A criterion for the inductive Alperin weight condition
aa r X i v : . [ m a t h . R T ] S e p A CRITERION FOR THE INDUCTIVE ALPERIN WEIGHTCONDITION
JULIAN BROUGH AND BRITTA SP ¨ATH
Abstract.
We give a criterion that simplifies the checking of the inductive Alperin weightcondition for the remaining open cases of simple groups of Lie type (see Theorem 3.3below). It is strongly related in form to the criterion of the second author for the inductiveMcKay conditions (see [Sp¨a12, 2.12]) that has proved very useful. The proof follows froma Clifford theory for weights intrinsically present in the proof of reduction theorems of theAlperin weight conjecture given by Navarro–Tiep and the second author. We also give arelated criterion for the inductive blockwise Alperin weight condition (Theorem 4.5). Introduction
The Alperin weight conjecture and its blockwise version together form one of the deepremaining problems for the representation theory of finite groups. These conjectures aimto count the irreducible modular representations of a group through certain local objects.If G is a finite group and p a prime, then a p - weight of G is a pair ( Q, θ ), where Q is a p -subgroup of G and θ ∈ Irr(N G ( Q ) /Q ) such that θ (1) p = | N G ( Q ) /Q | p . Conjecture 1.1. [Alp87]
Let G be a finite group and p a prime. Then the number ofirreducible p -Brauer characters of G equals the number of G -conjugacy classes of p -weightsof G . The blockwise Alperin weight conjecture forms a refinement of the previous conjectureby taking into account p -blocks of a finite group, which provide a partition of both theirreducible p -Brauer characters and p -weights. Conjecture 1.2. [Alp87]
Let G be a finite group, p a prime and B a p -block of G . Then thenumber of irreducible p -Brauer characters of B equals the number of G -conjugacy classesof p -weights of B . Since the statement of these conjectures, there have been multiple results towards provingthem for large classes of finite groups.In [NT11], Navarro and Tiep presented an approach which reduced the non-blockwiseversion to a certain set of conditions (the inductive AW condition) for every quasi-simplegroup. The second author in [Sp¨a13b] refined the methods of Navarro and Tiep to providea similar reduction for the blockwise version to the so-called inductive BAW condition.Furthermore, these papers established the respective inductive condition for the case ofsimple groups of Lie type for p the defining characteristic. E-mail address : [email protected], [email protected] .The first author thanks the Humbold Foundation for its support.Both authors like to thank the research training group GRK 2240: Algebro-Geometric Methods inAlgebra, Arithmetic and Topology , funded by the DFG.
The inductive AW condition and BAW condition bear a strong resemblence to the induc-tive condition for the McKay conjecture and its blockwise counterpart the Alperin–McKayconjecture [IMN07, Sp¨a13a]. Moreover, the second author has verified the inductive McKaycondition for most of the families of groups of Lie type, using as a key ingredient Propo-sition 2.12 of [Sp¨a12]. We propose here an analogue of this statement for the inductiveAlperin weight condition, see Theorem 3.3. Mimicking ideas of the reduction theoremsgiven in [NT11] and [Sp¨a13b] we develop a kind of Clifford theory for weights between agroup and a normal subgroup (see also [C13] for an early use of projective representationsin that context). This uses in an essential way properties proven in [NS14b] about theDade–Glauberman–Nagao correspondence.During the preparation of this note various authors used our criterion already verify-ing the inductive Alperin weight condition and the inductive blockwise Alperin weightcondition for families of simple groups of Lie type, see [FLZ20a, Li20, FLZ20b, FM20].
Acknowledgement:
We thank Marc Cabanes and Lucas Ruhstorfer for various insight-ful conversations. We are also indebted to Conghui Li and Gunter Malle for a thoroughreading of an earlier version of this paper.2.
Preliminaries
We start by recalling some results from [NT11] and [Sp¨a13b] on weights in general. Wealso recall the so-called Dade-Glauberman-Nagao correspondence (written as the DGN-correspondence). In the next part of this section we give some Clifford theory-like resultsfor weights and conclude this section by highlighting some consequences of those results.2.A.
Recall.
We use in general the notation of [Nav98] for p -blocks, characters and p -Brauer characters. We assume a prime p is fixed throughout the paper. Notation 2.1.
Let G be a finite group. Then Irr( G ) denotes the set of irreducible charac-ters of G , while Lin( G ) := { χ ∈ Irr( G ) | χ (1) = 1 } is seen here as a multiplicative group.Let F be the field of characteristic p defined as in [Nav98, p. 87]. The set of irreducible p -Brauer characters of G coming from representations over F is denoted by IBr( G ) (defined on G := G p ′ , the set of p -regular elements of G ), while LinBr( G ) := { ϕ ∈ IBr( G ) | ϕ (1) = 1 } .Let Bl p ( G ) denote the set of p -blocks of G . The p -block containing a given θ ∈ IBr( G ) isdenoted by bl( θ ) ∈ Bl p ( G ). Lastly dz( G ) denotes the set of p -defect zero characters of G ,that is dz( G ) := { χ ∈ Irr( G ) | χ (1) p = | G | p } . The map χ χ induces a bijection between Lin( G ) p ′ and LinBr( G ), where χ denotesthe restriction of χ to G . Notation 2.2.
Let N ⊳ G and χ ∈ Irr( G ). ThenIrr( N | χ ) := { η ∈ Irr( N ) | h χ N , η i N = 0 } . Similarly for η ∈ Irr( N ), one defines Irr( G | η ) := { χ ∈ Irr( G ) | h χ N , η i N = 0 } . DefineIBr( N | ϕ ) and IBr( G | θ ) for ϕ ∈ IBr( G ), θ ∈ IBr( N ), in an analogous way. Definition 2.3.
Let G be a finite group and p a prime. One denotes by Alp ( G ) the setof all p -weights of G , that is pairs ( Q, θ ) with θ ∈ dz(N G ( Q ) /Q ) for Q a p -subgroup of G .The G -orbit of ( Q, θ ) is denoted by (
Q, θ ) and Alp( G ) := Alp ( G ) / ∼ G . CRITERION FOR THE INDUCTIVE ALPERIN WEIGHT CONDITION 3
For ν ∈ Lin(Z( G )) p ′ we denote by Alp ( G | ν ) the pairs ( Q, θ ) ∈ Alp ( G ), such that θ lifts to a character in Irr(N G ( Q ) | ν ). Note that this passes to the G -orbit and so Alp( G | ν )is defined analagously. Lemma 2.4.
