A Module-Theoretic Approach

to Clifford Theory for Blocks

S. J. Witherspoon^{∗} Department of Mathematics, University of Toronto Toronto, Ontario M5S 3G3, Canada sjw@math.toronto.edu

Abstract

This work concerns a generalization of Clifford theory to blocks of group-graded algebras. A module-theoretic approach is taken to prove a one-to-one correspondence between the blocks of a fully group-graded algebra lying over a given block of its identity component, and conjugacy classes of blocks of a twisted group algebra. In particular, this applies to blocks of a finite group lying over blocks of a normal subgroup.

Introduction

In [8], Dade developed the block theory of group-graded algebras, in particular extending the Clifford correspondence to blocks of group-graded algebras. The Clifford correspondence for representations is a one-to-one correspondence between the irreducible representations of a group G having a given constituent when restricted to a normal subgroup, and the irreducible projective representations (with respect to a particular 2-cocycle) of a certain subgroup of the corresponding quotient group [3]. Dade’s generalization is achieved by an analysis of centralizer algebras, and he has recently used it to make progress towards solving conjectures of Alperin and Dade [11, 12]. In this work we reinterpret these ideas, treating blocks as bimodules to provide a new proof of the Clifford correspondence for blocks of group-graded algebras. This approach clarifies the connection between the Clifford correspondence for blocks and that for representations or modules.

Specifically, let G be a finite group and A a fully G-graded algebra over an algebraically closed field k. That is, A = ∑_{g∈G} A_{g} is a direct sum of subspaces A_{g} with A_{g}A_{h} = A_{gh} for all g, h ∈ G. Here A_{g}A_{h} denotes the set of all finite sums of products xy, where x ∈ A_{g} and y ∈ A_{h}. The first example of a fully G-graded algebra is A = kΓ for a group Γ with normal subgroup N and G = Γ/N. In this case a subspace A_{g} is spanned by the elements of the coset g. As the identity component is A_{1} = kN, comparing blocks of A_{1} to blocks of A amounts to comparing blocks of N to blocks of Γ.

∗Research supported in part by NSERC grant # OGP0170281.

We consider A to be a right A^{op} ⊗ A-module, where A^{op} is the opposite algebra to A, via left and right multiplication by elements of A. A block of A is an indecomposable direct summand of the A^{op} ⊗ A-module A. A block B̃ of A lies over a block B of A_{1} if B is a direct summand of the module B̃ restricted to A_{1}^{op} ⊗ A_{1}. That this definition is equivalent to that of Dade [8, p. 218] follows from the ideas contained in the proof of Lemma 1.1. The group G acts by conjugation on the blocks of A_{1}, an element g ∈ G sending B to A_{g^{−1}}BA_{g}. In Section 3 we assume B is G-invariant and derive a oneto-one correspondence between the blocks of A lying over a given block B of A_{1} and G-conjugacy classes of blocks of a twisted group algebra for a subgroup of G, as stated in the theorem below. A reduction to this invariant case is given in Section 1.

A starting point for the Clifford correspondence for blocks is provided by a more general correspondence for indecomposable modules discussed at the start of Section 2. In this context, we let B ↑^{A_{1}^{op}⊗A} denote the induced A_{1}^{op} ⊗A-module B ⊗_{A_{1}^{op}⊗A_{1}} (A_{1}^{op} ⊗A) and E = End_{A_{1}^{op}⊗A}(B ↑^{A_{1}^{op}⊗A}). Then E is a G-graded algebra; that is E = ∑_{g∈G} E_{g} is a direct sum of subspaces E_{g} with E_{g}E_{h} ⊆ E_{gh} for all g, h ∈ G. In Section 2 we describe the Miyashita action of G on E which permutes the blocks. Let J_{G}(E) denote the graded Jacobson radical of E, the intersection of all maximal graded right ideals. We show in Section 3 that E/J_{G}(E) is a twisted group algebra for a subgroup of G.

Theorem (Clifford correspondence) Let B be a G-invariant block of A_{1}. The blocks of A lying over B correspond one-to-one with G-conjugacy classes of blocks of the twisted group algebra E/J_{G}(E).

In order to prove the theorem, in Section 2 we relate blocks of A lying over B to blocks of the endomorphism algebra E in the following way. Let Δ be the diagonal subalgebra ∑_{g∈G}(A_{g^{−1}})^{op} ⊗ A_{g} of A^{op} ⊗ A. The identity component A_{1} of A is naturally a Δmodule; we denote this Δ-module by (A_{1})_{Δ}. The A^{op} ⊗ A-module (A_{1})_{Δ} ↑^{A^{op}⊗A} induced from Δ is isomorphic to A. If the block B of A_{1} is G-invariant, then B is naturally a Δ-module, and so we obtain an ideal direct summand B_{Δ} ↑^{A^{op}⊗A}≅ ABA of A. We show in Section 1 that the blocks of A lying over B are precisely the indecomposable direct summands of the A^{op} ⊗ A-module B_{Δ} ↑^{A^{op}⊗A}. In turn, these direct summands are in one-to-one correspondence with the blocks of another endomorphism algebra, End_{A^{op}⊗A}(B_{Δ} ↑^{A^{op}⊗A}). The algebra E^{G} of fixed points of E = End_{A_{1}^{op}⊗A}(B ↑^{A_{1}^{op}⊗A}) is isomorphic to End_{A^{op}⊗A}(B_{Δ} ↑^{A^{op}⊗A}), and blocks of E^{G} correspond to G-conjugacy classes of blocks of E, as we show in Section 2.

