Dade's ordinary conjecture implies the Alperin-McKay conjecture

We show that Dade's ordinary conjecture implies the Alperin-McKay conjecture. We remark that some of the methods can be used to identify a canonical height zero character in a nilpotent block.

The following result is a special case of a result due to Murai; we include a proof for convenience.
Lemma 3 (cf. [9,Theorem 4.4]). Let G be a finite group, B be a block of OG, and P a defect group of B. Let Z be a central p-subgroup of G and let η be an irreducible character of Z such that Irr 0 (B|η) = ∅. Then η extends to P .
Proof. By replacing K by a suitable finite extension we may assume that K is a splitting field for all subgroups of G. Let i ∈ B P be a source idempotent of B and let V be a KG-module affording an element of Irr 0 (B|η). Then n := dim K (iV ) is prime to p. Since i commutes with P , iV is a KP -module via x · iv = ixv, where x ∈ P, v ∈ V . Let ρ : P → GL n (K) be a corresponding representation and let δ : P → K × be the determinantal character of ρ. Then δ| Z = η n . The result follows since n is prime to p. Lemma 4. Let G be a finite group, let B be a block of OG with a defect group P , and let Z be a central p-subgroup of G. Then | Irr 0 (B)| equals the product of | Irr 0 (B|1 Z )| with the number of distinct linear characters η of Z which extend to P .
Proof. Let F = F (P,e P ) (G, B) be the fusion system of B with respect to a maximal B-Brauer pair (P, e P ), and let η be a linear character of Z which extends to P . Since Z ≤ Z(F ), by Lemma 2 there exists a linear characterη of P such thatη| Z = η and foc(F ) ≤ Ker(η). By the properties of the Broué-Puig * -construction [1], [16] the map χ →η * χ is a bijection between Irr 0 (B|1 Z ) and Irr 0 (B|η). The result follows by Lemma 3.
Slightly strengthening the terminology in [10], we say that a pair (G, B) consisting of a finite group G and a block B of OG is a minimal counterexample to the Alperin-McKay conjecture if B is a counterexample to the Alperin-McKay conjecture and if G is such that first |G : Z(G)| is smallest possible and then |G| is smallest possible. Let F be a saturated fusion system on a finite p-group P , and let C be a full subcategory of F which is upwardly closed; that is, if Q, R are subgroups of P such that Q belongs to C and if Hom F (Q, R) is nonempty, then also R belongs to C. Drawing upon notation and facts from [7, §5], S ⊳ (C) is the category having as objects nonempty chains σ = Q 0 < Q 1 < · · · < Q m of subgroups Q i of P belonging to C such that m ≥ 0 and Q i is normal in Q m , for 0 ≤ i ≤ m. Morphisms in S ⊳ (C) are given by certain 'obvious' commutative diagrams of morphisms in F ; see [7, 2.1, 4.1] for details. With this notation, the length of of a chain σ in S ⊳ (C) is the integer |σ| = m. The chain σ is called fully normalised if Q 0 is fully F -normalised and if either m = 0 or the chain σ ≥1 = Q 1 < Q 2 < · · · < Q m is fully N F (Q 0 )-normalised. Every chain in S ⊳ (C) is isomorphic (in the category S ⊳ (C)) to a fully normalised chain. There is an involution n on the set of fully normalised chains which fixes the chain of length zero P and which sends any other fully normalised chain σ to a fully normalised chain n(σ) of length |σ| ± 1. This involution is defined as follows. If σ = P , then set n(σ) = σ. If σ = Q 0 < Q 1 < · · · < Q m is a fully normalised chain different from P such that Q m = N P (σ), then define σ by removing the last term Q m ; if Q m < N P (σ), then define σ by adding N P (σ) as last term to the chain. Then n(σ) is fully normalised, and n(n(σ)) = σ. Denote by [S ⊳ (C)] the partially ordered set of isomorphism classes of chains in S ⊳ (C), and for each chain σ by [σ] its isomorphism class. We have a partition Proof. The hypothesis on f implies that the contributions from chains in B cancel those from chains in n(B), whence the result. Let C be the full subcategory of F consisting of all nontrivial subgroups of P . We briefly describe the standard translation process between chains in a fusion system of a block and the associated chains of Brauer pairs. The map sending a chain σ = Q 0 < Q 1 < · · · < Q m in S ⊳ (C) to the unique chain of nontrivial B-Brauer pairs τ = (Q 0 , e 0 ) < (Q 1 , e 1 ) < · · · < (Q m , e m ) contained in (P, e) induces a bijection between isomorphism classes of chains in S ⊳ (C) and the set of G-conjugacy classes of normal chains of nontrivial B-Brauer pairs (cf. [7, 2.5]). If σ is fully normalised, then the corresponding chain of Brauer pairs τ = (Q 0 , e 0 ) < (Q 1 , e 1 ) < · · · < (Q m , e m ) has the property that e τ = e m remains a block of N G (τ ), and by [7, 5.14], N P (σ) = N P (τ ) is a defect group of e τ as a block of N G (τ ). Denote by n(τ ) the chain of Brauer pairs corresponding to n(σ).
