The Alperin-McKay conjecture for simple groups of type A Julian - - PowerPoint PPT Presentation

The Alperin-McKay conjecture for simple groups

• f type A

Julian Brough joint work with Britta Sp¨ ath Bergische Universit¨ at Wuppertal June 12th, 2019

The Alperin-McKay conjecture

Notation:

• G a finite group and ℓ a prime with ℓ | |G|.
• Irr(G) the set of ordinary irreducible characters of G.
• B an ℓ-block of G with defect group D
• b the Brauer correspondent of B, an ℓ-block of NG(D)

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The Alperin-McKay conjecture

Notation:

• G a finite group and ℓ a prime with ℓ | |G|.
• Irr(G) the set of ordinary irreducible characters of G.
• B an ℓ-block of G with defect group D
• b the Brauer correspondent of B, an ℓ-block of NG(D)

Conjecture (Alperin-McKay conjecture) |Irr0(B)| = |Irr0(b)|, where Irr0(B) = {χ ∈ Irr(B) | χ(1)ℓ|D| = |G|ℓ}.

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The reduction theorem

Theorem (Sp¨ ath ’13) The Alperin-McKay conjecture holds for all groups if the so-called inductive Alperin-McKay condition (iAM) holds for all blocks of quasi-simple groups.

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The reduction theorem

Theorem (Sp¨ ath ’13) The Alperin-McKay conjecture holds for all groups if the so-called inductive Alperin-McKay condition (iAM) holds for all blocks of quasi-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

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The reduction theorem

Theorem (Sp¨ ath ’13) The Alperin-McKay conjecture holds for all groups if the so-called inductive Alperin-McKay condition (iAM) holds for all blocks of quasi-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

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The reduction theorem

Theorem (Sp¨ ath ’13) The Alperin-McKay conjecture holds for all groups if the so-called inductive Alperin-McKay condition (iAM) holds for all blocks of quasi-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 to

G ✁ G ⋊ Aut(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 ℓ ∤ q. (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath)

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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 ℓ ∤ q. (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath) For G = SLn(q), we have Aut(G) = GE

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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 ℓ ∤ q. (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath) For G = SLn(q), we have Aut(G) = GE where G := GLn(q) and E is generated by the automorphisms F0((ai,j)) = (ap

i,j) and

γ((ai,j)) = ((ai,j)Tr)−1

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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 ℓ ∤ q. (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath) For G = SLn(q), we have Aut(G) = GE where G := GLn(q) and E is generated by the automorphisms F0((ai,j)) = (ap

i,j) and

γ((ai,j)) = ((ai,j)Tr)−1 In general:

• G = GF, for G a connected reductive algebraic group over Fq with

Frobenius endomorphism F : G → G.

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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 ℓ ∤ q. (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath) For G = SLn(q), we have Aut(G) = GE where G := GLn(q) and E is generated by the automorphisms F0((ai,j)) = (ap

i,j) and

γ((ai,j)) = ((ai,j)Tr)−1 In general:

• G = GF, for G a connected reductive algebraic group over Fq with

Frobenius endomorphism F : G → G.

• Aut(G) is induced from

G a regular embedding of G and E the group 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 ℓ ∤ q. (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath) For G = SLn(q), we have Aut(G) = GE where G := GLn(q) and E is generated by the automorphisms F0((ai,j)) = (ap

i,j) and

γ((ai,j)) = ((ai,j)Tr)−1 In general:

• G = GF, for G a connected reductive algebraic group over Fq with

Frobenius endomorphism F : G → G.

• Aut(G) is induced from

G a regular embedding of G and E the group generated by graph and field automorphisms.

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A criterion tailored to groups of Lie type

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A criterion tailored to groups of Lie type

Theorem (B., Cabanes, Sp¨ ath ’19) Let Z be an abelian ℓ-group and set B = {B ∈ Bl(G) | Z is a maximal abelian normal subgroup of D}

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A criterion tailored to groups of Lie type

Theorem (B., Cabanes, Sp¨ ath ’19) Let Z be an abelian ℓ-group and set B = {B ∈ Bl(G) | Z is a maximal abelian normal subgroup of D} For M = NG(Z), M = NGE(Z), M = N

G(Z) and B′ ⊂ Bl(M) the set of

Brauer correspondents to B assume that

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A criterion tailored to groups of Lie type

