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

The Alperin-McKay conjecture for simple groups of 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 N G ( D ) 2 / 8

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 N G ( D ) Conjecture (Alperin-McKay conjecture) | Irr 0 ( B ) | = | Irr 0 ( b ) | , where Irr 0 ( B ) = { χ ∈ Irr ( B ) | χ (1) ℓ | D | = | G | ℓ } . 2 / 8

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. 3 / 8

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 3 / 8

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 : Irr 0 ( B ) → Irr 0 ( b ) , and 3 / 8

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 : Irr 0 ( B ) → Irr 0 ( b ) , and • Ω preserves the Clifford theory of characters with respect to G ✁ G ⋊ Aut ( G ) B , D 3 / 8

Validating the iAM -condition for quasi-simple groups Main open case : G is a group of Lie type over F q with ℓ ∤ q . (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath) 4 / 8

Validating the iAM -condition for quasi-simple groups Main open case : G is a group of Lie type over F q with ℓ ∤ q . (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath) For G = SL n ( q ), we have Aut ( G ) = � GE 4 / 8

Validating the iAM -condition for quasi-simple groups Main open case : G is a group of Lie type over F q with ℓ ∤ q . (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath) For G = SL n ( q ), we have Aut ( G ) = � GE where � G := GL n ( q ) and E is generated by the automorphisms F 0 (( a i , j )) = ( a p i , j ) and γ (( a i , j )) = (( a i , j ) Tr ) − 1 4 / 8

Validating the iAM -condition for quasi-simple groups Main open case : G is a group of Lie type over F q with ℓ ∤ q . (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath) For G = SL n ( q ), we have Aut ( G ) = � GE where � G := GL n ( q ) and E is generated by the automorphisms F 0 (( a i , j )) = ( a p i , j ) and γ (( a i , j )) = (( a i , j ) Tr ) − 1 In general : • G = G F , for G a connected reductive algebraic group over F q with Frobenius endomorphism F : G → G . 4 / 8

Validating the iAM -condition for quasi-simple groups Main open case : G is a group of Lie type over F q with ℓ ∤ q . (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath) For G = SL n ( q ), we have Aut ( G ) = � GE where � G := GL n ( q ) and E is generated by the automorphisms F 0 (( a i , j )) = ( a p i , j ) and γ (( a i , j )) = (( a i , j ) Tr ) − 1 In general : • G = G F , for G a connected reductive algebraic group over F q 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. 4 / 8

Validating the iAM -condition for quasi-simple groups Main open case : G is a group of Lie type over F q with ℓ ∤ q . (Due to results from Breuer, Cabanes, Denoncin, Koshitani, Malle, Schaeffer-Fry and Sp¨ ath) For G = SL n ( q ), we have Aut ( G ) = � GE where � G := GL n ( q ) and E is generated by the automorphisms F 0 (( a i , j )) = ( a p i , j ) and γ (( a i , j )) = (( a i , j ) Tr ) − 1 In general : • G = G F , for G a connected reductive algebraic group over F q 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. 4 / 8

A criterion tailored to groups of Lie type 5 / 8

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 } 5 / 8

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 } G ( Z ) and B ′ ⊂ Bl ( M ) the set of For M = N G ( Z ) , � M = N GE ( Z ) , � M = N � Brauer correspondents to B assume that 5 / 8

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 } G ( Z ) and B ′ ⊂ Bl ( M ) the set of For M = N G ( Z ) , � M = N GE ( Z ) , � M = N � Brauer correspondents to B assume that 1 there is an Irr ( � M / M ) ⋊ � M-equivariant bijection Ω : Irr ( � � G | Irr 0 ( B )) → Irr ( � M | Irr 0 ( B ′ )) , compatible with Brauer correspondence; 5 / 8

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 } G ( Z ) and B ′ ⊂ Bl ( M ) the set of For M = N G ( Z ) , � M = N GE ( Z ) , � M = N � Brauer correspondents to B assume that 1 there is an Irr ( � M / M ) ⋊ � M-equivariant bijection Ω : Irr ( � � G | Irr 0 ( B )) → Irr ( � M | Irr 0 ( B ′ )) , compatible with Brauer correspondence; 2 there is a GE-stable � G-transversal in Irr 0 ( B ) and a � M-stable � M-transversal in Irr 0 ( B ′ ) . 5 / 8

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 } G ( Z ) and B ′ ⊂ Bl ( M ) the set of For M = N G ( Z ) , � M = N GE ( Z ) , � M = N � Brauer correspondents to B assume that 1 there is an Irr ( � M / M ) ⋊ � M-equivariant bijection Ω : Irr ( � � G | Irr 0 ( B )) → Irr ( � M | Irr 0 ( B ′ )) , compatible with Brauer correspondence; 2 there is a GE-stable � G-transversal in Irr 0 ( B ) and a � M-stable � M-transversal in Irr 0 ( B ′ ) . If B ∈ B and for B 0 the � G-orbit of B either |B 0 | = 1 or Out ( G ) B 0 is abelian, then the iAM -condition holds for B. 5 / 8

Application to SL ǫ n ( q ) Theorem (B., Sp¨ ath ’19) Let ℓ be a prime with ℓ ∤ 6 q ( q − ǫ ) . 1 If B is a GL ǫ n ( q ) -stable collection of blocks of SL ǫ n ( q ) with Out ( SL ǫ n ( q )) B abelian, then the iAM -condition holds for each B ∈ B . 6 / 8

Application to SL ǫ n ( q ) Theorem (B., Sp¨ ath ’19) Let ℓ be a prime with ℓ ∤ 6 q ( q − ǫ ) . 1 If B is a GL ǫ n ( q ) -stable collection of blocks of SL ǫ n ( q ) with Out ( SL ǫ 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 C G ( D ) is a d-split Levi subgroup, then the inductive blockwise Alperin weight condition holds for B. 6 / 8

Application to SL ǫ n ( q ) Theorem (B., Sp¨ ath ’19) Let ℓ be a prime with ℓ ∤ 6 q ( q − ǫ ) . 1 If B is a GL ǫ n ( q ) -stable collection of blocks of SL ǫ n ( q ) with Out ( SL ǫ 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 C G ( 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 ∤ 6 q ( q − 1) . 1 The Alperin-McKay conjecture holds for all ℓ -blocks of SL ǫ n ( q ) . 2 The Alperin weight conjecture holds for all ℓ -blocks of SL ǫ n ( q ) with abelian defect. 6 / 8

Parametrising irreducible characters 1 Replace Z by S a Φ d -torus, d = o ( q ) mod( ℓ ). 7 / 8

Parametrising irreducible characters 1 Replace Z by S a Φ d -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 ). 7 / 8