The passage between formulations in terms of normalisers of chains of Brauer pairs rather than normalisers of chains of p-subgroups is well-known; see e.g. [6, 4.5], [15]. If |n(σ)| = |σ|+1, then, setting H = N G (τ ), we have N G (n(τ )) = N H (N P (τ ), e n(τ ) ); that is, (N P (τ ), e n(τ ) ) is a maximal (H, e τ )-Brauer pair. By the assumptions, the Alperin-McKay conjecture holds for the block e τ of H. This translates to the equality f ([σ]) = f ([n(σ)]). That is, the function f satisfies the hypotheses of Lemma 6. Thus the above alternating sum is equal to f ([P ]), which by definition is k d (N G (P, e), e), and thus the Alperin-McKay conjecture holds for B.
Theorem 1 follows now immediately from combining Propositions 5 and 7.
Remark 8. By work of Dade [2] and Okuyama and Wajima [12], the Alperin-McKay conjecture holds for blocks of finite p-solvable groups. G. R. Robinson pointed out that Proposition 5 yields another short proof of this fact.
Remark 9. Let G be a finite group, B a block algebra of OG, (P, e P ) a maximal (G, B)-Brauer pair with associated fusion system F on P , and let Z be a central p-subgroup of G. Let η be a linear character of Z, and suppose that η extends to a linear characterη of P satisfying foc(F ) ≤ Ker(η). The proof of Lemma 4 is based on the fact that the * -construction χ →η * χ yields a bijection Irr(B|1 Z ) → Irr(B|η).
There is some slightly more structural background to this. For χ ∈ Irr(B), denote by e(χ) the corresponding central primitive idempotent in K ⊗ O B. Set We conclude this note with an observation regarding canonical height zero characters in nilpotent blocks, based in part on some of the above methods.
Let G be a finite group, B a block algebra of OG, P a defect group of B and i ∈ B P a source idempotent of B. Denote by F the fusion system of B on P determined by the choice of i. Suppose that K is a splitting field for all subgroups of G. Proposition 10. With the notation above, let χ ∈ Irr(B) and η ∈ Irr(P/foc(P )). Regard η as a linear character of P . We have ∆ η * χ,P,i = η χ(i) ∆ χ,P,i .
Proof. The statement makes sense as the value of χ on an idempotent is a positive integer. Let V be an O-free OG-module affording χ. By [8, Theorem 1.1] there exists an O-algebra automorphism α of B such that the module V α (obtained from twisting V by α) affords η * χ and such that α(ui) = η(u)ui for all u ∈ P . Since in particular α(i) = i, it follows that ∆ V α ,P,i (u) = ∆ V.P,i (η(u)u) for all u ∈ P . The result follows as rank O (iV ) = χ(i).
Denote by Irr ′ (B) the set of all χ ∈ Irr(B) such that ∆ χ,P,i is the trivial map (sending all elements in P to 1). Set Irr ′ 0 (B) = Irr ′ (B) ∩ Irr 0 (B). The maximal local pointed groups on B are G-conjugate. Thus if P ′ is any other defect group of B and i ′ ∈ B P ′ a source idempotent, then there exist g ∈ G and c ∈ (B P ′ ) × such that P ′ = gP g −1 and i ′ = cgig −1 c −1 . Therefore the map ∆ V,P,i is trivial if and only if the map ∆ V,P ′ ,i ′ is trivial, and hence the sets Irr ′ (B) and Irr ′ 0 (B) are independent of the choice of P and i. The following is immediate.
Proposition 11. The sets Irr ′ (B) and Irr ′ 0 (B) are invariant under any automorphism of G which stabilises B.
The next result shows that if B is nilpotent, then Irr ′ 0 (B) consists of a single element.
Proposition 12. Suppose that B is nilpotent. Then | Irr ′ 0 (B)| = 1. Moreover, if p is odd, then the unique element of Irr ′ 0 (B) is the unique p-rational height zero character in B.
Suppose that p is odd. Let χ 0 be the unique p-rational character in Irr 0 (B). Let W (k) be the ring of Witt vectors in O. By the structure theory of nilpotent blocks (see [14]) there exists a W (k)G-module V affording χ 0 . Since the source idempotent i can be chosen to be in W (k)G, we have that ∆ χ,P,i takes values in W (k). Since p is odd, it follows that the trivial character of P is the unique linear character of P which takes values in W (k).