Our results also hold in the more general situation where k is replaced by a pmodular system (K, R, k) for a prime p, where k is algebraically closed. Here R is a complete discrete valuation ring with maximal ideal 𝔭, quotient field K, and residue class field k = R/𝔭 of characteristic p. The basic theory needed includes the Krull-Schmidt-Azumaya Theorem [6, Proposition 56.4] and existence of projective covers of modules which follows from [5, Theorem 6.23] and [6, Propositions 56.2 and 56.4]. In this situation, A is a fully G-graded R-algebra (free and finitely generated as an R-module), and the twisted group algebra E/J_{G}(E) of the Clifford correspondence is a k-algebra as J_{G}(E) ⊇ 𝔭E. For simplicity however, we state our results in the case of a single base field k. All algebras and modules will be finite dimensional over k, and tensor products will be over k unless otherwise indicated.

The author thanks J. L. Alperin for suggesting this work and P. R. Boisen, E. C. Dade, and H. Ellers for helpful conversations.

1 An endomorphism algebra

In this section we give a correspondence between blocks lying over a given G-invariant block B of A_{1} and blocks of the endomorphism algebra End_{A^{op}⊗A}(B_{Δ} ↑^{A^{op}⊗A}), at the same time giving a reduction to this invariant case. Let G_{B} be the subgroup of G fixing an arbitrary block B of A_{1}, that is

G_{B} = {g ∈ G | A_{g^{−1}}BA_{g} = B}.

Let A_{G_{B}} = ∑_{g∈G_{B}} A_{g} and Δ_{G_{B}} = ∑_{g∈G_{B}}(A^{op})_{g} ⊗ A_{g}, where (A^{op})_{g} = (A_{g^{−1}})^{op}, a fully δ(G_{B})-graded algebra where δ(G_{B}) = {(g, g) | g ∈ G_{B}} ⊆ G × G. Note that B is naturally a Δ_{G_{B}}-module, denoted B_{Δ_{G_{B}}}. We observe that if B̃ is a block of A lying over B, so that B is a direct summand of B̃ ↓_{A_{1}^{op}⊗A_{1}}, then B̃ lies over all G-conjugates of B: The restricted module B̃ ↓_{A_{1}^{op}⊗A_{1}} is fixed by the G-action, as B̃ is. So for any g ∈ G, B^{g} = A_{g^{−1}}BA_{g} is also a direct summand of B̃ ↓_{A_{1}^{op}⊗A_{1}}. In addition B̃ lies over no other blocks of A_{1}, a consequence of the proof of the following lemma. We refer the reader to [2], [9], or [16] for facts about conjugate modules and induced modules for a fully group-graded ring.

Lemma 1.1 The blocks of A lying over B correspond one-to-one with the blocks of End_{A^{op}⊗A}(B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}).

Proof: First we prove that the blocks of A lying over B correspond one-to-one with the indecomposable summands of the A^{op} ⊗A-module B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}≅ ABA. Write A_{1} ≅ B_{1}⊕⋯⊕B_{k}⊕M as A_{1}^{op}⊗A_{1}-modules, where B = B_{1}, …, B_{k} are the distinct conjugates of B, and M is the sum of the remaining blocks of A_{1}. Let Δ = ∑_{g∈G}(A^{op})_{g} ⊗ A_{g}. The induced A^{op} ⊗ A-module (A_{1})_{Δ} ↑^{A^{op}⊗A} is isomorphic to A [2, Lemma 3.3]; this isomorphism is given simply by sending an element ∑_{i} a_{i} ⊗ (b_{i} ⊗ c_{i}) of (A_{1})_{Δ} ↑^{A^{op}⊗A} to ∑_{i} b_{i}a_{i}c_{i}. As B_{1} ⊕ ⋯ ⊕ B_{k} and M are naturally Δ-submodules of (A_{1})_{Δ}, we have

A ≅ (B_{1} ⊕ ⋯ ⊕ B_{k})_{Δ} ↑^{A^{op}⊗A} ⊕M_{Δ} ↑^{A^{op}⊗A}

as A^{op}⊗A-modules. The Δ-module (B_{1}⊕⋯⊕B_{k})_{Δ} is isomorphic to B_{Δ_{G_{B}}} ↑^{Δ} by an argument similar to that used in [2, Lemma 3.3]. Therefore B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}≅ B_{Δ_{G_{B}}} ↑^{Δ}↑^{A^{op}⊗A}≅ (B_{1} ⊕ ⋯ ⊕ B_{k})_{Δ} ↑^{A^{op}⊗A}, and so A ≅ B_{Δ_{G_{B}}} ↑^{A^{op}⊗A} ⊕M_{Δ} ↑^{A^{op}⊗A}.

