The Alperin-McKay conjecture for simple groupsof type A

Julian Broughjoint work with Britta Spath

Bergische Universitat Wuppertal

June 12th, 2019

The Alperin-McKay conjecture

Notation:

• G a finite group and ` a prime with ` | |G |.• Irr(G ) the set of ordinary irreducible characters of G .

• B an `-block of G with defect group D

• b the Brauer correspondent of B, an `-block of NG (D)

Conjecture (Alperin-McKay conjecture)

|Irr0(B)| = |Irr0(b)|,

where Irr0(B) = χ ∈ Irr(B) | χ(1)`|D| = |G |`.

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The Alperin-McKay conjecture

Notation:

• G a finite group and ` a prime with ` | |G |.• Irr(G ) the set of ordinary irreducible characters of G .

• B an `-block of G with defect group D

• b the Brauer correspondent of B, an `-block of NG (D)

Conjecture (Alperin-McKay conjecture)

|Irr0(B)| = |Irr0(b)|,

where Irr0(B) = χ ∈ Irr(B) | χ(1)`|D| = |G |`.

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The reduction theorem

Theorem (Spath ’13)

The Alperin-McKay conjecture holds for all groups if the so-calledinductive Alperin-McKay condition (iAM) holds for all blocks ofquasi-simple groups.

Assume G is a quasi-simple group.Recall: For B ∈ Bl(G ) with defect group D and Brauer correspondent b,iAM-condition holds if

• there exists an Aut(G )B,D -equivariant bijection

Ω : Irr0(B)→ Irr0(b), and

• Ω preserves the Clifford theory of characters with respect toG G oAut(G )B,D

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The reduction theorem

Theorem (Spath ’13)

The Alperin-McKay conjecture holds for all groups if the so-calledinductive Alperin-McKay condition (iAM) holds for all blocks ofquasi-simple groups.

Assume G is a quasi-simple group.Recall: For B ∈ Bl(G ) with defect group D and Brauer correspondent b,iAM-condition holds if

• there exists an Aut(G )B,D -equivariant bijection

Ω : Irr0(B)→ Irr0(b), and

• Ω preserves the Clifford theory of characters with respect toG G oAut(G )B,D

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The reduction theorem

Theorem (Spath ’13)

The Alperin-McKay conjecture holds for all groups if the so-calledinductive Alperin-McKay condition (iAM) holds for all blocks ofquasi-simple groups.

Assume G is a quasi-simple group.Recall: For B ∈ Bl(G ) with defect group D and Brauer correspondent b,iAM-condition holds if

• there exists an Aut(G )B,D -equivariant bijection

Ω : Irr0(B)→ Irr0(b), and

• Ω preserves the Clifford theory of characters with respect toG G oAut(G )B,D

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The reduction theorem

Theorem (Spath ’13)

The Alperin-McKay conjecture holds for all groups if the so-calledinductive Alperin-McKay condition (iAM) holds for all blocks ofquasi-simple groups.

Assume G is a quasi-simple group.Recall: For B ∈ Bl(G ) with defect group D and Brauer correspondent b,iAM-condition holds if

• there exists an Aut(G )B,D -equivariant bijection

Ω : Irr0(B)→ Irr0(b), and

• Ω preserves the Clifford theory of characters with respect toG G oAut(G )B,D

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Validating the iAM-condition for quasi-simple groups

Main open case: G is a group of Lie type over Fq with ` - q.(Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle,Schaeffer-Fry and Spath)

For G = SLn(q), we have Aut(G ) = GE where G := GLn(q) and E isgenerated by the automorphisms F0((ai,j)) = (api,j) and

γ((ai,j)) = ((ai,j)Tr)−1

In general:

• G = GF , for G a connected reductive algebraic group over Fq withFrobenius endomorphism F : G→ G.

• Aut(G ) is induced from G a regular embedding of G and E thegroup generated by graph and field automorphisms.