Let G be a finite group. Then Lin( G ) p ′ (and therefore LinBr( G ) ) acts on Alp( G ) by µ. ( Q, θ ) = (
Q, θ.µ ) where µ ∈ Lin(N G ( Q ) /Q ) is the restriction of µ ∈ Lin( G ) p ′ to N G ( Q ) seen as a linear character with Q in its kernel.Proof. If Q is any p -subgroup, then Q ≤ ker( µ ) as µ has p ′ -order. Thus for ( Q, θ ) ∈ Alp ( G ) the restriction of µ to N G ( Q ) passes to the quotient N G ( Q ) /Q , say µ N G ( Q ) /Q .Hence µ · ( Q, θ ) := (
Q, θµ N G ( Q ) /Q ) is also a weight in Alp ( G ). The lemma now follows asthis action commutes with G -conjugation. (cid:3) Clifford theory for weights.
Let us now recall a crucial ingredient, the Dade–Glauberman–Nagao correspondence (see [NS14a]).The context is as follows. Let K ⊳ G , M/K a p -subgroup of G/K and b a G -invariantblock of K with defect group D ≤ Z( M ). Let D be a defect group of the unique block B of M covering b , so that M = KD and K ∩ D = D . Let B ′ ∈ Bl p (N M ( D )) be theBrauer correspondent of B and b ′ the unique block of N K ( D ) covered by B ′ . Then thereis an N G ( D )-equivariant bijection π D : Irr( b ) −→ Irr( b ′ )according to [NS14b, Thm 5.2]. More precisely [NS14b, Thm 5.2] defines π D only onthe set Irr D ( b ) of D -invariant characters in Irr( b ), but Irr( b ) and Irr( b ′ ) are in naturalcorrespondence with Irr( D ) (for example according to [Nav98, Thm. 9.12]) and henceIrr( b ) = Irr D ( b ) and Irr( b ′ ) = Irr D ( b ′ ). This bijection π D is called the Dade-Glauberman-Nagao (DGN) correspondence .When G ⊳ e G is a normal inclusion of finite groups then the weights of G and e G can berelated via the DGN-correspondence. In this section we establish this relation and considerthe Clifford theory for weights (see Remark 2.8).Now assume G ⊳ e G and ( Q, θ ) ∈ Alp ( G ). Set N := N G ( Q ), e N := N e G ( Q ) θ . Take M/N a p -subgroup of e N /N . For the unique block of
M/Q which covers the block bl( θ )of N/Q fix D a defect group. Then as above D ∩ ( N/Q ) = 1,
M/Q = (
N/Q ) D andN M/Q ( D ) = C N/Q ( D ) D , a direct product. Note that π D ( θ ) ∈ dz(C N/Q ( D )). Definition 2.5.
Let π D ( θ ) be the character in Irr(N M/Q ( D ) /D ) which lifts to π D ( θ ) × D ∈ Irr(N
M/Q ( D )).For D = e Q/Q it follows that M = N e Q and N N ( e Q ) = N G ( e Q ). Thus N M/Q ( D ) /D ∼ =N G ( e Q ) e Q/ e Q is normal in N e G ( e Q ) / e Q . Definition 2.6.
With the above notation, ( e Q, e θ ) ∈ Alp ( e G ) is said to cover ( Q, θ ) ∈ Alp ( G ) if e θ ∈ dz(N e G ( e Q ) / e Q | π D ( θ )). We write Alp ( e G | ( Q, θ )) for the set of thoseweights.Assume that ( e Q, e θ ) covers some ( Q, θ ) ∈ Alp ( G ). By Clifford theory and the equiv-ariance of the map π D we observe: ( e Q, e θ ) covers ( Q, θ ′ ) if and only if ( Q, θ ′ ) lies in theN e G ( e Q )-orbit of ( Q, θ ). Moreover, if g ∈ e G , then ( e Q g , e θ g ) covers ( Q g , θ g ). As an analogue, A CRITERION FOR THE INDUCTIVE ALPERIN WEIGHT CONDITION we say that ( e Q, e θ ) ∈ Alp( e G ) covers ( Q, θ ) ∈ Alp( G ) if ( e Q, e θ ) ∈ Alp ( e G | ( Q, θ ) g ) for some g ∈ e G . Lemma 2.7.
Let G ⊳ e G . The elements in Alp( G ) covered by ( e Q, e θ ) ∈ Alp( e G ) form anon-empty e G -orbit.Remark . With the notation from Definition 2.6 this implies a Clifford-like partitioningof weights: Alp( e G ) = G ( Q,θ ) Alp( e G | ( Q, θ )) , where ( Q, θ ) runs through a e G -transversal in Alp( G ).It turns out that the sets Alp( e G | ( Q, θ )) are non-empty (see Corollary 2.12 below).We require an application of a result from [NT11] about the consequences of DGN-correspondence. Let us first recall the following notation:
Notation 2.9. If G ⊳ e G and χ ∈ Irr( G ) we setrdz( e G | χ ) := ( e χ ∈ Irr( e G | χ ) (cid:12)(cid:12) (cid:18) e χ (1) χ (1) (cid:19) p = | e G/G | p ) , the set of so-called relative p -defect zero characters. Theorem 2.10.
Let G ⊳ e G , ( Q, θ ) ∈ Alp ( G ) , N := N G ( Q ) , e N := N e G ( Q ) θ and let M/N be a p -subgroup of e N /N . Set e Q/Q to be a defect group of the unique block of
M/Q covering bl ( θ ) and π e Q/Q ( θ ) ∈ Irr(N
N/Q ( e Q/Q )) the DGN-correspondent.There exists a N G ( M ) -invariant extension b θ of θ to M/Q and a
Lin(N e N ( M ) /M ) - equi-variant bijection ∆ θ : rdz(N e N ( M ) | b θ ) −→ dz(N e G ( e Q ) / e Q | π e Q/Q ( θ )) , where bl( ϕ ) = bl( ϕ ′ ) N e N ( M ) for every ϕ ∈ rdz(N e N ( M ) | b θ ) , whenever ϕ ′ ∈ Irr(N e G ( e Q )) is thelift of ∆ θ ( ϕ ) .Proof. This is essentially [NT11, Thm 4.2] and [Sp¨a13b, Thm 3.8]. Note that for thestatement about block induction we apply [Sp¨a13b, Cor. 2.11].By [Sp¨a13b, Thm 3.8] there is a bjiection∆ θ : rdz(N e N ( M ) /Q | b θ ) −→ dz(N N e N ( M ) ( e Q ) / e Q | π e Q/Q ( θ ))However N e G ( e Q ) ≤ N e G ( M ) ∩ N e G ( Q ) and thusN N e N ( M ) ( e Q ) / e Q = (cid:16) N N e G ( M ) ∩ N e G ( Q ) ( e Q ) / e Q (cid:17) θ = N e G ( e Q ) θ / e Q. Both N G ( Q ) and N G ( e Q ) are N e G ( e Q )-stable and thus the equivariance of the DGN-correspondence (see [NS14b, 5.2]) shows thatN e G ( e Q ) θ = N e G ( e Q ) π e Q/Q ( θ ) = N e G ( e Q ) π e Q/Q ( θ ) Hence Clifford correspondence, given by character induction, provides a bijectiondz (cid:16) N N e N ( M ) ( e Q ) / e Q | π e Q/Q ( θ ) (cid:17) ↔ dz (cid:16) N e G ( e Q ) / e Q | π e Q/Q ( θ ) (cid:17) . (cid:3) CRITERION FOR THE INDUCTIVE ALPERIN WEIGHT CONDITION 5
Remark . The construction of the (non-unique) map ∆ θ is e G -equivariant in the sensethat (∆ θ ) x gives a map ∆ θ x with the above properties. Corollary 2.12.
