Wiley, 1981.

[10] Curtis, C. W.; Reiner, I., Methods of representation theory II, Wiley, 1987.

[11] Fröhlich, A., Galois module structure of algebraic integers, Ergebnisse der Math. (3) 1, Springer, 1983.

[12] Karpilovsky, G., Symmetric and G-algebras, Math. and Its Appl. 60, Kluwer, 1990.

[13] Külshammer, B., Bemerkungen über die Gruppenalgebra als symmetrische Algebra, J. Alg. 72, p. 1-7, 1981.

[14] Künzer, M., Ties for the integral group ring of the symmetric group, thesis, http://www.mathematik.uni-bielefeld.de/∼kuenzer, Bielefeld, 1999.

[15] Nakayama, T.; Shoda, K., Über die Darstellung einer endlichen Gruppe durch halblineare Transformationen, Jap. J. Math. 12, 1935.

[16] Neukirch, J., Algebraische Zahlentheorie, Springer, 1992.

[17] Noether, E., Normalbasis bei Körpern ohne höhere Verzweigung, J. Reine Angew. Math. 167, p. 147-152, 1932 (cf. Zbl. Math. 3, p. 146).

[18] Noether, E., Zerfallende verschränkte Produkte und ihre Maximalordnungen, Actualités scientifiques et industrielles 148, p. 5-15, 1934.

[19] Plesken, W., Group Rings of Finite Groups Over p-adic Integers, SLN 1026, 1983.

[20] Reiner, I., Maximal Orders, Academic Press, London, 1975.

[21] Ringe, M., The C Meat Axe, GAP package, 1993.

[22] Serre, J.P., Linear representations of finite groups, Springer GTM 42, 1977.

[23] Speiser, A., Gruppendeterminante und Körperdiskriminante, Math. Ann. 77, p. 546-562, 1916.

[24] Thevenaz, J., G-Algebras and Modular Representation Theory, Oxford Math. Monographs, 1995.

[25] Weber, H., Integral group rings for a series of p-groups, abstract of forthcoming thesis, http://at.yorku.ca/cgi-bin/amca/cafe-47, 2000.

[26] Williamson, S., Crossed products and hereditary orders, Nagoya Math. J. 23, p. 103-120, 1963.

[27] Williamson, S., Crossed products and maximal orders, Nagoya Math. J. 25, p. 165-174, 1965.

[28] Williamson, S., Crossed products and ramification, Nagoya Math. J. 28, p. 85-111, 1966.

Matthias Künzer Universität Ulm

Abteilung Reine Mathematik

D-89069 Ulm

kuenzer@mathematik.uni-ulm.de