EXTENDED CONDITIONS FOR THE CONSEQUENCES OF A PERFECT ISOMETRY

MORTON E. HARRIS

Abstract. In an important paper [1], M. Broué defines the fundamental concept of a perfect isometry between two p-blocks f and e of finite groups H and G that posits numerical conditions on a generalized character of G × H. Then, in one of the fundamental results of [1], he shows, in [1, Théorème 1.5], that such a perfect isometry implies several equivalences between basic invariants of the blocks f and e. In this paper, we extend the conditions for a perfect isometry (that includes Broué’s definition) and demonstrate that all of the conclusions of [1, Théorème 1.5] hold in our more general setting. We seek applications of these results.

1. Introduction and Statements of Results

The purpose of this paper is to present an extension of M. Broué’s concept of a perfect isometry between two p-blocks f and e of finite groups H and G that includes Broué’s definition and to demonstrate that all of the equivalences between basic invariants of f and e of [1, Théorème 1.5] can be proved under our more general hypotheses using the methods of [1].

This section contains the statements of our results. The required proofs are given in Section 2. In our setting the methods of [1] yield the required proofs.

Throughout this section G will be a finite group, p will be a prime integer and all rings will have identities. Our notation and terminology are standard and tend to follow [1], [2] and [3]. For example, ccl(G) denotes the set of conjugacy classes of G.

Let R be a subring of the commutative ring S (with 1_{S} ∈ R).

We shall use, without explicit reference, the easy to prove:

Lemma 1.1. Let e be an idempotent of RG. Then (RG)e = (RG) ∩ ((SG)e).

As usual, F (G, R) denotes the set of functions from G to R and via R-linearity we shall identify F (G, R) with Hom_{R}(RG, R).

For each subset X of G, let δ_{X} ∈ F (G, R) denote the characteristic function of X so that:

δ_{X}(g) = \begin{cases}0 if g ∉ X \\ 1 if g ∈ X. \end{cases}

If g ∈ G and X = {g}, set δ_{g} = δ_{X}.

Let A_{G}: RG → Hom_{R}(RG, R) denote the R-linear homomorphism such that g ↦ δ_{g^{−1}} for all g ∈ G and let B_{G}: Hom_{R}(RG, R) → RG denote the R-linear homomorphism such that δ_{g} ↦ g^{−1} for all g ∈ G.

Let α ∈ Aut(G). Clearly α ∈ Aut_{R}(RG) in the usual way and α ∈ Aut_{R}(Hom_{R}(RG, R)) where (αf)(g) = f(α^{−1}(g)) for all g ∈ G and all f ∈ Hom_{R}(RG, R). Thus αδ_{g} = δ_{α(g)} for all g ∈ G.

As usual, RG is a left RG-module and Hom_{R}(RG, R) is a left RGmodule where, if φ ∈ RG, f ∈ Hom_{R}(RG, R) and γ ∈ RG, then (φf)(γ) = f(γφ). Thus, if g, h ∈ G, then gδ_{h} = δ_{hg^{−1}}.

One can easily prove:

Lemma 1.2. (a) A_{G} and B_{G} are inverse R-linear isomorphisms;

(b) If α ∈ Aut(G), u ∈ RG and f ∈ Hom_{R}(RG, R), then (α · A_{G})(u) = A_{G}(α(u)) and B_{G}(αf) = αB_{G}(f); and

(c) If u, v ∈ RG and f ∈ Hom_{R}(RG, R), then A_{G}(vu) = vA_{G}(u) and B_{G}(vf) = vB_{G}(f).

Let CF (G, R) = {f ∈ F (G, R) | f is constant on the elements of 𝒞 for each 𝒞 ∈ ccl(G)}.