Let ( Q, θ ) ∈ Alp ( G ) and G ⊳ e G . Then Alp ( e G | ( Q, θ )) is non-empty.Proof. (See also [Sp¨a13b, 5.10] and its proof.) By the classical construction (see [Isa94,Ch. 11]) the invariant character θ gives rise to a central extension K of N e G ( Q ) θ / N G ( Q ),with the quotient map ǫ : K −→ N e G ( Q ) θ / N G ( Q ) of kernel a central p ′ -group Z , togetherwith a faithful ν ∈ Irr( Z ). Any weight ( M, κ ) of K above ν corresponds via ǫ to some κ ′ ∈ rdz( H | e θ ), where H/ N G ( Q ) = ǫ (N K ( M )). Via ∆ θ from Theorem 2.10, κ ′ defines aweight of e G above ( Q, θ ). (cid:3) Clifford theory of weights in a particular situation.
In this section we assumethat G and e G from before satisfy additional properties, that hold for example when e G/G is cyclic (see Hypothesis 2.13). For applications in Section 3, we describe the structure ofstabilisers of weights and Brauer characters under the following assumptions.
Hypothesis 2.13.
Let G ⊳ e G with abelian e G/G such that(1) every ϕ ∈ IBr( G ) extends to e G ϕ , and(2) for every ( Q, θ ) ∈ Alp( G ) the character θ extends to (N e G ( Q ) /Q ) θ .According to Theorem 2.10, Hypothesis 2.13(2) is equivalent to:(2’) for every ( e Q, e θ ) ∈ Alp( e G ), any θ ∈ IBr(N G ( e Q ) e Q/ e Q | e θ ) extends to (N e G ( e Q ) / e Q ) θ .In [Sp¨a12] the following observation for ordinary characters was exploited: If χ ∈ Irr( G )extends to e G χ , e G/G is abelian and e χ ∈ Irr( e G | χ ), Clifford theory shows that e G χ = \ µ ∈ Irr( e G/G ) e χ ker( µ )and Irr( e G | χ ) is a Lin( e G/G )-orbit in Irr( e G ). We give an analogous description for Brauercharacters. Lemma 2.14.
Let ϕ ∈ IBr( G ) , e ϕ ∈ IBr( e G | ϕ ) and assume Hypothesis 2.13. Set J G ( e ϕ ) := e G ϕ e G p ′ , where e G p ′ /G is the Hall p ′ -subgroup of e G/G . Then(a)
IBr( e G | ϕ ) is a LinBr( e G/G ) -orbit, and(b) e G ϕ = \ µ ∈ LinBr( e G/G ) e ϕ ker( µ ) ∩ J G ( e ϕ ) . Proof.
Let e ϕ ∈ IBr( e G ϕ | ϕ ) with e ϕ e G = e ϕ . Then e ϕ is an extension of ϕ by Hypothesis 2.13.According to [Nav98, 8.20] we have IBr( e G | ϕ ) = { ( e ϕ µ ′ ) e G | µ ′ ∈ IBr( e G ϕ /G ) } . This provespart (a).Because of e ϕ e G = e ϕ we see that e G ϕ ≤ T µ ∈ LinBr( e G/G ) e ϕ ker( µ ). By definition J G ( e ϕ ) satisfies e G ϕ ≤ J G ( e ϕ ) and p ∤ | J G ( e ϕ ) : e G ϕ | . Since | T µ ∈ LinBr( e G/G ) e ϕ ker( µ ) : e G ϕ | is a power of p , thisimplies (b). (cid:3) A CRITERION FOR THE INDUCTIVE ALPERIN WEIGHT CONDITION
The above statement has an analogue for a weight (
Q, θ ) ∈ Alp( G ), i.e. e G ( Q,θ ) andAlp( e G | ( Q, θ )) can be described in a similar way.
Lemma 2.15.
Let ( Q, θ ) ∈ Alp( G ) , ( e Q, e θ ) ∈ Alp( e G | ( Q, θ )) and assume Hypothesis 2.13.Set J G (( e Q, e θ )) := e G ( Q,θ ) e G p ′ , where e G p ′ /G is the Hall p ′ -subgroup of e G/G . Denote N :=N G ( Q ) and e N := N e G ( Q ) θ . Then(a) Alp( e G | ( Q, θ )) forms an LinBr( e G/G ) -orbit in Alp( e G ) , and(b) e G ( Q,θ ) = \ µ ∈ LinBr( e G/G ) ( e Q, e θ ) ker( µ ) ∩ J G (( e Q, e θ )) . Proof.
Let M := e QN . Note that e N /N ∼ = G e N /G is abelian and by Hypothesis 2.13 allcharacters of Irr( e N /Q | θ ) are extensions of θ . Let ∆ θ be the map from Theorem 2.10for the extension b θ of θ to M/Q . Then the character e θ corresponds to some character inrdz(N e N ( M ) | b θ ) via the bijection ∆ θ . The set rdz(N e N ( M ) | b θ ) is only non-empty if M/N is the Sylow p -subgroup of e N /N .Note that all extensions of θ to M/Q lie in the same block and have e Q/Q as a de-fect group. Hence every weight in Alp( e G | ( Q, θ )) is of the form ( e Q, e θ ′ ) for some e θ ′ ∈ dz(N e N ( e Q ) / e Q | π e Q/Q ( θ )). The bijection ∆ θ from Theorem 2.10 is Lin( e N /M )-compatibleand rdz(N e N ( M ) | b θ ) forms a Lin( e N /M )-orbit. Accordingly dz(N e N ( e Q ) / e Q | π e Q/Q ( θ )) is aLin( e N /M )-orbit as well. This proves part (a).For part (b) we apply the arguments from Lemma 2.14(b) to e G ( Q,θ ) = G N e G ( Q ) θ = G e N and use that ∆ θ is Lin( e N /M )-compatible. (cid:3)
Lemma 2.16.