Let B̃ be a block of A contained in B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}. By Mackey’s Theorem for groupgraded algebras [16],

B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}↓_{A_{1}^{op}⊗A_{1}} ≅ ∑_{(s,t)∈δ(G_{B})\G×G} B ⊗_{A_{1}^{op}⊗A_{1}} ((A^{op})_{s} ⊗ A_{t}).

A similar argument to [2, Lemma 3.3] shows that the A_{1}^{op} ⊗ A_{1}-module A_{s^{−1}}BA_{t} is isomorphic to the conjugate module B⊗_{A_{1}^{op}⊗A_{1}}((A^{op})_{s}⊗A_{t}). As B is indecomposable and A is fully graded, these conjugate modules A_{s^{−1}}BA_{t} are indecomposable as well. Since B̃ divides B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}, the Krull-Schmidt Theorem now implies that some A_{s^{−1}}BA_{t} divides B̃ ↓_{A_{1}^{op}⊗A_{1}}. Therefore B is a summand of A_{s}(B̃ ↓_{A_{1}^{op}⊗A_{1}})A_{t^{−1}} ≅ B̃ ↓_{A_{1}^{op}⊗A_{1}}; that is, B̃ lies over B.

Now assume B̃ is a block of A lying over B, but does not divide B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}. Then B̃ divides M_{Δ} ↑^{A^{op}⊗A}, and so B divides M_{Δ} ↑^{A^{op}⊗A}↓_{A_{1}^{op}⊗A_{1}}. As before, we apply Mackey’s Theorem to M_{Δ} ↑^{A^{op}⊗A}↓_{A_{1}^{op}⊗A_{1}}, and conclude that B divides A_{s^{−1}}MA_{t} for some s, t ∈ G. This implies that the conjugate A_{1}^{op} ⊗ A_{1}-module B^{s^{−1}} = A_{s}BA_{s^{−1}} divides A_{s}A_{s^{−1}}MA_{t}A_{s^{−1}} = MA_{ts^{−1}}. Letting e be the primitive central idempotent of A_{1} corresponding to the block B^{s^{−1}}, we derive a contradiction, as eB^{s^{−1}} ≠ 0 and eM = 0. Therefore the blocks of A lying over B are exactly the indecomposable summands of B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}.

As a summand of A, B_{Δ_{G_{B}}} ↑^{A^{op}⊗A} is an algebra, and End_{A^{op}⊗A}(B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}) ≅ Z(B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}), the center of B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}. Thus blocks of B_{Δ_{G_{B}}} ↑^{A^{op}⊗A} correspond one-to-one with blocks of End_{A^{op}⊗A}(B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}). □

We next show that the blocks of A lying over B correspond one-to-one with the blocks of A_{G_{B}} lying over B, a reduction allowing us to consider only the G-invariant case from now on. This is a version of the Fong-Reynolds Theorem [15, V.2.5].

Lemma 1.2 The blocks of A lying over B correspond one-to-one with the blocks of A_{G_{B}} lying over B. Further, this correspondence is given by induction of blocks of A_{G_{B}}, as A_{G_{B}}^{op} ⊗ A_{G_{B}}-modules, to A^{op} ⊗ A-modules.

Proof: We prove the first statement by showing that there is an isomorphism of algebras

End_{A_{G_{B}}^{op} ⊗A_{G_{B}}} (B_{Δ_{G_{B}}} ↑^{A_{G_{B}}^{op} ⊗A_{G_{B}}}) ≅ End_{A^{op}⊗A}(B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}),

and applying Lemma 1.1. By [1, Proposition 2.8.3] we have two natural isomorphisms

End_{A_{G_{B}}^{op} ⊗A_{G_{B}}} (B_{Δ_{G_{B}}} ↑^{A_{G_{B}}^{op} ⊗A_{G_{B}}}) ≅ Hom_{Δ_{G_{B}}} (B_{Δ_{G_{B}}} , B_{Δ_{G_{B}}} ↑^{A_{G_{B}}^{op} ⊗A_{G_{B}}} ↓_{Δ_{G_{B}}})

and

End_{A^{op}⊗A}(B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}) ≅ Hom_{Δ_{G_{B}}} (B_{Δ_{G_{B}}} , B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}↓_{Δ_{G_{B}}}).