Hypothesis C. For the finite group G, let K be a finite extension of the rational numbers Q in the complex numbers ℂ such that K is “large enough” for all subgroups of G and such that K is invariant under complex conjugation. Let I denote the integral closure of the integers Z in K and let M be a maximal ideal of I such that pI ≤ M. Thus Z ∩ M = pZ and k = I/M is a field of characteristic p. Set O = {αβ^{−1} | α ∈ I and β ∈ I − M}. Thus O is a local subring of K, I ≤ O ≤ K, M = {αβ^{−1} | α ∈ M and β ∈ I − M} is the unique maximal ideal of O. Also I ∩ M = M, Z ∩ M = pZ, and K is the field of fractions of I and of O. As in [3, Chapter 15], the natural ring epimorphism ∗: I → k = I/M extends to the natural ring epimorphism ∗: O → k by defining (αβ^{−1})^{∗} = α^{∗}(β^{∗})^{−1} for all α ∈ I and all β ∈ I − M.

Let e be an idempotent of Z(OG); possibly e = 1. Let I(G) denote the set of irreducible K-characters of G and let I(e) = {χ ∈ I(G) | eχ = χ}. Also let I Br(G) denote the set of irreducible Brauer characters of G (cf. [3, Chapter 15]) (canonically extended to take the value zero on p-singular elements and let I Br(e) = {φ ∈ I Br(G) | eφ = φ}. Also let F_{p′}(G, O) = {f ∈ F (G, O) | f(G − G_{p′}) = 0} and similarly let F_{p′}(G, K) = {f ∈ F (G, K) | f(G − G_{p′}) = 0}. As usual CF (G, O) = {f ∈ F (G, O) | f is constant on each 𝒞 ∈ ccl(G)} and CF (G, K) = {f ∈ F (G, K) | f is constant on each 𝒞 ∈ ccl(G). Set CF_{p′}(G, O) = CF (G, O) ∩ F_{p′}(G, O) and CF_{p′}(G, K) = CF (G, K) ∩ F_{p′}(G, K).

For the remainder of this section we adapt the inner product of [3, pp. 20–21] on CF (G, K).

Clearly:

(1.1) F (G, O) = Hom_{O}(OG, O) = ⨁_{g∈G}(Oδ_{g}) in O-mod;

and

F (G, K) = Hom_{K}(KG, K) = ⨁_{g∈G}(Kδ_{g}).

Set F (G, e, O)=e Hom_{O}(OG, O), F (G, e, K)=e Hom_{K}(KG, K). Similarly set CF (G, e, O) = eCF (G, O) and CF (G, e, K) = eCF (G, K).

Thus:

(1.2)  CF (G, O) = ⨁_{𝒞∈ccl(G)}(Oδ_{𝒞}) in O-mod, CF (G, K)

= ⨁_{𝒞∈ccl(G)}(Kδ_{𝒞}) = ⨁_{χ∈I(G)}(Kχ)

and

CF (G, e, K) = ⨁_{χ∈I(e)}(Kχ) in K-mod.

Also set R_{O}(G)= ⨁_{χ∈I(G)}(Oχ) in O-mod and R_{O}(G, e)= ⨁_{χ∈I(e)}(Oχ)

in O-mod.

For each φ ∈ I Br(G), let Φ_{φ} denote the associated projective indecomposable characters of φ. Thus

(1.3) CF_{p′}(G, K) = ⨁_{φ∈I Br(G)}(Kφ) = ⨁_{φ∈I Br(G)}(KΦ_{φ}) in K-mod, and
CF_{p′}(G, e, K) = ⨁_{φ∈I Br(e)}(Kφ) = ⨁_{φ∈I Br(e)}(KΦ_{φ}) in K-mod.

(cf. [2, IV, Lemma 3.4]).

Lemma 1.3.

CF_{p′}(G, O) = ⨁_{φ∈I Br(G)}(Oφ) in O-mod,

and

CF_{p′}(G, e, O) = ⨁_{φ∈I Br(e)}(Oφ) in O-mod.

Definition 1.4. CF^{pr}(G, e, O) = {π ∈ CF (G, e, K) | [π, α]_{G} ∈ O for all α ∈ CF (G, e, O)} and CF_{p′}^{pr}(G, e, O) = CF^{pr}(G, e, O) ∩ F_{p′}(G, K).