Assume G ⊳ e G satisfies Hypothesis 2.13. Then there is a bijection Π : Alp( e G ) / ∼ LinBr( e G/G ) −→ Alp( G ) / ∼ e G/G , which sends each LinBr( e G/G ) -orbit containing ( e Q, e θ ) to the e G -orbit of weights of G coveredby ( e Q, e θ ) .Proof. For every (
Q, θ ) ∈ Alp( G ) we define Π by mapping any element of Alp( e G | ( Q, θ ))to the e G -orbit containing ( Q, θ ). According to Lemma 2.7 and Lemma 2.15 this map iswell-defined and injective. Following Corollary 2.12 the map is surjective. (cid:3) The inductive AW condition
The following definition is equivalent to the one stated in [NT11, §
3] (see also [KS16b]),although the version presented here uses the notion of weight, instead of sets IBr( G | Q ). Definition 3.1.
Let p be a prime, S a finite non-abelian simple group, G a p ′ -coveringgroup of S (that is a maximal perfect group of which S is a central quotient by a p ′ -group).We say that the inductive AW condition holds for S and p if the following statementsare satisfied: CRITERION FOR THE INDUCTIVE ALPERIN WEIGHT CONDITION 7 (i) There exists an Aut( G )-equivariant bijectionΩ : IBr( G ) → Alp( G )such that Ω(IBr( G | ν )) = Alp( G | ν ) for every ν ∈ IBr(Z( G )).(ii) For every ϕ ∈ IBr( G ) and ( Q, θ ) = Ω( ϕ ), there exists a finite group A := A ( ϕ, Q )and characters e ϕ ∈ IBr( A ) and e θ ∈ IBr(N A ( Q )) such that(1) The group A satisfies G ⊳ A , A/ C A ( G ) ∼ = Aut( G ) ϕ , C A ( G ) = Z ( A ) and p ∤ | Z ( A ) | .(2) e ϕ ∈ IBr( A ) is an extension of ϕ .(3) e θ ∈ IBr(N A ( Q )) is an extension of Ω( ϕ ) .(4) IBr(Z( A ) | e ϕ ) = IBr(Z( A ) | e θ ). Remark . In [Sp¨a17a] the inductive conditions for some of the local-global conjectureswere rephrased in terms of character triples. In particular, by [Sp¨a17a, Thm 4.3], therequirement on ( ϕ, Ω( ϕ )) in Definition 3.1(ii) is equivalent to showing that( G ⋊ Aut( G ) ϕ , G, ϕ ) ≻ F,c (( G ⋊ Aut( G )) Q,θ , N G ( Q ) , θ )in the notation of [Sp¨a17a, 3.1].If, in addition G ⋊ Aut( G ) ϕ = G ( G ⋊ Aut( G )) ( Q,θ ) then [Sp¨a17a, Proposition 3.7] provesthis property of character triples has an equivalent formulation in terms of projectiverepresentations. Namely, there exists two modular projective representations P and P ′ of G ⋊ Aut( G ) ϕ and ( G ⋊ Aut( G )) Q,θ associated with ϕ and θ in the sense of [Sp¨a17a, § • the factor sets of P and P ′ coincide on ( G ⋊ Aut( G )) Q,θ × ( G ⋊ Aut( G )) Q,θ , and • for x ∈ C G ⋊ Aut( G ) ϕ ( G ) the matrices P ( x ) and P ′ ( x ) are associated with the samescalar.We now state our main result. Theorem 3.3.
Let S be a finite non-abelian simple group and p a prime dividing | S | . Let G be a p ′ -covering group of S . Assume we have a semi-direct product e G ⋊ E , such that thefollowing conditions hold:(i) • G = [ e G, e G ] , • C e G ⋊ E ( G ) = Z( e G ) and e GE/ Z( e G ) ∼ = Aut( G ) by the natural map, • every ϕ ∈ IBr( G ) extends to e G ϕ , • for every ( Q, θ ) ∈ Alp( G ) the character θ extends to (N e G ( Q ) /Q ) θ .(ii) There exists some LinBr( e G/G ) ⋊ E -equivariant bijection e Ω : IBr( e G ) −→ Alp( e G ) such that(1) e Ω(IBr( e G | ν )) = Alp( e G | ν ) for every ν ∈ Lin(Z( e G )) p ′ , and(2) J G ( e ϕ ) = J G ( e Ω( e ϕ )) for every e ϕ ∈ IBr( e G ) (using the notation of Lemmas 2.14and 2.15 above).(iii) For every e ϕ ∈ IBr( e G ) there exists some ϕ ∈ IBr( G | e ϕ ) such that • ( e G ⋊ E ) ϕ = e G ϕ ⋊ E ϕ , and • ϕ extends to G ⋊ E ϕ .(iv) In every e G -orbit on Alp( G ) there exists a weight ( Q, θ ) such that • ( e GE ) ( Q,θ ) = e G ( Q,θ ) E ( Q,θ ) , and A CRITERION FOR THE INDUCTIVE ALPERIN WEIGHT CONDITION • θ extends to ( GE ) Q,θ /Q .Then the inductive AW condition from [NT11, Section 3] holds for S and p . Lemma 3.4.
Let ϕ ∈ IBr( G ) and ( Q, θ ) ∈ Alp( G ) such that the following holds:(i) ( e G ⋊ E ) ϕ = e G ϕ ⋊ E ϕ and ϕ extends to G ⋊ E ϕ and to e G ϕ ,(ii) ( e GE ) ( Q,θ ) = e G ( Q,θ ) E ( Q,θ ) and θ extends to ( G ⋊ E ) Q,θ /Q and to N e G ( Q ) θ /Q ,(iii) ( e GE ) ϕ = ( e GE ) ( Q,θ ) , and(iv) there exist e ϕ ∈ IBr( e G | ϕ ) and ( e Q, e θ ) ∈ Alp( e G | ( Q, θ )) such that there exists forevery x ∈ E ϕ some µ ∈ IBr( e G/G ) with µ · ( e Q, e θ ) = ( e Q, e θ ) x and e ϕ x = e ϕµ, and IBr(Z( e G ) | e ϕ ) = IBr(Z( e G ) | e θ ) .Let Z := ker( ϕ ) ∩ Z( G ) . Then (cid:16) ( e GE ) ϕ /Z, G/Z, ϕ (cid:17) ≻ F,c (cid:16) ( e GE ) Q,θ /Z, N G ( Q ) /Z, θ (cid:17) . Proof.