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Validating the iAM-condition for quasi-simple groups

Main open case: G is a group of Lie type over Fq with ` - q.(Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle,Schaeffer-Fry and Spath)

For G = SLn(q), we have Aut(G ) = GE

where G := GLn(q) and E isgenerated by the automorphisms F0((ai,j)) = (api,j) and

γ((ai,j)) = ((ai,j)Tr)−1

In general:

• G = GF , for G a connected reductive algebraic group over Fq withFrobenius endomorphism F : G→ G.

• Aut(G ) is induced from G a regular embedding of G and E thegroup generated by graph and field automorphisms.

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Validating the iAM-condition for quasi-simple groups

Main open case: G is a group of Lie type over Fq with ` - q.(Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle,Schaeffer-Fry and Spath)

For G = SLn(q), we have Aut(G ) = GE where G := GLn(q) and E isgenerated by the automorphisms F0((ai,j)) = (api,j) and

γ((ai,j)) = ((ai,j)Tr)−1

In general:

• G = GF , for G a connected reductive algebraic group over Fq withFrobenius endomorphism F : G→ G.

• Aut(G ) is induced from G a regular embedding of G and E thegroup generated by graph and field automorphisms.

4 / 8

Validating the iAM-condition for quasi-simple groups

Main open case: G is a group of Lie type over Fq with ` - q.(Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle,Schaeffer-Fry and Spath)

For G = SLn(q), we have Aut(G ) = GE where G := GLn(q) and E isgenerated by the automorphisms F0((ai,j)) = (api,j) and

γ((ai,j)) = ((ai,j)Tr)−1

In general:

• G = GF , for G a connected reductive algebraic group over Fq withFrobenius endomorphism F : G→ G.

• Aut(G ) is induced from G a regular embedding of G and E thegroup generated by graph and field automorphisms.

4 / 8

Validating the iAM-condition for quasi-simple groups

Main open case: G is a group of Lie type over Fq with ` - q.(Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle,Schaeffer-Fry and Spath)

For G = SLn(q), we have Aut(G ) = GE where G := GLn(q) and E isgenerated by the automorphisms F0((ai,j)) = (api,j) and

γ((ai,j)) = ((ai,j)Tr)−1

In general:

• G = GF , for G a connected reductive algebraic group over Fq withFrobenius endomorphism F : G→ G.

• Aut(G ) is induced from G a regular embedding of G and E thegroup generated by graph and field automorphisms.

4 / 8

Validating the iAM-condition for quasi-simple groups

Main open case: G is a group of Lie type over Fq with ` - q.(Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle,Schaeffer-Fry and Spath)

For G = SLn(q), we have Aut(G ) = GE where G := GLn(q) and E isgenerated by the automorphisms F0((ai,j)) = (api,j) and

γ((ai,j)) = ((ai,j)Tr)−1

In general:

• G = GF , for G a connected reductive algebraic group over Fq withFrobenius endomorphism F : G→ G.

• Aut(G ) is induced from G a regular embedding of G and E thegroup generated by graph and field automorphisms.

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A criterion tailored to groups of Lie type

Theorem (B., Cabanes, Spath ’19)

Let Z be an abelian `-group and set

B = B ∈ Bl(G ) | Z is a maximal abelian normal subgroup of D

For M = NG (Z ), M = NGE (Z ), M = NG (Z ) and B′ ⊂ Bl(M) the set ofBrauer correspondents to B assume that

1 there is an Irr(M/M) o M-equivariant bijection

Ω : Irr(G | Irr0(B))→ Irr(M | Irr0(B′)),

compatible with Brauer correspondence;

2 there is a GE -stable G -transversal in Irr0(B)

and a M-stable M-transversal in Irr0(B′).If B ∈ B and for B0 the G -orbit of B either |B0| = 1 or Out(G )B0 isabelian, then the iAM-condition holds for B.