Lemma 1.5. Let π ∈ CF (G, e, K). Then the following four conditions are equivalent:

(a) π ∈ CF^{pr}(G, e, O);

(b) [π, α]_{G} ∈ O for all α ∈ CF (G, O);

(c) [π, δ_{𝒞}] ∈ O for all 𝒞 ∈ ccl(G); and

(d) \frac{π(g))}{|(C_{G}(g)|} ∈ O for all g ∈ G.

Consequently CF^{pr}(G, e, O) is an O-submodule of CF (G, e, O) and CF_{p′}^{pr}(G, e, O) is an O-submodule of CF_{p′}(G, e, O).

Lemma 1.6. CF_{p′}^{pr}(G, e, O) = ⨁_{φ∈I Br(e)}(OΦ_{φ}).

Let H also be a finite group.

For the remainder of this section, assume Hypothesis C for H × G, let f be and idemotent of Z(OH) and let e be an idempotent of Z(OG). Thus f ⊗_{O} e is an idempotent of O(H × G), I(f ⊗ e) = {ψχ | ψ ∈ I(f) and χ ∈ I(e)} and the results above hold for H × G, K, O, f ⊗_{O} e, etc.

As in [1, Section 1], each

μ ∈ CF (H × G, f ⊗_{O} e, K) = ⨁_{\begin{matrix}χ∈I(f) \\ χ∈I(e) \end{matrix}}(Kψχ)

determines unique K-linear homomorphisms

I_{μ}: CF (H, f, K) → CF (G, e, K)

and R_{μ}: CF (G, e, K) → CF (H, f, K) where if ψ ∈ I(f), then

I_{μ}(ψ)(g) = \frac{1}{|H|} ∑_{h∈H}(μ(g, h^{−1})ψ(h))

for all g ∈ G and if χ ∈ I(e), then

R_{μ}(χ)(h) = \frac{1}{|G|} ∑_{g∈G}(μ(g^{−1}, h)χ(g))

for all h ∈ H.

Analogously, each μ ∈ CF (H × G, f ⊗_{O} e, K) determines unique K-linear homomorphisms

I_{μ}^{0} : Z((KH)f) → Z((KG)e) and
R_{μ}^{0} : Z((KG)e) → Z((KH)f) such that

if α = ∑_{h∈H} y(h)h ∈ Z((KH)f), then

I_{μ}^{0}(α) = ∑_{g∈G} (\frac{1}{|H|} (∑_{h∈H} μ(g^{−1}, h)y(h)) g)
and if β = ∑_{g∈G} x(g)g ∈ Z((KG)e), then

R_{μ}^{0}(β) = ∑_{h∈H} (\frac{1}{|G|} (∑_{g∈G} μ(g, h^{−1})x(g)) h) .

The K-linear isomorphism A_{H}: F (H, K) → KK such that δ_{h} ↦ h^{−1} where δ_{h} is the characteristic function of {h} on H for all h ∈ H induces a K-linear isomorphism A_{H}: CF (H, f, K) → Z((KH)f) and similarly we have the K-linear isomorphism A_{G}: CF (G, e, K) → Z((KG)e).

It is easy to see that the following diagram of K-linear homomorphisms commutes:

\begin{matrix}CF (H, f, K) & A_{H} & [figure] & I_{μ} & [figure] & Z((KH)f) & I_{μ}^{0} & [figure] & CF (G, e, K) & A_{G} & [figure] & R_{μ} & [figure] & Z((KG)e) & R_{μ}^{0} & [figure] & CF (H, f, K) & A_{H} & [figure] & Z((KH)f) \\ (1.4) \end{matrix}

The next two lemmas restate [1, Proposition 4.1]:

Lemma 1.7. The following two conditions are equivalent:

(a) I_{μ}(CF (H, f, O)) ≤ CF (G, e, O); and

(b) \frac{α(g, h)}{|C_{H}(h)|} ∈ O for all g ∈ G and h ∈ H.