As in the proof of [Sp¨a12, Lem. 2.13] the assumptions allow the construction ofmodular projective representations P and P ′ of ( e G/ZE ) ϕ and ( e G/ZE ) Q,θ associated withthe Brauer characters ϕ and θ whose factor sets agree on ( e G/ZE ) Q,θ × ( e G/ZE ) Q,θ andsuch that for every x ∈ C ( e G/ZE ) ϕ ( G/Z ) the matrices P ( x ) and P ′ ( x ) are scalar matricesassociated with the same scalar. (cid:3) Proof of Theorem 3.3.
Let A be an LinBr( e G/G ) ⋊ E -transversal in Alp( e G ). By the as-sumption (i) and Lemma 2.15, each LinBr( e G/G )-orbit in Alp( e G ) coincides with a setAlp( e G | ( Q, θ )) for some (
Q, θ ) ∈ Alp( G ). For each weight ( e Q, e θ ) ∈ A we find accordingto Lemma 2.7 a weight ( Q, θ ) of G that is covered by it and by assumption (iv) we canchoose it such that ( e GE ) Q,θ = e G Q,θ ( GE ) Q,θ and θ extends to ( GE ) Q,θ . Let A be the set of( Q, θ ) constructed this way, which forms a e G ⋊ E -transversal in Alp( G ), see Remark 2.11and Lemma 2.15.Since e Ω is LinBr( e G/G ) ⋊ E -equivariant by (ii) the set G := e Ω − ( A ) is a LinBr( e G/G ) ⋊ E -transversal in IBr( e G ). For each e ϕ ∈ G we fix some ϕ ∈ IBr( G | e ϕ ) satisfying Condition(iv) of the theorem and let G denote the set of such ϕ . By standard Clifford theory G isa e G ⋊ E -transversal in IBr( G ).Fix e ϕ ∈ G . Let ( e Q, e θ ) := e Ω( e ϕ ) ∈ A , ϕ ∈ G be covered by e ϕ and ( Q, θ ) ∈ A be coveredby ( e Q, e θ ) ∈ A . Then ϕ and ( Q, θ ) are uniquely defined. We set Ω( ϕ ) = ( Q, θ ).Then according to Lemma 2.14, e G ϕ = (cid:16)T µ ∈ LinBr( e G/G ) e ϕ ker( µ ) (cid:17) ∩ J G ( e ϕ ). Analogously e G ( Q,θ ) = (cid:18)T µ ∈ LinBr( e N/N ) ( e Q, e θ ) ker( µ ) (cid:19) ∩ J G ( e Q , e θ ), see Lemma 2.15. By using the LinBr( e G/G )-equivariance of e Ω and the assumption J G ( e ϕ ) = J G (( e Q , e θ )) we see e G ϕ = e G ( Q,θ ) .Recall ϕ ∈ IBr( G ) with ( e GE ) ϕ = e G ϕ E ϕ . This means that the e G -orbit and the E -orbit of ϕ intersect only at ϕ , thus implying easily E ϕ = E e G -orbit( ϕ ) = E IBr( e G | ϕ ) . CRITERION FOR THE INDUCTIVE ALPERIN WEIGHT CONDITION 9
Similarly for (
Q, θ ) ∈ Alp( G ) we see E ( Q,θ ) = E e G -orbit(( Q,θ )) = E Alp( e G | ( Q,θ )) according to Remark 2.11. Then e Ω maps IBr( e G | ϕ ), the LinBr( e G/G )-orbit containing e ϕ ,to Alp( e G | ( Q, θ )), the LinBr( e G/G )-orbit containing e Ω( e ϕ ). Since e Ω is E -equivariant thisimplies E ϕ = E IBr( e G | ϕ ) = E Alp( e G | ( Q,θ )) = E ( Q,θ ) . This proves ( e G ⋊ E ) ϕ = ( e G ⋊ E ) ( Q,θ ) and hence there exists a unique e G ⋊ E -equivariant bijection Ω : IBr( G ) → Alp( G ) with thegiven values on G . Note that this map is indeed a bijection thanks to Corollary 2.12 andLemma 2.7. The bijection Ω satisfies the condition in 3.1(ii), by giving ordered charactertriples according to Lemma 3.4, which in turn via Remark 3.2 implies 3.1(ii) according tothe Butterfly Theorem [Sp¨a18, 2.16] for Brauer characters. (cid:3) The inductive BAW condition
The inductive BAW condition for a set of blocks.
We now turn to the versionof Alperin’s conjecture with blocks, see Conjecture 1.2 above. All blocks are p -blocks for achosen prime p . Definition 3.2 of [KS16b] provides an inductive BAW condition related toa single block. Notation 4.1.
For a block B ∈ Bl p ( G ), Alp( B ) denotes the set of weights ( Q, θ ) ∈ Alp( G )such that bl( θ ′ ) G = B for θ ′ ∈ Irr(N G ( Q )) the lift of θ . Note that this is well defined asbl( θ ′ g ) G = bl( θ ′ ) G for all g ∈ G . We extend the notation to sets B of blocks of G byAlp( B ) = ∪ B ∈B Alp( B ). Remark . Let ( e Q, e θ ) be a weight of e G and ( Q, θ ) a weight of G covered by ( e Q, e θ ). Set e b tobe the block of N e G ( e Q ) which dominates bl( e θ ), and b the block of N G ( Q ) which dominatesbl( θ ). Then by applying [KS15, Thm B] with the results in [NS14b, Section 5] it followsthat e b e G covers b G . Definition 4.3.