Theorem (B., Cabanes, Sp¨ ath ’19) Let Z be an abelian ℓ-group and set B = {B ∈ Bl(G) | Z is a maximal abelian normal subgroup of D} For M = NG(Z), M = NGE(Z), M = N

G(Z) and B′ ⊂ Bl(M) the set of

Brauer correspondents to B assume that

1 there is an Irr(

M/M) ⋊ M-equivariant bijection

• Ω : Irr(

G | Irr0(B)) → Irr( M | Irr0(B′)), compatible with Brauer correspondence;

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A criterion tailored to groups of Lie type

Theorem (B., Cabanes, Sp¨ ath ’19) Let Z be an abelian ℓ-group and set B = {B ∈ Bl(G) | Z is a maximal abelian normal subgroup of D} For M = NG(Z), M = NGE(Z), M = N

G(Z) and B′ ⊂ Bl(M) the set of

Brauer correspondents to B assume that

1 there is an Irr(

M/M) ⋊ 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′).

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A criterion tailored to groups of Lie type

Theorem (B., Cabanes, Sp¨ ath ’19) Let Z be an abelian ℓ-group and set B = {B ∈ Bl(G) | Z is a maximal abelian normal subgroup of D} For M = NG(Z), M = NGE(Z), M = N

G(Z) and B′ ⊂ Bl(M) the set of

Brauer correspondents to B assume that

1 there is an Irr(

M/M) ⋊ 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 is abelian, then the iAM-condition holds for B.

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Application to SLǫ

n(q) Theorem (B., Sp¨ ath ’19) Let ℓ be a prime with ℓ ∤ 6q(q − ǫ).

1 If B is a GLǫ n(q)-stable collection of blocks of SLǫ n(q) with

Out(SLǫ

n(q))B abelian, then the iAM-condition holds for each

B ∈ B.

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Application to SLǫ

n(q) Theorem (B., Sp¨ ath ’19) Let ℓ be a prime with ℓ ∤ 6q(q − ǫ).

1 If B is a GLǫ n(q)-stable collection of blocks of SLǫ n(q) with

Out(SLǫ

n(q))B abelian, then the iAM-condition holds for each

B ∈ B.

2 If in addition the defect group D of B is abelian and CG(D) is a

d-split Levi subgroup, then the inductive blockwise Alperin weight condition holds for B.

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Application to SLǫ

n(q) Theorem (B., Sp¨ ath ’19) Let ℓ be a prime with ℓ ∤ 6q(q − ǫ).

1 If B is a GLǫ n(q)-stable collection of blocks of SLǫ n(q) with

Out(SLǫ

n(q))B abelian, then the iAM-condition holds for each

B ∈ B.

2 If in addition the defect group D of B is abelian and CG(D) is a

d-split Levi subgroup, then the inductive blockwise Alperin weight condition holds for B. Theorem (B., Sp¨ ath ’19) Let ℓ be a prime with ℓ ∤ 6q(q − 1).

1 The Alperin-McKay conjecture holds for all ℓ-blocks of SLǫ n(q). 2 The Alperin weight conjecture holds for all ℓ-blocks of SLǫ n(q) with

abelian defect.

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Parametrising irreducible characters

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

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Parametrising irreducible characters

1 Replace Z by S a Φd-torus, d = o(q) mod(ℓ). 2 Characters of N G(S):

• Each character of C

G(S) extends to its inertial subgroup in N G(S).

• Clifford theory then parametrises the irreducible characters of N

G(S).

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Parametrising irreducible characters

1 Replace Z by S a Φd-torus, d = o(q) mod(ℓ). 2 Characters of N G(S):

• Each character of C

G(S) extends to its inertial subgroup in N G(S).

• Clifford theory then parametrises the irreducible characters of N

G(S).

3 Characters of

G can be parametrised via Jordan decomposition and d-Harish-Chandra theory.

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Parametrising irreducible characters

1 Replace Z by S a Φd-torus, d = o(q) mod(ℓ). 2 Characters of N G(S):

• Each character of C

G(S) extends to its inertial subgroup in N G(S).

• Clifford theory then parametrises the irreducible characters of N

G(S).

3 Characters of

G can be parametrised via Jordan decomposition and d-Harish-Chandra theory.

4 The parametrisations yield a bijection as required for the previous

theorem.

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Thank you for your attention

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