arXiv:1802.05367v1 [math.RT] 15 Feb 2018

DADE’S ORDINARY CONJECTURE IMPLIES THE
ALPERIN-MCKAY CONJECTURE
RADHA KESSAR AND MARKUS LINCKELMANN
Abstract. We show that Dade’s ordinary conjecture implies the Alperin-McKay
conjecture.

Dade proved in [3] that his projective conjecture [3, 15.5] implies the AlperinMcKay conjecture. Navarro showed in [9, Theorem 9.27] that the group version of
Dade’s ordinary conjecture implies the McKay conjecture. We show here that Dade’s
ordinary conjecture [2, 6.3] implies the Alperin-McKay conjecture. Let p be a prime
number.
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 [12] and formal properties of chains
of subgroups in fusion systems from [6]. Let (K, O, k) be a p-modular system. We
assume that k is algebraically closed and that K is a splitting field for all finite groups
considered in the paper. For a finite group G, a block B of OG, let Irr0 (B) denote the
set of K-valued irreducible height zero characters of G in the block B. For a central
p-subgroup Z of G and a linear K-valued character η of Z, let Irr0 (B|η) denote the
subset of Irr0 (B) consisting of those characters which cover the character η. The
following lemma is implicit in [12].
Lemma 2. Let P be a finite group, let F be a saturated fusion system on P and let
Z ≤ Z(F ). Suppose that η is a linear character of P . There exists a linear character
η ∗ of P such that η ∗ |Z = η|Z and foc(F ) ≤ Ker(η ∗ ).
Proof. First consider the case that η|Z is faithful. Then Z ∩ [P, P ] = 1. Hence
by [4, Lemma 4.3], foc(F ) ∩ Z = 1. The result is now immediate. Now suppose
Z0 = Ker(η|Z ) and let F̄ = F /Z0. By the previous argument, applied to P/Z0 and F̄ ,
there exists a character η ∗ of P/Z0 such that η ∗ |Z/Z0 = η|Z/Z0 and foc(F̄) ≤ Ker(η ∗ ).
Denote also by η ∗ the pull back of η ∗ to to P . Then η ∗ has the required properties
since foc(F̄ ) = foc(F )Z0 /Z0.

2010 Mathematics Subject Classification. 20C20.
Key words and phrases. Dade Ordinary Conjecture, Alperin-McKay conjecture.
This note was written while the authors were visiting the Mathematical Sciences Research Institute, Berkeley and they would like to thank the Institute for its hospitality. The second author
acknowledges support from EPSRC grant EP/M02525X/1.
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The following result is a special case of a result due to Murai; we include a proof
for convenience.
Lemma 3 (cf. [7, 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 Irr0 (B|η) 6= ∅. Then η extends to P .
Proof. Let i ∈ B P be a source idempotent of B and let V be a KG-module affording
an element of Irr0 (B|η). Then n := dimK (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 → GLn (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 and let B be a block of OG. Let F = F(P,eP ) (G, B)
be the fusion system of B with respect to a maximal B-Brauer pair (P, eP ) and let Z
be a central p-subgroup of G. Then | Irr0 (B)| equals the product of | Irr0 (B|1Z )| with
the number of distinct linear characters η of Z which extend to P .
Proof. Let η be a linear character of Z which extends to P . Since Z ≤ Z(F ), by
Lemma 2 there exists a linear character η ∗ of P such that η ∗ |Z = η and foc(F ) ≤
Ker(η ∗ ). By the properties of the Broué-Puig ∗-construction [1], [11] the map χ 7→
η ∗ ∗ χ is a bijection between Irr(B|1Z ) and Irr(B|η). The result follows by Lemma
3.

Slightly strengthening the terminology in [8], 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 AlperinMcKay 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.
Proposition 5. Let (G, B) be a minimal counterexample to the Alperin-McKay conjecture. Then Op (G) = 1.
Proof. By a result of Murai [8], we have that Z := Op (G) is central in G. Let P be a
defect group of B and let C be the block of ONG (P ) in Brauer correspondence with
B. By Lemma 4, | Irr0 (B)| = | Irr0 (C)| if and only if | Irr0 (B̄)| = | Irr0 (C̄)| where
B̄ (respectively C̄) is the block of OG/Z (respectively ONG (P )/Z) dominated by B
(respectively C). The result follows since NG/Z (P/Z) = NG (P )/Z and B̄ and C̄ 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 HomF (Q, R) is nonempty, then also R belongs to C. The
following notation and facts are from [6, §5]. We denote by S⊳ (C) the category of
nonempty chains σ = Q0 < Q1 < · · · < Qm of subgroups Qi of P belonging to C
such that m ≥ 0 and Qi is normal in Qm , for 0 ≤ i ≤ m. With this notation, the