We may consider B_{Δ_{G_{B}}} ↑^{A_{G_{B}}^{op} ⊗A_{G_{B}}} ↓_{Δ_{G_{B}}} as a direct summand of B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}↓_{Δ_{G_{B}}} by Mackey’s Theorem [16]. In order to achieve the desired isomorphism, we need only show that all Δ_{G_{B}}-homomorphisms from B_{Δ_{G_{B}}} to B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}↓_{Δ_{G_{B}}} in fact have image contained in B_{Δ_{G_{B}}} ↑^{A_{G_{B}}^{op} ⊗A_{G_{B}}} ↓_{Δ_{G_{B}}}. Consider a summand A_{s}BA_{t} = A_{s}BA_{s^{−1}}A_{st} = B^{s^{−1}}A_{st} of B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}↓_{Δ_{G_{B}}} as an A_{1}^{op}⊗A_{1}-module. If s ∉ G_{B}, then the primitive central idempotent e of A_{1} associated with B yields the identity map on B but 0 on B^{s^{−1}} ≠ B. In this case there are no A_{1}^{op} ⊗ A_{1}-homomorphisms, and so no Δ_{G_{B}}-homomorphisms from B_{Δ_{G_{B}}} to A_{s}BA_{t}. A similar argument works if s ∈ G_{B} and t ∉ G_{B}. Therefore we have proved the first statement of the lemma.

By the proof of Lemma 1.1, blocks of A_{G_{B}} lying over B are the indecomposable A_{G_{B}}^{op} ⊗ A_{G_{B}}-direct summands of B_{Δ_{G_{B}}} ↑^{A_{G_{B}}^{op} ⊗A_{G_{B}}}, while blocks of A lying over B are the indecomposable A^{op} ⊗ A-direct summands of B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}≅ (B_{Δ_{G_{B}}} ↑^{A_{G_{B}}^{op} ⊗A_{G_{B}}}) ↑^{A^{op}⊗A}. Suppose that B_{Δ_{G_{B}}} ↑^{A_{G_{B}}^{op} ⊗A_{G_{B}}} ≅ B̃_{1} ⊕ ⋯ ⊕ B̃_{n} is a decomposition into indecomposable A_{G_{B}}^{op} ⊗ A_{G_{B}}-modules. Then

B_{Δ_{G_{B}}} ↑^{A^{op}⊗A}≅ B̃_{1} ↑^{A^{op}⊗A} ⊕ ⋯ ⊕ B̃_{n} ↑^{A^{op}⊗A}

as A^{op} ⊗ A-modules. If some B̃_{i} ↑^{A^{op}⊗A} were not indecomposable, there would be a contradiction to the first statement of the lemma. Therefore each B̃_{i} ↑^{A^{op}⊗A} is a block of A. □

2 Another endomorphism algebra

In this section we discuss a Clifford correspondence for indecomposable modules which is implicit in Dade’s work in [13]. This provides a starting point for the Clifford correspondence for blocks. We show how an endomorphism algebra arising in this indecomposable module situation is related to the endomorphism algebra of Lemma 1.1. We assume B is a G-invariant block of A_{1}, so that G_{B} = G.

Let V be an indecomposable right A_{1}-module, and E = End_{A}(V ↑^{A}). The induced module V ↑^{A}= V ⊗_{A_{1}} A is a graded module [18], and so E is a G-graded algebra with identity component E_{1} ≅ End_{A_{1}}(V). One way to see this is to consider the isomorphisms of vector spaces

E ≅ Hom_{A_{1}}(V, V ↑^{A}↓_{A_{1}}) ≅ ∑_{g∈G} Hom_{A_{1}}(V, V^{g})

that follow from [1, Proposition 2.8.3] and Mackey’s Theorem [16]. Here V^{g} is the A_{1}- module V ⊗_{A_{1}} A_{g}, and E_{g} ≅ Hom_{A_{1}}(V, V^{g}) as a vector space. If V is G-invariant, so that V^{g} ≅ V for all g ∈ G, then E is fully G-graded, as may be seen from the above decomposition of E or [18, I.5.2]. If V is also irreducible then Schur’s Lemma implies that End_{A}(V ↑^{A}) is a twisted group algebra. This is the twisted group algebra of the classical Clifford correspondence. In case V is indecomposable but not irreducible, or in case k is replaced by a complete discrete valuation ring R, the fact that E_{1} ≅ End_{A_{1}}(V) is local [5, Proposition 6.10] replaces Schur’s Lemma as follows: Let J_{G}(E) denote the graded Jacobson radical of E, the intersection of all maximal graded right ideals. By [4, Theorem 4.4] J_{G}(E) is contained in the ordinary Jacobson radical J(E). In case V is G-invariant, [13, Proposition 2.19], [5, Proposition 5.22], and [7, Lemma 14.2] imply that E/J_{G}(E) is a twisted group algebra for a subgroup of G. For the sake of completeness, we will give these arguments in greater detail later for the special case where V is a block.

