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.


Theorem 1. If Dade's ordinary conjecture holds for all p-blocks of finite groups, then the Alperin-McKay conjecture holds for all p-blocks of finite groups.
The proof combines arguments from Sambale [17] and formal properties of chains of subgroups in fusion systems from [7]. Let (K, O, k) be a p-modular system. We assume that k is algebraically closed, and letK be an algebraic closure of K. By a character of a finite group, we will mean aK-valued character. For a finite group G and a block B of OG, let Irr(B) denote the set of irreducible characters of G in the block B, and let Irr 0 (B) denote the set of irreducible height zero characters of G in B. For a central p-subgroup Z of G and a character η of Z, let Irr 0 (B|η) denote the subset of Irr 0 (B) consisting of Proof. By a result of Murai [10], we have that Z := O p (G) is central in G. Let P be a defect group of B and let C be the block of ON G (P ) in Brauer correspondence with B. By Lemma 4, | Irr 0 (B)| = | Irr 0 (C)| if and only if | Irr 0 (B)| = | Irr 0 (C)| whereB (resp.C) is the block of OG/Z (resp. ON G (P )/Z) dominated by B (resp. C). The result follows since N G/Z (P/Z) = N G (P )/Z andB andC are in Brauer correspondence.
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 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 where B is the set of isomorphism classes of fully normalised chains σ satisfying |n(σ)| = |σ| + 1. The following Lemma is a very special case of a functor cohomological statement [7,Theorem 5.11].

Lemma 6. With the notation above, let f : [S (C)] → Z be a function on the set of isomorphism classes of chains in
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).
Let  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. [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 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 Δ η * χ,P,i = η χ(i) Δ 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 Δ V α ,P,i (u) = Δ 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 Δ 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