Lemma 1.8. The following two conditions are equivalent:

(a) I_{μ}(CF_{p′}(H, f, K)) ≤ CF_{p′}(G, e, K); and

(b) if g ∈ G, h ∈ H_{p′} and μ(g, h) ≠ 0, then g ∈ G_{p′}.

For the remainder of this section we assume that J is a bijection J: I(f) → I(e) and that for each ψ ∈ I(f) an element ϵ_{ψ} ∈ O^{×} has been chosen such that ϵ_{ψ}\overline{e}_{ψ} = 1.

Set μ = ∑_{x∈I(f)} ϵ_{ψ}(ψJ(ψ) and μ = ∑_{ψ∈I(f)} \overline{ϵ}_{ψ}ψJ(ψ). Thus μ and ω are

elements of R_{O}(G × H),

I_{μ}: CF (H, f, K) → CF (G, e, K),
I_{ω}: CF (H, f, K) → CF (G, e, K),
R_{μ}: CF (G, e, K) → CF (H, f, K), and
R_{ω}: CF (G, e, K) → CF (H, f, K)

are K-linear isomorphisms that are isometries such that I_{μ}(ψ) = ϵ_{ψ}J(e), I_{ω}(ψ) = \overline{ϵ}_{ψ}J(ψ), R_{μ}(J(ψ)) = ϵ_{ψ}ψ and R_{ω}(J(ψ)) = \overline{ϵ}_{ψ}ψ for all ψ ∈ I(f).

Thus we clearly have:

Lemma 1.9. (a) (I_{μ}, R_{ω}) and (I_{ω}, R_{μ}) are adjoint pairs of inverse

K-linear isometries;

(b) I_{μ}(CF (H, f, K)) = CF (G, e, K), R_{μ}(CF (G, e, K)) = CF (H, f, K), I_{ω}(CF (H, f, K)) = CF (G, e, K) and R_{ω}(CF (G, e, K)) = CF (H, f, K); and

(c) I_{μ}(R_{O}(H, f)) = R_{O}(G, e), R_{μ}(G, e) = R_{O}(H, f), I_{ω}(R_{O}(H, f)) = R_{O}(G, e) and R_{ω}(G, e) = R_{O}(H, f).

For the remainder of this section, we assume:

Hypothesis PI. {ϵ_{ψ} | ψ ∈ I(f)} has been chosen such that:

I_{μ}: CF (H, f, K) → CF (G, e, K) and
R_{ω}: CF (G, e, K) → CF (H, f, K)

satisfy both equivalent conditions of Lemmas 1.7 and 1.8.

Theorem 1.10. (a) I_{μ}(CF (H, f, O)) = CF (G, e, O) and

R_{ω}(CF (G, e, O)) = CF (H, f, O);

(b) I_{μ}(CF_{p′}(H, f, K)) = CF_{p′}(G, e, K) and R_{ω}(CF_{p′}(G, e, K)) = CF_{p′}(H, f, K);

(c) I_{μ}(CF_{p′}(H, f, O)) = CF_{p′}(G, e, O) and R_{ω}(CF_{p′}(G, e, O)) = CF_{p′}(H, f, O);

(d) I_{μ}(CF^{pr}(H, f, O)) = CF^{pr}(G, e, O) and R_{ω}(CF^{pr}(G, e, O)) = CF^{pr}(H, f, O); and (e) I_{μ}(CF_{p′}^{pr}(H, f, O)) = CF_{p′}^{pr}(G, e, O) and

R_{ω}(CF_{p′}^{pr}(G, e, O)) = CF_{p′}(H, f, O).

Remark 1.11. If ϵ_{ψ} = ±1 for all ψ ∈ I(f), then μ = ω and I_{μ} is a perfect isometry in the sense of [1, Definition 1.4].

Let I_{μ}^{0} : KH → KG and R_{ω}^{0} : KG → KH be the K-linear maps such that:

I_{μ}^{0}(h) = ∑_{g∈G} (\frac{1}{|H|}μ(g^{−1}, h)g) for all h ∈ H and

R_{ω}^{0}(g) = ∑_{h∈H} (\frac{1}{|H|}ω(g, h^{−1})h)) for all g ∈ G as above.