Whether or not V is G-invariant, there is a one-to-one correspondence between indecomposable A-direct summands of V ↑^{A} and indecomposable right E/J_{G}(E)-direct summands of E/J_{G}(E): Indecomposable summands of V ↑^{A} correspond one-to-one with indecomposable right summands of E by sending a summand of V ↑^{A} to the corresponding projection endomorphism, an idempotent of E [5, Proposition 6.3]. As J_{G}(E) ⊆ J(E), indecomposable right E-direct summands of E correspond with indecomposable right E/J_{G}(E)-direct summands of E/J_{G}(E) [5, Corollaries 6.22 and 6.25].

In the block situation, we replace A by the fully G-graded algebra A_{1}^{op} ⊗ A, A_{1} by A_{1}^{op} ⊗ A_{1}, and V by the block B of A_{1}. The relevant G-graded endomorphism algebra is then

E = End_{A_{1}^{op}⊗A}(B ↑^{A_{1}^{op}⊗A}),

and the Clifford correspondence for indecomposable modules described above applies here. It will need to be modified to yield the Clifford correspondence for blocks given in Theorem 3.5. We caution that, although B is G-invariant as a block, it may not be 1 × G-invariant as an A_{1}^{op} ⊗ A_{1}-module. Thus it does not follow from the above arguments that E/J_{G}(E) is a twisted group algebra; in Section 3 we show that this fact follows from a characterization of the graded Jacobson radical given in [13]. In addition, we are interested in A^{op} ⊗ A-modules rather than A_{1}^{op} ⊗ A-modules. This is where a G-action on E arises. The following proposition is Theorem 1.3 of [17]; see also [10, Theorem 2.1].

Proposition 2.1 (Miyashita action) Let G be a finite group and S a fully G-graded algebra. Let M and N be S-modules, g ∈ G, and ϕ ∈ Hom_{S_{1}}(M, N). Then there is a unique element ϕ^{g} ∈ Hom_{S_{1}}(M, N) such that ϕ^{g}(ms_{g}) = ϕ(m)s_{g} for all m ∈ M and s_{g} ∈ S_{g}. If M = N, then G acts as algebra automorphisms of End_{S_{1}}(M).

We give the definition of ϕ^{g}, which will be needed in the next section. As S is fully G-graded, we have S_{g^{−1}}S_{g} = S_{1} for all g ∈ G. As 1 ∈ S_{1} [18], there are elements α_{i} ∈ S_{g^{−1}}, β_{i} ∈ S_{g} such that ∑_{i=1}^{n} α_{i}β_{i} = 1. Then for all m ∈ M,

ϕ^{g}(m) = ∑_{i=1}^{n} ϕ(mα_{i})β_{i}.

In the block situation, we let S = A^{op} ⊗ A be the fully G-graded algebra with components S_{g} = (A^{op})_{g}⊗A. The proposition gives a G-action on End_{A_{1}^{op}⊗A}(B_{Δ} ↑^{A^{op}⊗A}), where Δ = ∑_{g∈G}(A^{op})_{g} ⊗ A_{g}, and B_{Δ} is the Δ-module B. This provides the connection between E = End_{A_{1}^{op}⊗A}(B ↑^{A_{1}^{op}⊗A}) and the endomorphism algebra End_{A^{op}⊗A}(B_{Δ} ↑^{A^{op}⊗A}) of Lemma 1.1, as we see next. We note that B_{Δ} ↑^{A^{op}⊗A}↓_{A_{1}^{op}⊗A}≅ B ↑^{A_{1}^{op}⊗A}, where the latter module is induced from A_{1}^{op} ⊗ A_{1}. This follows from an application of Mackey’s Theorem [16], as there is only one δ(G), 1×G-double coset in G×G, and B_{Δ} ↓_{A_{1}^{op}⊗A_{1}}= B. Therefore

End_{A_{1}^{op}⊗A}(B_{Δ} ↑^{A^{op}⊗A}) ≅ End_{A_{1}^{op}⊗A}(B ↑^{A_{1}^{op}⊗A}) = E.

Thus the above proposition yields an action of G as automorphisms of E. The fixed point subalgebra is E^{G} ≅ End_{A^{op}⊗A}(B_{Δ} ↑^{A^{op}⊗A}), the endomorphism algebra of Lemma 1.1. We mention that if we had taken S_{g} = A^{op} ⊗A_{g} instead, we would have gotten a different Ggraded endomorphism algebra, but upon taking G-fixed points, we would have obtained the same A^{op} ⊗ A-endomorphism algebra.