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A criterion tailored to groups of Lie type

Theorem (B., Cabanes, Spath ’19)

Let Z be an abelian `-group and set

B = B ∈ Bl(G ) | Z is a maximal abelian normal subgroup of D

For M = NG (Z ), M = NGE (Z ), M = NG (Z ) and B′ ⊂ Bl(M) the set ofBrauer correspondents to B assume that

1 there is an Irr(M/M) o M-equivariant bijection

Ω : Irr(G | Irr0(B))→ Irr(M | Irr0(B′)),

compatible with Brauer correspondence;

2 there is a GE -stable G -transversal in Irr0(B)

and a M-stable M-transversal in Irr0(B′).If B ∈ B and for B0 the G -orbit of B either |B0| = 1 or Out(G )B0 isabelian, then the iAM-condition holds for B.

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A criterion tailored to groups of Lie type

Theorem (B., Cabanes, Spath ’19)

Let Z be an abelian `-group and set

B = B ∈ Bl(G ) | Z is a maximal abelian normal subgroup of D

For M = NG (Z ), M = NGE (Z ), M = NG (Z ) and B′ ⊂ Bl(M) the set ofBrauer correspondents to B assume that

1 there is an Irr(M/M) o M-equivariant bijection

Ω : Irr(G | Irr0(B))→ Irr(M | Irr0(B′)),

compatible with Brauer correspondence;

2 there is a GE -stable G -transversal in Irr0(B)

and a M-stable M-transversal in Irr0(B′).If B ∈ B and for B0 the G -orbit of B either |B0| = 1 or Out(G )B0 isabelian, then the iAM-condition holds for B.

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A criterion tailored to groups of Lie type

Theorem (B., Cabanes, Spath ’19)

Let Z be an abelian `-group and set

B = B ∈ Bl(G ) | Z is a maximal abelian normal subgroup of D

For M = NG (Z ), M = NGE (Z ), M = NG (Z ) and B′ ⊂ Bl(M) the set ofBrauer correspondents to B assume that

1 there is an Irr(M/M) o M-equivariant bijection

Ω : Irr(G | Irr0(B))→ Irr(M | Irr0(B′)),

compatible with Brauer correspondence;

2 there is a GE -stable G -transversal in Irr0(B)

and a M-stable M-transversal in Irr0(B′).If B ∈ B and for B0 the G -orbit of B either |B0| = 1 or Out(G )B0 isabelian, then the iAM-condition holds for B.

5 / 8

A criterion tailored to groups of Lie type

Theorem (B., Cabanes, Spath ’19)

Let Z be an abelian `-group and set

B = B ∈ Bl(G ) | Z is a maximal abelian normal subgroup of D

For M = NG (Z ), M = NGE (Z ), M = NG (Z ) and B′ ⊂ Bl(M) the set ofBrauer correspondents to B assume that

1 there is an Irr(M/M) o M-equivariant bijection

Ω : Irr(G | Irr0(B))→ Irr(M | Irr0(B′)),

compatible with Brauer correspondence;

2 there is a GE -stable G -transversal in Irr0(B)

and a M-stable M-transversal in Irr0(B′).

If B ∈ B and for B0 the G -orbit of B either |B0| = 1 or Out(G )B0 isabelian, then the iAM-condition holds for B.

5 / 8

A criterion tailored to groups of Lie type

Theorem (B., Cabanes, Spath ’19)

Let Z be an abelian `-group and set

B = B ∈ Bl(G ) | Z is a maximal abelian normal subgroup of D

For M = NG (Z ), M = NGE (Z ), M = NG (Z ) and B′ ⊂ Bl(M) the set ofBrauer correspondents to B assume that

1 there is an Irr(M/M) o M-equivariant bijection

Ω : Irr(G | Irr0(B))→ Irr(M | Irr0(B′)),

compatible with Brauer correspondence;

2 there is a GE -stable G -transversal in Irr0(B)

and a M-stable M-transversal in Irr0(B′).If B ∈ B and for B0 the G -orbit of B either |B0| = 1 or Out(G )B0 isabelian, then the iAM-condition holds for B.