Theorem 1.12. (a) I_{μ}^{0}(KH) ≤ Z(KG) and R_{ω}^{0}(KG) ≤ Z(KH);

(b) If h ∈ H, then

I_{μ}^{0}(h) = I_{μ}^{0}(fh)

= \frac{|G|}{|H|} ∑_{ψ∈I(f)} (\frac{ϵ_{ψ}ψ(h)}{J(ψ)(1)}e_{J(ψ)})

= I_{μ}^{0}(h)e

and if g ∈ G, then

R_{ω}^{0}(g) = R_{ω}^{0}(eg)

= \frac{|H|}{|G|} ∑_{ψ∈I(f)} (\overline{ϵ}_{ψ} \frac{J(ψ)(g)}{ψ(1)}e_{ψ})

= R_{ω}^{0}(g)f;

(c) If ψ ∈ I(f), then

I_{μ}^{0}(e_{ψ}) = ϵ_{ψ} \frac{ψ(1)}{|H|} \frac{|G|}{J(ψ)(1)}e_{J(ψ)}

and

R_{ω}^{0}(e_{J(ψ)}) = \overline{ϵ}_{ψ} \frac{|H|}{ψ(1)} \frac{J(ψ)(1)}{|G|}e_{ψ};

(d) I_{μ}^{0} : Z((KH)f) → Z((KG)e) and R_{ω}^{0} : Z((KG)e) → Z((KH)f) are K-linear inverse maps;

(e) I_{μ}^{0} : Z((OH)f) → Z((OG)e) and R_{ω}^{0} : Z((OG)e) → Z((OH)f) are O-linear inverse maps; (f) σ_{μ} = R_{ω}^{0}(e)I_{μ}^{0} : Z((KH)f) → Z((KG)e) is a K-linear map such

that σ_{μ}(e_{ψ}) = e_{J(ψ)} for all ψ ∈ I(f) and

τ_{ω} = I_{μ}^{0}(f)R_{ω}^{0} : Z((KG)e) → Z(KHf)

is a K-linear map such that τ_{ω}(e_{J(ψ)}) = e_{ψ} for all ψ ∈ I(f). Thus σ_{μ} and t_{ω} are inverse K-algebra isomorphisms; and

(g) σ_{μ}: Z((OH)f) → Z((OG)e) and I_{ω}: Z(OGe) → Z((OH)f) are inverse O-algebra isomorphisms.

Remark 1.13. Let R ∈ {O, K}. Then the following diagram of Rlinear isomorphisms commutes:

\xymatrix{fCF (H, R) = CF (H, f, R) \ar[r]^{A_{H}} \ar[d]_{I_{μ}} & Z((RH)f) = Z(RH)f \ar[d]_{I_{μ}^{0}} \\ eCF (G, R) = CF (G, e, R) \ar[r]^{A_{G}} \ar[d]_{R_{ω}} & Z((RG)e) = Z(RG)e \ar[d]_{R_{ω}^{0}} \\ fCF (H, R) = CF (H, f, R) \ar[r]^{A_{H}} & Z((RH)f) = Z(RH)f}

By Theorem 1.12(g), there is an α ∈ Z((OH)f) such that

I_{μ}^{0}(αR_{ω}^{0}(e)) = I_{μ}^{0}(f).

Thus αR_{ω}^{0}(e) = f and R_{ω}^{0}(e) is an invertible element of Z((OH)f). Let β ∈ Bℓ((OH)f) and let ψ ∈ I(β). Then ω_{ψ}: Z(OH) → O is an Oalgebra homomorphism such that ω_{ψ}(f) = 1. Thus ω_{ψ}(α)ω_{ψ}(R_{ω}^{0}(e)) =

1 where ω_{ψ}(R_{ω}^{0}(e)) = \overline{ϵ}_{ψ} \frac{|H|}{ψ(1)} \frac{J(ψ)(1)}{|G|} ∈ O^{×}. Similarly, since J(ψ) ∈

σ_{μ}(β) ∈ Bℓ((OG)e), we have ϵ_{ψ} \frac{|G|}{J(ψ)(1)} \frac{ψ(1)}{|G|} ∈ O^{×}.