We now give the relationship between our endomorphism algebras and the centralizer algebras of [8]. Let C_{A}(A_{1}) denote the centralizer in A of A_{1}. Let e be the primitive central idempotent of A_{1} corresponding to the block B. Then as graded algebras, E is isomorphic to eC_{A}(A_{1}): We identify B ↑^{A_{1}^{op}⊗A} with ∑_{g∈G} BA_{g} as discussed in the proof of Lemma 1.1. Given ϕ ∈ E, ϕ is determined by ϕ(e), as ϕ(b) = ϕ(eb) = ϕ(e)b for any b ∈ B ↑^{A_{1}^{op}⊗A}. Further, ϕ(e) ∈ eC_{A}(A_{1}) as eϕ(e) = ϕ(e^{2}) = ϕ(e) and ϕ ∈ E implies that ϕ(e) commutes with elements of A_{1}. Conversely, any element of eC_{A}(A_{1}) defines an element of E in this way. The G-grading on eC_{A}(A_{1}) inherited from A corresponds to that of E, and the G-action on E provided by Proposition 2.1 gives rise to a G-action on eC_{A}(A_{1}).

Lemma 2.2 The subalgebras E^{G} and E_{1} of E are both contained in the center of E.

Proof: Let ϕ ∈ E and ψ ∈ E_{g} for some g ∈ G, and suppose ψ(e) = r_{g} ∈ eC_{A}(A_{1}), so that ψ(b) = r_{g}b for all b ∈ B ↑^{A_{1}^{op}⊗A}. Applying Proposition 2.1, ϕ^{g^{−1}} ∘ψ(e) = ϕ^{g^{−1}}(r_{g}e) = r_{g}ϕ(e) = ψ ∘ ϕ(e), so that ϕ^{g^{−1}} ∘ ψ = ψ ∘ ϕ. If ϕ ∈ E^{G} then, ϕ is in the center of E. If ψ ∈ E_{1}, then ϕ ∘ ψ = ψ ∘ ϕ for all ϕ ∈ E, so that ψ is in the center of E. □

Next we prove that blocks of E^{G} correspond one-to-one with G-conjugacy classes of blocks of E. Combined with Lemma 1.1 for G = G_{B}, this shows that blocks of A lying over B correspond one-to-one with G-conjugacy classes of blocks of E.

Lemma 2.3 The blocks of E^{G} ≅ End_{A^{op}⊗A}(B_{Δ} ↑^{A^{op}⊗A}) correspond one-to-one with Gconjugacy classes of blocks of E.

Proof: A G-conjugacy class of blocks of E corresponds to a G-conjugacy class e_{1}, …, e_{k} of primitive central idempotents of E. Their sum e′ = e_{1} + ⋯ + e_{k} is a central idempotent of E^{G}. If e′ were not primitive, then it would decompose into primitive central idempotents of E^{G}, each of which is a central idempotent of E by Lemma 2.2. Thus each decomposes into primitive central idempotents of E, and by uniqueness these must be the e_{1}, …, e_{k}. This contradicts the assumption that e_{1}, …, e_{k} is a single G-conjugacy class. Conversely, a block of E^{G} corresponds to a primitive central idempotent of E^{G}, which is a central idempotent of E, and decomposes into primitive central idempotents of E. The G-action must permute these primitive central idempotents transitively. □

3 The Clifford correspondence

Let B be a G-invariant block of A_{1}. In this section we show that blocks of E = End_{A_{1}^{op}⊗A}(B ↑^{A_{1}^{op}⊗A}) correspond one-to-one with blocks of E/J_{G}(E), a version of [8, Lemma 3.1 and Theorem 3.5]. Together with the results of Sections 1 and 2, this yields the Clifford correspondence in Theorem 3.5. In order to achieve this correspondence, it seems necessary to introduce a fully group-graded subalgebra E[B] of E, as is done for centralizer algebras in [8].

Let G[B] be the subgroup of G defined by g ∈ G[B] if E_{g} contains a unit, and E[B] = E_{G[B]} = ∑_{g∈G[B]} E_{g}. Then E[B] is a fully G[B]-graded algebra, as for each g ∈ G[B], E_{g} = u_{g}E_{1} for a unit u_{g} ∈ E_{g}. There are examples for which G[B] ≠ G [14, Lemma 3.11].

The graded Jacobson radical J_{G}(E) of any group-graded algebra E is a graded twosided ideal of E [18, Lemma I.7.4]. We give the characterization of J_{G}(E) in [13], providing details for the sake of completeness. The components of J_{G}(E) are given by

J_{G}(E)_{g} = {ϕ ∈ E_{g} | ϕE_{g^{−1}} ⊆ J(E_{1})}.