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length of σ is the integer |σ| = m. The chain σ is called fully normalised if Q0 is
fully F -normalised and if either m = 0 or the chain σ≥1 = Q1 < Q2 < · · · < Qm is
fully NF (Q0 )-normalised. Every chain in S⊳ (C) is isomorphic 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 = Q0 < Q1 < · · · < Qm is a fully normalised chain
different from P such that Qm = NP (σ), then define σ by removing the last term Qm ;
if Qm < NP (σ), then define σ by adding NP (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
[S⊳ (C)] = {[P ]} ∪ B ∪ n(B) ,
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 [6, 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 S⊳ (C) satisfying f ([σ]) = f ([n(σ)]) for any fully
normalised chain σ in S⊳ (C). Then
X
(−1)|σ| f ([σ]) = f ([P ]) .
[σ]∈[S⊳ (C)]

Proof. The hypothesis on f implies that the contributions from chains in B cancel
those from chains in n(B), whence the result.

Proposition 7. Let G be a finite group such that Op (G) = {1}, and let B be a
block of OG with nontrivial defect groups. Suppose that Dade’s ordinary conjecture
holds for B and that the Alperin-McKay conjecture holds for any block of any proper
subgroup of G. Then the Alperin-McKay conjecture holds for the block B.
Proof. Let (P, e) be a maximal B-Brauer pair, and denote by F the associated fusion
system on P . For d a positive integer, denote by kd (G, B) the number of ordinary
irreducible characters in B of defect d. If pd = |P |, then kd (G, B) is the number of
height zero characters, and if pd > |P |, then kd (G, B) = 0.
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 =
Q0 < Q1 < · · · < Qm in S⊳ (C) to the unique chain of nontrivial B-Brauer pairs τ =
(Q0 , e0 ) < (Q1 , e1 ) < · · · < (Qm , em ) 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. [6, 2.5]). If σ is fully normalised, then the

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corresponding chain of Brauer pairs τ = (Q0 , e0 ) < (Q1 , e1 ) < · · · < (Qm , em ) has the
property that eτ = em remains a block of NG (τ ), and by [6, 5.14], NP (σ) = NP (τ ) is
a defect group of eτ as a block of NG (τ ). Denote by n(τ ) the chain of Brauer pairs
corresponding to n(σ).
Fix d > 0 such that pd = |P |. Define a function f on S⊳ (C) by setting
f ([σ]) = kd (NG (τ ), eτ )
for any fully normalised chain σ and corresponding chain τ of Brauer pairs. If NP (σ)
is a proper subgroup of P , then f ([σ]) = 0, and if NP (σ) = P , then f ([σ]) is the
number of height zero characters of the block eτ of NG (τ ). Dade’s ordinary conjecture
for B, reformulated here in terms of chains of Brauer pairs, asserts that kd (G, B) is
equal to the alternating sum
X
(−1)|σ| f ([σ]) .
[σ]∈S⊳ (C)

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. [5, 4.5], [10].
If |n(σ)| = |σ|+1, then, setting H = NG (τ ), we have NG (n(τ )) = NH (NP (τ ), en(τ ) );
that is, (NP (τ ), en(τ ) ) is a maximal (H, eτ )-Brauer pair. By the assumptions, the
Alperin-McKay conjecture holds for the block eτ 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 kd (NG (P, e), e),
and thus the Alperin-McKay conjecture holds for B.

Theorem 1 follows now immediately from combining the Propositions 5 and 7.
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[10] G. R. Robinson, Weight conjectures for ordinary characters, J. Algebra 276 (2004), 761–
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[11] G. R. Robinson, On the focal defect group of a block, characters of height zero, and lower
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Department of Mathematics, City, University of London EC1V 0HB, United Kingdom
E-mail address: radha.kessar.1@city.ac.uk
Department of Mathematics, City, University of London EC1V 0HB, United Kingdom
E-mail address: markus.linckelmann.1@city.ac.uk