Lemma 1.14. Let β ∈ Bℓ((OH)f) and let ψ ∈ I(β). Then: (a) β and σ_{μ}(β) have the same defect;

(b) ψ and J(ψ) have the same height; and

(c) if ψ^{∗} ∈ I(β) also, then

\overline{ϵ}_{ψ} \frac{|H|}{ψ(1)} \frac{\overline{J}(ψ)(1)}{|G|} ≡ \overline{ϵ}_{ψ^{∗}} \frac{|H|}{ψ^{∗}(1)} \frac{(J(ψ^{∗})(1)}{|G|} (mod M)

Corollary 1.15. Let β ∈ Bℓ((OH)f), so that σ_{μ}(β) ∈ Bℓ((OG)e). Then:

(a) if ψ ∈ I(β), then J(ψ) ∈ I(σ_{μ}(β)); and

(b) if φ ∈ I Br(β) and Φ_{φ} is the associated indecomposable projective character of φ, then I_{μ}(φ) ∈ CF_{p′}(G, σ_{μ}(β), O) and I_{μ}(Φ_{φ}) ∈ CF_{p′}^{pr}(G, σ_{μ}(β), O).

CONSEQUENCES OF A PERFECT ISOMETRY 9

From the above, we clearly have: Corollary 1.16. All of the conclusions of [1, Théorème 1.5 and Lemme 1.6] hold for f and e.

2. Required Proofs

A Proof of Lemma 1.3. Clearly

⨁_{φ∈I Bℓ(G)}(Oφ) ≤ CF_{p′}(G, O) = ⨁_{\begin{matrix}𝒞∈ccl(O) \\ 𝒞⊆G_{p′} \end{matrix}}(Oδ_{𝒞})

≤ CF_{p′}(G, K) = ⨁_{φ∈I Br(e)}(Kφ).

Fix 𝒞 ∈ ccl(G) with 𝒞 ⊆ G_{p′} and let g ∈ 𝒞. Here δ_{𝒞} = ∑_{φ∈I Br(G)} k_{φ}φ for unique k_{φ} ∈ K for all φ ∈ I Br((G). Thus k_{φ} = [Φ_{φ}, δ_{𝒞}]_{G} =

\frac{1}{|G|}Φ_{φ}(g)|𝒞| = \frac{Φ_{φ}(g)}{|C_{G}(g)|} ∈ O and we are done.

A Proof of Lemma 1.5. Let π ∈ CF (G, e, K). If 𝒞 ∈ ccl(G) and g ∈ 𝒞, then [π, δ_{𝒞}]_{G} = \frac{π(g)}{|C_{G}(g)|}. Thus (b), (c) and (d) are equivalent and (b) implies (a). Assume (a) and let α ∈ CF (G, O). Then α = eα+(1−e)α and so [π, α]_{G} = [π, eα]_{G} ∈ O. Thus (b) holds and we are done. A Proof of Lemma 1.6. Clearly ⨁_{φ∈I Br(e)}(OΦ_{φ}) ≤ CF_{p′}^{pr}(G, e, O). Let π ∈ CF_{p′}^{pr}(G, e, O). Thus π = ∑_{φ∈I Br(e)} k_{φ}Φ_{φ} for unique k_{φ} ∈ K for each φ ∈ I Br(e). Consequently for φ ∈ I Br(e), [π, φ]_{G} = k_{φ} ∈ O and we are done.

A Proof of Lemma 1.7. Here CF (H, f, O) = f ( ⨁_{𝒞∈ccl(H)}(Oδ_{𝒞})). Let

h ∈ 𝒞 ∈ ccl(H) and let g ∈ G. Then I_{μ}(δ_{𝒞})(g) = \frac{1}{|H|}μ(g, h^{−1})|𝒞| =

\frac{μ(g, h^{−1})}{|C_{G}(h^{−1})|} and our proof is complete.