Let p be a prime, S a finite non-abelian simple group, G a p ′ -coveringgroup of S and B ∈ Bl( G ) such that Z ( G ) ∩ ker( ϕ ) is trivial for any ϕ ∈ IBr( B ). Wesay that the inductive BAW condition holds for B if the following statements aresatisfied:(i) There exists an Aut( G ) B -equivariant bijectionΩ B : IBr( B ) → Alp( B ) . (ii) For every ϕ ∈ IBr( B ) and ( Q, θ ) = Ω B ( ϕ ), there exist a finite group A := A ( ϕ, Q )and characters e ϕ ∈ IBr( A ) and e θ ∈ IBr(N A ( Q )) such that(1) The group A satisfies G ⊳ A , A/ C A ( G ) ∼ = Aut( G ) ϕ , C A ( G ) = Z ( A ) and p ∤ | Z ( A ) | ,(2) e ϕ ∈ IBr( A ) is an extension of ϕ ,(3) e θ ∈ IBr(N A ( Q )) is an extension of Ω B ( ϕ ) , (4) for every J with G ≤ J ≤ A the characters e ϕ and e θ satisfybl( e ϕ J ) = bl( e θ N J ( Q ) ) J . Remark . As with the inductive Alperin weight condition, an alternative formulationin terms of character triples and then also with projective representations was providedin [Sp¨a17a]. In particular, (ii) in Definition 4.3 will be verified by showing the followingequivalent property(( G ⋊ Aut( G )) ϕ , G, ϕ ) ≻ F,b (cid:0) ( G ⋊ Aut( G )) Q,θ , N G ( Q ) , θ (cid:1) , see also Theorem 4.4 of [Sp¨a17a]. The order relation on character triples given by “ ≻ F,b ”arises from “ ≻ F,c ” with an additional requirement related to Definition 4.3(ii)(4), see Def-inition 3.2 of [Sp¨a17a]. This relation holds if the projective representations P and P ′ fromRemark 3.2 can be chosen to satisfy additionally the condition described in Remark 4.5 of[Sp¨a18].For the verification one can also consider every block b B of the universal p ′ -covering group b G of S . Let b B ∈ Bl p ( b G ) and Z := Z( b G ) ∩ ker ϕ for ϕ ∈ IBr( b B ). Then the block B of G := b G/Z contained in b B satisfies the inductive BAW condition, if(1) there exists an Aut( b G ) B -equivariant bijection b Ω b B : IBr( b B ) → Alp( b B ),(2) for every b φ ∈ IBr( b B ), and ( b Q, b θ ) = b Ω b B ( φ ) the associated characters φ ∈ IBr( b G/Z )and θ ∈ Irr(N b G/Z ( b Q )) satisfy(( G ⋊ Aut( G )) ϕ , G, ϕ ) ≻ F,b (cid:16) ( G ⋊ Aut( G )) b Q,θ , N G ( b Q ) , θ (cid:17) or equivalently (cid:16) ( b G ⋊ Aut( b G )) b ϕ , b G, b ϕ (cid:17) ≻ F,b (cid:16) ( b G ⋊ Aut( b G )) b Q, b θ , N b G ( b Q ) , b θ (cid:17) . The equivalence of the two relations comes from the fact that Z is a p ′ -group and(the proof of) Corollary 4.5 of [Sp¨a17b].4.B. Criterion for the inductive BAW condition.
In this section we give a criterionfor the inductive BAW condition adapted to simple groups of Lie type. It is clear that thelast assumption on the orbit stabiliser will not hold for all blocks. It mimics the criterionprovided for the inductive Alperin–McKay condition in [BS20].
Theorem 4.5.
Let S be a finite non-abelian simple group and p a prime dividing | S | . Let G be a p ′ -covering group of S and B ⊆
Bl( G ) a e G -invariant subset with ( e GE ) B ≤ ( e GE ) B for all B ∈ B . Assume we have a semi-direct product e G ⋊ E , such that the followingconditions hold:(i) • G = [ e G, e G ] and E is abelian or isomorphic to the direct product of a cyclicgroup and S , • C e G ⋊ E ( G ) = Z( e G ) and e GE/ Z( e G ) ∼ = Aut( G ) B by the natural map, • every ϕ ∈ IBr( B ) extends to e G ϕ , • for every ( Q, θ ) ∈ Alp( B ) the characters θ extends to (N e G ( Q ) /Q ) θ .(ii) Let e B = Bl p ( e G | B ) . There exists a LinBr( e G/G ) ⋊ E -equivariant bijection e Ω : IBr( e B ) −→ Alp( e B ) CRITERION FOR THE INDUCTIVE ALPERIN WEIGHT CONDITION 11 with e Ω( e B ) = Alp( e B ) for all e B ∈ e B and J G ( e ϕ ) = J G ( e Ω( e ϕ )) for every e ϕ ∈ IBr( B ) .(iii) For every e ϕ ∈ IBr( e B ) there exists some ϕ ∈ IBr( G | e ϕ ) such that • ( e G ⋊ E ) ϕ = e G ϕ ⋊ E ϕ , and • every e G -conjugate of ϕ extends to its stabilizer in G ⋊ E .(iv) In every e G -orbit on Alp( B ) there exists a weight ( Q, θ ) such that • ( e GE ) ( Q,θ ) = e G ( Q,θ ) E ( Q,θ ) , and • θ extends to ( GE ) Q,θ /Q .Then the inductive BAW condition holds for all p -blocks B ∈ B with abelian Out( G ) e G -orbit of B . To prove this we use the following variant of Lemma 3.4 where we keep the group-theoretic assumption of the above statement.
Lemma 4.6.