This follows immediately from the one-to-one correspondence between all maximal Ggraded right ideals M of E and all maximal right ideals N of E_{1} given by sending M to N = M_{1}, and N to M where M_{g} = {ϕ ∈ E_{g} | ϕE_{g^{−1}} ⊆ N}. This correspondence is straightforward to verify. As B is an indecomposable A_{1}^{op} ⊗ A_{1}-module, E_{1} ≅ End_{A_{1}^{op}⊗A_{1}}(B) is local, and so E_{g} − J_{G}(E)_{g} is the set of units in E_{g} [13, Lemma 2.18]. It follows that

J_{G}(E)_{g} = \{ \begin{matrix}J(E_{1})E_{g} & if g ∈ G[B] \\ E_{g} & if g ∉ G[B] \end{matrix} ,

as in [13, Proposition 2.19]. These arguments also apply to E[B], so in that case J_{G[B]}(E[B]) = J(E_{1})E[B], and E[B]/J_{G[B]}(E[B]) is a twisted group algebra for G[B] by [7, Lemma 14.2]. The above characterizations of J_{G}(E) and J_{G[B]}(E[B]) also immediately imply the following lemma.

Lemma 3.1 There is an isomorphism of twisted group algebras

E/J_{G}(E) ≅ E[B]/J_{G[B]}(E[B]).

The proof of the next lemma, a reinterpretation of that of [8, Lemma 3.1], uses E_{1} ⊆ Z(E[B]) and the fact that J_{G[B]}(E[B]) = J(E_{1})E[B] in that Nakayama’s Lemma is applied to E_{1}-submodules of E[B]. This shows the necessity of dealing with E[B] rather than E.

Lemma 3.2 The blocks of E[B] correspond one-to-one with blocks of E[B]/J_{G[B]}(E[B]).

Proof: Let E[B]/J_{G[B]}(E[B]) = b_{1} ⊕⋯⊕b_{k} be a decomposition into blocks. Consider each b_{i} as a right E[B]-module via the canonical map from E[B] to E[B]/J_{G[B]}(E[B]). Let P (b_{i}) be the projective cover of b_{i} as a right E[B]-module. As J_{G[B]}(E[B]) ⊆ J(E[B]) [4, Theorem 4.4], E[B] itself is the projective cover of E[B]/J_{G[B]}(E[B]) as a right E[B]- module [5, Corollary 6.22]. As projective covers preserve direct sums we have E[B] ≅ P (b_{1}) ⊕ ⋯ ⊕ P (b_{k}) as a right E[B]-module. We identify the P (b_{i}) with right ideals of E[B] via this isomorphism. We will show that the P (b_{i}) are also left ideals, and that they are indecomposable E[B]^{op} ⊗ E[B]-modules; that is, the P (b_{i}) are blocks of E[B].

We let P′(b_{i}) be the projective cover of b_{i} as a left E[B]-module, so that E[B] ≅ P′(b_{1})⊕⋯⊕P′(b_{k}) as a left E[B]-module. Consider the sum m_{i} of all blocks b_{j} with j ≠ i. We have E[B] ≅ P (b_{i}) ⊕ P (m_{i}) as a right E[B]-module, and E[B] ≅ P′(b_{i}) ⊕ P′(m_{i}) as a left E[B]-module. Restrict these E[B]-modules to E_{1}-modules. As E_{1} is central in E[B] by Lemma 2.2, we do not distinguish between left and right modules. We have P (b_{i}) = P (b_{i})E[B] ≅ P (b_{i})P′(b_{i}) ⊕ P (b_{i})P′(m_{i}) and P (m_{i}) ≅ P (m_{i})P′(b_{i}) ⊕ P (m_{i})P′(m_{i}) as E_{1}- modules, since clearly this is true without the direct sums, but for example P (b_{i})P′(b_{i}) ⊆ P′(b_{i}), P (b_{i})P′(m_{i}) ⊆ P′(m_{i}) and P′(b_{i}) ∩ P′(m_{i}) = 0. Therefore

E[B] ≅ P (b_{i})P′(b_{i}) ⊕ P (b_{i})P′(m_{i}) ⊕ P (m_{i})P′(b_{i}) ⊕ P (m_{i})P′(m_{i})

as E_{1}-modules. As J_{G[B]}(E[B]) = J(E_{1})E[B], the right E_{1}-module P (b_{i})P′(m_{i}) becomes 0 upon passing to the quotient E[B]/J_{G[B]}(E[B]), and so P (b_{i})P′(m_{i})J(E_{1}) = P (b_{i})P′(m_{i}). By Nakayama’s Lemma, P (b_{i})P′(m_{i}) = 0, and similarly P (m_{i})P′(b_{i}) = 0. Therefore

E[B] ≅ P (b_{i})P′(b_{i}) ⊕ P (m_{i})P′(m_{i})

as E_{1}-modules, where P (b_{i}) = P (b_{i})P′(b_{i}) = P′(b_{i}), and P (m_{i}) = P (m_{i})P′(m_{i}) = P′(m_{i}). It follows that each P (b_{i}) is both a left and a right ideal of E[B].