A Proof of Lemma 1.8. Let g ∈ G and h ∈ 𝒞 ∈ ccl(G) with 𝒞 ⊆ H_{p′}. Then I_{μ}(δ_{p})(g) = \frac{α(g, h^{−1})}{|C_{H}(h^{−1})|}. Assume (a) and μ(g, h^{−1}) ≠ 0. Then g ∈ G_{p′} and (b) holds. The converse is clear and we are done.

A Proof of Theorem 1.10. Here I_{μ}(CF (H, f, O)) ≤ CF (G, e, O). Thus CF (H, f, O) ≤ R_{ω}(CF (G, e, O)) ≤ CF (H, f, O) and (a) holds. Similarly (b) follows and (c) is a consequence of (a) and (b). Let π ∈ CF^{pr}(H, f, O) and let α ∈ CF (G, e, O). Then

[I_{μ}(π), α]_{G} = [π, R_{ω}(α)]_{H} ∈ O

and so I_{μ}(CF^{pr}(H, f, O)) ≤ CF^{pr}(G, e, O). Thus (d) is proved, (e) follows from (c) and (d) and we are done.

A Proof of Theorem 1.12. Here (a) is clear and if h ∈ H, I_{μ}^{0}(h) =

∑_{g∈G} \frac{1}{|H|} ∑_{ψ∈I(f)}(ϵ_{ψ}ψ(h)J(ψ)(g^{−1})g) = \frac{|G|}{|H|} ∑_{ψ∈I(f)} (\frac{ϵ_{ψ}ψ(h)}{J(ψ)(1)}e_{J(ψ)}). Thus

(b) is proved and (c) follows similarly. Clearly (c) implies (d). If h ∈ 𝒞 ∈ ccl(H), then

I_{μ}^{0}(∑_{u∈𝒞} u) = \frac{1}{|H|} ∑_{g∈G}(μ(g^{−1}, h)|𝒞|g)

= ∑_{g∈G} (\frac{μ(g^{−1}, h)}{|C_{H}(h)|}g) ∈ Z((OG)e).

Thus (e), (f) and (g) hold and we are done.

A Proof of Lemma 1.14. Here (a) and (b) follow as on [1, page 65] and, since ω_{ψ}(α) ≡ ω_{ψ^{∗}}(α) (mod M), (c) follows.

A Proof of Corollary 1.15. Let ψ ∈ I(β). Then

A_{H}(ψ) = A_{H}(βψ) = βA_{H}(ψ) = \frac{|H|}{ψ(1)}e_{ψ} = β \frac{|H|}{ψ(1)}e_{ψ}.

Thus

σ_{μ}(A_{H}(ψ)) = σ_{μ}(β)σ_{μ}(A_{H}(ψ)) = \frac{|H|}{ψ(1)}e_{J(ψ)} = σ_{μ}(β)\frac{|H|}{ψ(1)}e_{J(ψ)}.

Thus σ_{μ}(β)e_{J(ψ)} = e_{J(ψ)} and (a) follows. Let φ ∈ I Br(β). Thus

I_{μ}(Φ_{φ}) ∈ CF_{p′}^{pr}(G, σ_{μ}(β), O) and Φ_{φ} = ∑_{ψ∈I Br(β)} c_{ψ}ψ for unique c_{ψ} ∈ Z

with c_{ψ} ≥ 0 for all ψ ∈ I Br(β). Since the Cartan matrix of β is invertible, I_{μ}(φ) ∈ CF_{p′}(G, e, O) ∩ CF_{p′}(G, σ_{μ}(β), K) = CF_{p′}(G, σ_{μ}(β), O) and we are done.

References

1. M. Broué, Isométries parfaites, Types de blocs, catégories dérivées, Astérisque 181–182 (1990), 61–92.

2. W. Feit, The representation theory of finite groups, North-Holland, Amsterdam, 3. I. M. Isaac, Character theory of finite groups, Academic Press, New York, 1970.

Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045

  E-mail address: harris@math.uic.edu