Assume that G , e G and E satisfy the group-theoretic assumptions made inTheorem 4.5. Let ϕ ∈ IBr( G ) and ( Q, θ ) ∈ Alp( G ) such that the following holds:(i) ( e G ⋊ E ) ϕ = e G ϕ ⋊ E ϕ , [( e G ⋊ E ) ϕ , e G ] ≤ e G ϕ , and all e G -conjugates of ϕ extend in G ⋊ E ϕ and e G ϕ .(ii) ( e GE ) ( Q,θ ) = e G ( Q,θ ) E ( Q,θ ) and θ extends to ( G ⋊ E ) Q,θ /Q and to N e G ( Q ) θ /Q .(iii) ( e GE ) ϕ = ( e GE ) ( Q,θ ) .(iv) There exist e ϕ ∈ IBr( e G | ϕ ) and ( e Q, e θ ) ∈ Alp( e G | ( Q, θ )) such that for every x ∈ E ϕ there exists some µ ∈ IBr( e G/G ) with µ · ( e Q, e θ ) = ( e Q, e θ ) x , e ϕ x = e ϕµ and bl( e ϕ ) = bl( e θ ) e G .Let Z := ker( ϕ ) ∩ Z( G ) . Then (cid:16) ( e GE ) ϕ /Z, G/Z, ϕ ′ (cid:17) ≻ F,b (cid:16) ( e GE ) Q,θ /Z, N G ( Q ) /Z, θ (cid:17) , for some e G -conjugate ϕ ′ of ϕ .Proof. Note that as bl( e ϕ ) = bl( e θ ) e G , it follows that e ϕ and e θ lie above the same centralcharacter. Because of [( e G ⋊ E ) ϕ , e G ] ≤ e G ϕ all e G -conjugates have ( e G × E ) ϕ as their stabilizerin e G ⋊ E . Thus by Lemma 3.4, it follows that (cid:16) ( e GE ) ϕ /Z, G/Z, ϕ ′ (cid:17) ≻ F,c (cid:16) ( e GE ) Q,θ /Z, N G ( Q ) /Z, θ (cid:17) for all e G -conjugates ϕ ′ of ϕ . (Here ϕ ′ denotes the character of G/Z associated to ϕ ′ .)Let ( e Q, e θ ) a representative of the weight ( e Q, e θ ) and b θ be an extension of θ to N e G ( Q ) θ /Q such that via induction and a map ∆ θ it corresponds to ( e Q, e θ ). Let ϕ ′ be some e G -conjugateof ϕ such that some extension b ϕ ′ of ϕ ′ to e G ϕ satisfies ( b ϕ ′ ) e G = e ϕ and bl( b ϕ ) = bl( b θ ) e G ϕ , where b θ is an extension of θ to N e G ( Q ) θ /Q . The existence of b θ and b ϕ ′ follows from Clifford theoryand the facts that e G/G is abelian and θ extends to N e G ( Q ) /Q by (ii).This implies (cid:16) e G ϕ /Z, G/Z, ϕ ′ (cid:17) ≻ F,b (cid:16) e G Q,θ /Z, N G ( Q ) /Z, θ (cid:17) for this particular character ϕ ′ . According to [CS15, Lem. 3.2], see also its application inthe proof of [BS20, Thm. 2.4], this implies (cid:16) ( e GE ) ϕ /Z, G/Z, ϕ ′ (cid:17) ≻ F,b (cid:16) ( e GE ) Q,θ /Z, N G ( Q ) /Z, θ (cid:17) for the case of an abelian E .If E is non-abelian, then it is isomorphic to S × C for some finite cyclic group C .When the considerations above are applied to b ϕ ′ and b θ , Lemma 3.2 of [CS15] is replacedby the result [Ruh20, Lem. 6.13] for Brauer characters. Note that in the proof of [Ruh20,Lem. 6.13] ordinary characters can be replaced by Brauer characters without additionalrequirements. Again we obtain (cid:16) ( e GE ) ϕ /Z, G/Z, ϕ ′ (cid:17) ≻ F,b (cid:16) ( e GE ) Q,θ /Z, N G ( Q ) /Z, θ (cid:17) . (cid:3) Proof of Theorem 4.5.
Let e B be the set of blocks of e G covering some block of B . Thenthe set Alp( e B ) consist of all the weights of e G covering some weight of Alp( B ) according toRemark 4.2, and this set is E ′ := E B -stable. We can choose A to be a LinBr( e G/G ) ⋊ E ′ -transversal in Alp( e B ). As in the proof of Theorem 3.3 we choose A to be a set of weightssuch that each weight of G in A is covered by exactly one of A and it satisfies 4.5(iv). Wesee that A is then a e G ⋊ E ′ -transversal in Alp( B ).Like in the proof of Theorem 3.3 let G := e Ω − ( A ). We see that it is a LinBr( e G/G ) ⋊ E ′ -transversal.Let e ϕ ∈ G . Then there exist ( Q, θ ) ∈ A and ( e Q, e θ ) ∈ A such that e Ω( e ϕ ) = ( e Q, e θ ) and( e Q, e θ ) covers ( Q, θ ). As Out( G ) e G − -orbit of B is abelian, [( e G ⋊ E ) ϕ , e G ] ≤ e G ϕ holds for every ϕ ∈ IBr( B ). Hence we can apply Lemma 4.6 and obtain that there is some ϕ ∈ IBr( G | e ϕ )such that (cid:16) ( e GE ) ϕ /Z, G/Z, ϕ (cid:17) ≻ F,b (cid:16) ( e GE ) Q,θ /Z, N G ( Q ) /Z, θ (cid:17) . For each e ϕ ∈ G we choose one character ϕ that way. Let G be the set of characters ϕ thusobtained.We define Ω first on G : for ϕ ∈ G one fixes Ω( ϕ ) to be the unique element in A suchthat e Ω(IBr( e G | ϕ )) = Alp( e G | Ω( ϕ )). As in the proof of Theorem 3.3 the stabilizers of ϕ and Ω( ϕ ) in e G ⋊ E ′ coincide and Ω is well-defined as a e G ⋊ E ′ -equivariant bijectionbetween IBr( B ) and Alp( B ) giving ordered character triples. Since ≻ F,b is preserved viaconjugation, Ω has all the required properties. (cid:3)
In view of the possible applications the requirement 4.5(iii) can be lightened, by onlyrequiring that ϕ extends to its stabilizer in G ⋊ E . When for G , e G and E certain groups arechosen then it is sufficient to require that one character extends to its stabilizer in G ⋊ E .This aspect was implicitly used in the proof of [BS20, Thm. 2.4]. Lemma 4.7.