Suppose P (b_{i}) = B̃_{1} ⊕ B̃_{2} as E[B]^{op} ⊗ E[B]-modules. Under the canonical map from E[B] to E[B]/J_{G[B]}(E[B]), the image of one of B̃_{1} or B̃_{2} must be 0. Considering B̃_{1} and B̃_{2} as E_{1}-modules, another application of Nakayama’s Lemma implies that one of B̃_{1} or B̃_{2} is 0. □

Next we see that G-conjugacy classes of blocks of E/J_{G}(E) correspond one-to-one with G-conjugacy classes of blocks of E, resulting in the Clifford correspondence. For this, we need a lemma.

Lemma 3.3 The subgroup G[B] is normal in G, and (E_{h})^{g} = E_{h^{g}} for all g, h ∈ G.

Proof: Let g, h ∈ G, and ϕ ∈ E_{h}. As stated in the text following Proposition 2.1, ϕ^{g}(b) = ∑_{i} β_{i}ϕ(α_{i}b) for all b ∈ B ↑^{A_{1}^{op}⊗A}, where α_{i} ∈ A_{g}, β_{i} ∈ A_{g^{−1}}, and ∑_{i} β_{i}α_{i} = 1. Let e be the primitive central idempotent of A_{1} corresponding to the block B, so ϕ(e) ∈ A_{h} as discussed in the text preceding Lemma 2.2. As e is a G-invariant element of C_{A}(A_{1}), e is central in A_{1} so ϕ^{g}(e) = ∑_{i} β_{i}ϕ(α_{i}e) = ∑_{i} β_{i}ϕ(e)α_{i}, since ϕ is an A_{1}^{op} ⊗ A-map. But ∑_{i} β_{i}ϕ(e)α_{i} ∈ A_{g^{−1}}A_{h}A_{g} = A_{h^{g}}. Therefore ϕ^{g} ∈ E_{h^{g}}, and (E_{h})^{g} ⊆ E_{h^{g}}. Conversely, let ψ ∈ E_{h^{g}}, and let ϕ = ψ^{g^{−1}} ∈ (E_{h^{g}})^{g^{−1}} ⊆ E_{h} by the above argument. Then ψ = ϕ^{g} ∈ (E_{h})^{g}, and so E_{h^{g}} = (E_{h})^{g}.

To see that G[B] is normal in G, let h ∈ G[B], g ∈ G, and u_{h} a unit in E_{h}. Then (u_{h})^{g} is a unit in E_{h^{g}}. □

By the lemma, the G-action on E yields a G-action on E[B]. Further, G fixes the ideal J_{G[B]}(E[B]) = J(E_{1})E[B], as G fixes E_{1} by the lemma and so permutes the maximal ideals of E_{1}. Therefore the G-action on E[B] induces a G-action on E[B]/J_{G[B]}(E[B]).

Lemma 3.4 The G-conjugacy classes of blocks of E correspond one-to-one with Gconjugacy classes of blocks of E/J_{G}(E).

Proof: By Lemmas 3.1 and 3.2, we need only show that G-conjugacy classes of blocks of E[B] correspond one-to-one with G-conjugacy classes of blocks of E. Given a Gconjugacy class of blocks of E[B], there is a corresponding G-invariant central idempotent e of E[B], which is also then a G-invariant central idempotent of E, as E^{G} is central in E. Consider the image \overline{e} of e in E/J_{G}(E) ≅ E[B]/J_{G[B]}(E[B]). If e may be decomposed in E as the sum of two nontrivial G-invariant central idempotents, this decomposition gives such a decomposition of \overline{e}, contradicting Lemma 3.2.

Conversely, a G-conjugacy class of blocks of E corresponds to a G-invariant central idempotent e of E. This corresponds to an idempotent \overline{e} of E/J_{G}(E) ≅ E[B]/J_{G[B]}(E[B]), which by Lemma 3.2 corresponds to an idempotent e′ of E[B]. As J_{G[B]}(E[B]) ⊆ J_{G}(E), e and e′ are central idempotents of E having the same image modulo J_{G}(E), and so e = e′. □

The Clifford correspondence now follows immediately.

Theorem 3.5 The blocks of A lying over B correspond one-to-one with G-conjugacy classes of blocks of the twisted group algebra E/J_{G}(E).

Proof: By Lemma 1.1 for G = G_{B} and Lemma 2.3, the blocks of A lying over B correspond one-to-one with G-conjugacy classes of blocks of E. By Lemma 3.4, Gconjugacy classes of blocks of E correspond one-to-one with G-conjugacy classes of blocks of E/J_{G}(E). □

We close by returning to the general case where B is a block of A_{1} that is not necessarily G-invariant, G_{B} is the set of all elements g of G such that A_{g^{−1}}BA_{g} = B, and E = End_{A_{1}^{op}⊗A_{G_{B}}} (B ↑^{A_{1}^{op}⊗A_{G_{B}}}). By Lemma 1.2 and Theorem 3.5 with G replaced by G_{B}, we have the following.

Corollary 3.6 The blocks of A lying over B correspond one-to-one with G_{B}-conjugacy classes of blocks of the twisted group algebra E/J_{G_{B}}(E).

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