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Application to SLεn(q)

Theorem (B., Spath ’19)

Let ` be a prime with ` - 6q(q − ε).1 If B is a GLεn(q)-stable collection of blocks of SLεn(q) with

Out(SLεn(q))B abelian, then the iAM-condition holds for eachB ∈ B.

2 If in addition the defect group D of B is abelian and CG (D) is ad-split Levi subgroup, then the inductive blockwise Alperin weightcondition holds for B.

Theorem (B., Spath ’19)

Let ` be a prime with ` - 6q(q − 1).

1 The Alperin-McKay conjecture holds for all `-blocks of SLεn(q).

2 The Alperin weight conjecture holds for all `-blocks of SLεn(q) withabelian defect.

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Application to SLεn(q)

Theorem (B., Spath ’19)

Let ` be a prime with ` - 6q(q − ε).1 If B is a GLεn(q)-stable collection of blocks of SLεn(q) with

Out(SLεn(q))B abelian, then the iAM-condition holds for eachB ∈ B.

2 If in addition the defect group D of B is abelian and CG (D) is ad-split Levi subgroup, then the inductive blockwise Alperin weightcondition holds for B.

Theorem (B., Spath ’19)

Let ` be a prime with ` - 6q(q − 1).

1 The Alperin-McKay conjecture holds for all `-blocks of SLεn(q).

2 The Alperin weight conjecture holds for all `-blocks of SLεn(q) withabelian defect.

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Application to SLεn(q)

Theorem (B., Spath ’19)

Let ` be a prime with ` - 6q(q − ε).1 If B is a GLεn(q)-stable collection of blocks of SLεn(q) with

Out(SLεn(q))B abelian, then the iAM-condition holds for eachB ∈ B.

2 If in addition the defect group D of B is abelian and CG (D) is ad-split Levi subgroup, then the inductive blockwise Alperin weightcondition holds for B.

Theorem (B., Spath ’19)

Let ` be a prime with ` - 6q(q − 1).

1 The Alperin-McKay conjecture holds for all `-blocks of SLεn(q).

2 The Alperin weight conjecture holds for all `-blocks of SLεn(q) withabelian defect.

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Parametrising irreducible characters

1 Replace Z by S a Φd -torus, d = o(q) mod(`).

2 Characters of NG (S):• Each character of CG (S) extends to its inertial subgroup in NG (S).• Clifford theory then parametrises the irreducible characters of NG (S).

3 Characters of G can be parametrised via Jordan decomposition andd-Harish-Chandra theory.

4 The parametrisations yield a bijection as required for the previoustheorem.

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Parametrising irreducible characters

1 Replace Z by S a Φd -torus, d = o(q) mod(`).

2 Characters of NG (S):• Each character of CG (S) extends to its inertial subgroup in NG (S).• Clifford theory then parametrises the irreducible characters of NG (S).

3 Characters of G can be parametrised via Jordan decomposition andd-Harish-Chandra theory.

4 The parametrisations yield a bijection as required for the previoustheorem.

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Parametrising irreducible characters

1 Replace Z by S a Φd -torus, d = o(q) mod(`).

2 Characters of NG (S):• Each character of CG (S) extends to its inertial subgroup in NG (S).• Clifford theory then parametrises the irreducible characters of NG (S).

3 Characters of G can be parametrised via Jordan decomposition andd-Harish-Chandra theory.

4 The parametrisations yield a bijection as required for the previoustheorem.

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Parametrising irreducible characters

1 Replace Z by S a Φd -torus, d = o(q) mod(`).

2 Characters of NG (S):• Each character of CG (S) extends to its inertial subgroup in NG (S).• Clifford theory then parametrises the irreducible characters of NG (S).

3 Characters of G can be parametrised via Jordan decomposition andd-Harish-Chandra theory.

4 The parametrisations yield a bijection as required for the previoustheorem.

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Thank you for your attention

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