Let G be some simply connected simple algebraic group G with an F q -structure (where q is a power of a prime that is not assumed to be p ) given by a Frobeniusendomorphism F : G → G and e G an algebraic group coming from a regular embeddingof G and acted on by F . Let E ≤ Aut( G F ) be the subgroup induced by field and graphautomorphisms of e G such that e G F ⋊ E induces the whole automorphism group of G F , (see [GLS98, 2.5.12] for the structure of Aut( G F ) ). CRITERION FOR THE INDUCTIVE ALPERIN WEIGHT CONDITION 13 If B is a e G F -orbit in Bl p ( G F ) with abelian Out( G F ) B and ϕ ∈ Irr( B ) satisfies Condition4.3(iii), then ϕ g satisfies Condition 4.3(iii) for any g ∈ e G F .Proof. Let G := G F and e G := e G F . Since Out( G ) B is abelian, ϕ ′ := ϕ g satisfies( e G ⋊ E ) ϕ = e G ϕ ′ ⋊ E ϕ ′ . We have to show that ϕ ′ extends to GE ϕ ′ = GE ϕ .Without loss of generality it can be assumed that E ϕ is non-cyclic and e G ϕ = e G , inparticular Z ( G ) = 1. This implies that G is of type A n ( q ), D n ( q ) or E ( q ) (all untwisted).Let γ, F ∈ Out( G ) be associated with a graph automorphism γ of G of order 2 and witha field automorphism F , such that E = h F , γ i . Since E is non-cyclic, F is of even order(and so 4 | ( q − q is odd).In type D m , field automorphisms centralise Diag( G ) := e G/ ( G.Z ( e G )) while the graphautomorphisms acts faithfully on it. Since Out( G ) B is abelian, E B only contains fieldautomorphisms and thus is cyclic. In types A n − , D m +1 , E the automorphism γ acts byinversion on Diag( G ). As E ϕ is non-cyclic, it follows that E ϕ = h F i , γ i ≤ E B for somepositive integer i and so Z ( G ) ∼ = Diag( G ) ∼ = C . This excludes type E where | Z ( G ) | = 3,also D m +1 because then | Z ( G ) | = gcd(4 , q −
1) and type A n − with n | Z ( G ) | = gcd( n, q − G is of type A n − with n ≡ | q − L F be the Lang map corresponding to F with G = G F , which commutes with F and γ . Additionally G ⊳ b G := L − F ( Z ( G )). Since F and γ come from endomorphisms of G ,there are automorphisms c F and b γ of b G extending the automorphisms F and γ of G . Fromthe equality e G = G Z ( e G ) and Lang’s theorem, it follows that there is an isomorphism e G/GZ ( e G ) ∼ = b G/GZ ( G )which preserves the conjugation action on G . Moreover, applying the Lang map yieldsisomorphisms b G/G ∼ = Z ( G ) and b G/GZ ( G ) ∼ = Z ( G ) / L F ( Z ( G )) . As G = SL n ( q ) with 4 | q − n ≡ L F ( Z ( G )) does notcontain an element of order 2, since L F acts on Z ( G ) by x x q − . Let x ∈ b G \ GZ ( G ).Then L F ( x ) is the central involution of G and so c F ( x ) x − , b γ ( x ) x − ∈ G . Moreover x induces the non-trivial diagonal automorphism of G and so ϕ g = ϕ x is the only characterof G that is e G -conjugate to ϕ but different from ϕ .It follows that ϕ x extends to G ⋊ E ϕ if and only if ϕ x extends to G h γ i and some extensionis E ϕ -invariant.Take e ϕ an F i -invariant extension of ϕ to G h γ i . Then e ϕ x is an extension of ϕ x to G ⋊ h γ i ∼ = G ⋊ h b γ i since b γ x ∈ G b γ . By definition e ϕ x is ( c F i ) x -invariant. Since ( c F i ) x ∈ G c F i it is also c F i -invariant and thus F i -invariant. This implies that ϕ x extends to GE ϕ = GE ϕ x . (cid:3) References [Alp87] J.-L. Alperin. Weights for finite groups. In
The Arcata Conference on Representations of FiniteGroups , volume 47 of
Proc. Sympos. Pure Math. , pages 369–379. Amer. Math. Soc., Providence,1987. [C13] M. Cabanes Two remarks on the reduction of Alperin’s weight conjecture. Bull. Lond. Math.Soc., 45:895–906, 2013.[BS20] J. Brough and B. Sp¨ath. On the Alperin–McKay conjecture for simple groups of type A.
J.Algebra , 558:221-259, 2020.[CS15] M. Cabanes and B. Sp¨ath. On the inductive Alperin-McKay condition for simple groups of type A . J. Algebra , 442:104–123, 2015.[FLZ20a] Z. Feng, C. Li, and J. Zhang. Equivariant correspondences and the inductive Alperin weightcondition for type A. (Preprint) arXiv:2008.05645, 2020.[FLZ20b] Z. Feng, C. Li, and J. Zhang. On the inductive blockwise Alperin weight condition for type A.(Preprint) arXiv:2008.06206, 2020.[FM20] Z. Feng, G. Malle. The inductive blockwise Alperin weight condition for type C and the prime2. (Preprint) arXiv:2007.13378, 2020.[GLS98] D. Gorenstein, R. Lyons, and R. Solomon.
The classification of the finite simple groups. Number3. Part I. Chapter A , volume 40 of
Mathematical Surveys and Monographs . American Mathe-matical Society, Providence, RI, 1998.[IMN07] I. M. Isaacs, G. Malle, and G. Navarro. A reduction theorem for the McKay conjecture.
Invent.Math. , 170(1):33–101, 2007.[Isa94] I. M. Isaacs.
Character theory of finite groups . Dover Publications, Inc., New York, 1994.[KS15] S. Koshitani and B. Sp¨ath. Clifford theory of characters in induced blocks.
Proc. Amer. Math.Soc. , 143(9):3687–3702, 2015.[KS16b] S. Koshitani and B. Sp¨ath. The inductive Alperin-McKay and blockwise Alperin weight con-ditions for blocks with cyclic defect groups and odd primes.
J. Group Theory , 19(5):777–813,2016.[Li20] C. Li. The inductive blockwise Alperin weight condition for Sp n ( q ) and odd primes. (Preprint)arXiv:2005.03916, 2020.[Nav98] G. Navarro. Characters and blocks of finite groups , volume 250 of
London Mathematical SocietyLecture Note Series . Cambridge University Press, Cambridge, 1998.[NS14a] G. Navarro and B. Sp¨ath. Character correspondences in blocks with normal defect groups.
J.Algebra , 398:396–406, 2014.[NS14b] G. Navarro and B. Sp¨ath. On Brauer’s height zero conjecture.
J. Eur. Math. Soc. (JEMS) ,16(4):695–747, 2014.[NT11] G. Navarro and P.H. Tiep. A reduction theorem for the Alperin weight conjecture.
Invent. Math. ,184, 529–565, 2011.[Ruh20] L. Ruhstorfer. Jordan decomposition for the Alperin–McKay conjecture. Dissertation, BergischeUniversit¨at Wuppertal (2020).[Sp¨a12] B. Sp¨ath. Inductive McKay condition in defining characteristic.
Bull. Lond. Math. Soc. ,44(3):426–438, 2012.[Sp¨a13a] B. Sp¨ath. A reduction theorem for the Alperin-McKay conjecture.
J. Reine Angew. Math. ,680:153–189, 2013.[Sp¨a13b] B. Sp¨ath. A reduction theorem for the blockwise Alperin weight conjecture.
J. Group Theory ,16(2):159–220, 2013.[Sp¨a17a] B. Sp¨ath. Inductive conditions for counting conjectures via character triples.
Eur. Math. Soc. ,EMS Ser. Congr. Rep.:665–680, 2017.[Sp¨a17b] B. Sp¨ath. A reduction theorem for Dade’s projective conjecture.
J. Eur. Math. Soc. , 19: 1071–1126, 2017.[Sp¨a18] B. Sp¨ath. Reduction theorems for some global-local conjectures.
Eur. Math. Soc. , EMS Ser.Lect. Math.: 23–61, 2018., EMS Ser.Lect. Math.: 23–61, 2018.