Morita equivalence classes of blocks with defect group(C2)5
Cesare Giulio Ardito
University of Manchester
Groups, Rings and Associated Structures 2019
Spa, Belgium
Cesare Giulio Ardito University of Manchester 1 / 10
A classification of kG -modules
Given a field k and a finite group G , we can form the group algebra kG .
A priori, we want to classify indecomposable kG -modules. But...
Maschke’s theorem
The group algebra kG is semisimple if and only if char(k) = p does notdivide the order of G
When p divides the order of G , most group algebras have wildrepresentation type, so in particular there is an infinite number of familiesof isomorphism classes of indecomposable kG -modules.
(exceptions: Groups with a cyclic Sylow p-subgroup.(exceptions: When p = 2, groups with a dihedral, semidihedral or(exceptions: generalised quaternion Sylow 2-subgroup.)
Cesare Giulio Ardito University of Manchester 2 / 10
A classification of kG -modules
Given a field k and a finite group G , we can form the group algebra kG .A priori, we want to classify indecomposable kG -modules. But...
Maschke’s theorem
The group algebra kG is semisimple if and only if char(k) = p does notdivide the order of G
When p divides the order of G , most group algebras have wildrepresentation type, so in particular there is an infinite number of familiesof isomorphism classes of indecomposable kG -modules.
(exceptions: Groups with a cyclic Sylow p-subgroup.(exceptions: When p = 2, groups with a dihedral, semidihedral or(exceptions: generalised quaternion Sylow 2-subgroup.)
Cesare Giulio Ardito University of Manchester 2 / 10
A classification of kG -modules
Given a field k and a finite group G , we can form the group algebra kG .A priori, we want to classify indecomposable kG -modules. But...
Maschke’s theorem
The group algebra kG is semisimple if and only if char(k) = p does notdivide the order of G
When p divides the order of G , most group algebras have wildrepresentation type, so in particular there is an infinite number of familiesof isomorphism classes of indecomposable kG -modules.
(exceptions: Groups with a cyclic Sylow p-subgroup.(exceptions: When p = 2, groups with a dihedral, semidihedral or(exceptions: generalised quaternion Sylow 2-subgroup.)
Cesare Giulio Ardito University of Manchester 2 / 10
A classification of kG -modules
Given a field k and a finite group G , we can form the group algebra kG .A priori, we want to classify indecomposable kG -modules. But...
Maschke’s theorem
The group algebra kG is semisimple if and only if char(k) = p does notdivide the order of G
When p divides the order of G , most group algebras have wildrepresentation type, so in particular there is an infinite number of familiesof isomorphism classes of indecomposable kG -modules.
(exceptions: Groups with a cyclic Sylow p-subgroup.(exceptions: When p = 2, groups with a dihedral, semidihedral or(exceptions: generalised quaternion Sylow 2-subgroup.)
Cesare Giulio Ardito University of Manchester 2 / 10
Blocks of finite groups
Ingredients
A finite group GFor a prime p, a p-modular system (K ,O, k), where
- K is a field with characteristic 0.- O is a complete discrete valuation ring such that K = FOF (O).- k = O/J(O) is an algebraically closed field of characteristic p.
We can choose primitive orthogonal central idempotents ei ∈ Z (kG )such that 1 = e1 + · · ·+ en. Then
kG = kGe1 ⊕ · · · ⊕ kGen
Bi = kGei is called a p-block of kG .
Cesare Giulio Ardito University of Manchester 3 / 10
Blocks of finite groups
Ingredients
A finite group GFor a prime p, a p-modular system (K ,O, k), where
- K is a field with characteristic 0.- O is a complete discrete valuation ring such that K = FOF (O).- k = O/J(O) is an algebraically closed field of characteristic p.
We can choose primitive orthogonal central idempotents ei ∈ Z (kG )such that 1 = e1 + · · ·+ en. Then
kG = kGe1 ⊕ · · · ⊕ kGen
Bi = kGei is called a p-block of kG .
Cesare Giulio Ardito University of Manchester 3 / 10
We can study blocks instead of modules
kG = kGe1 ⊕ · · · ⊕ kGen
We can decompose any kG -module M as
M = Me1 ⊕ · · · ⊕Men
so for any indecomposable module there is a unique i such that Mei 6= 0.We say that M belongs to the block Bi .
Remark
It is also possible to partition ordinary irreducible characters (over K ) andBrauer characters (over k) into blocks.
Cesare Giulio Ardito University of Manchester 4 / 10
We can study blocks instead of modules
kG = kGe1 ⊕ · · · ⊕ kGen
We can decompose any kG -module M as
M = Me1 ⊕ · · · ⊕Men
so for any indecomposable module there is a unique i such that Mei 6= 0.We say that M belongs to the block Bi .
Remark
It is also possible to partition ordinary irreducible characters (over K ) andBrauer characters (over k) into blocks.
Cesare Giulio Ardito University of Manchester 4 / 10
Studying blocks: important concepts
Defect group: D
A defect group of a block B of kG is a minimal p-subgroup D of OG thatcontains a vertex of every indecomposable module in B.
It measures the complexity of a block, or how “far” it is from being projective.
If D = 1 the block contains a single simple projective module.
If B contains the trivial module (the “principal” block) then D = Sylp(G ).
Inertial quotient: E
If B is a block of kG with defect group D, and bD is a block of kCG (D)corresponding to B, then E = NG (D, bD)/DCG (D) is the inertial quotient of B.
Important: E is a subgroup of Out(D) with odd order.
Cesare Giulio Ardito University of Manchester 5 / 10
Studying blocks: important concepts
Defect group: D
A defect group of a block B of kG is a minimal p-subgroup D of OG thatcontains a vertex of every indecomposable module in B.
It measures the complexity of a block, or how “far” it is from being projective.
If D = 1 the block contains a single simple projective module.
If B contains the trivial module (the “principal” block) then D = Sylp(G ).
Inertial quotient: E
If B is a block of kG with defect group D, and bD is a block of kCG (D)corresponding to B, then E = NG (D, bD)/DCG (D) is the inertial quotient of B.
Important: E is a subgroup of Out(D) with odd order.
Cesare Giulio Ardito University of Manchester 5 / 10
Studying blocks: important concepts
Defect group: D
A defect group of a block B of kG is a minimal p-subgroup D of OG thatcontains a vertex of every indecomposable module in B.
It measures the complexity of a block, or how “far” it is from being projective.
If D = 1 the block contains a single simple projective module.
If B contains the trivial module (the “principal” block) then D = Sylp(G ).
Inertial quotient: E
If B is a block of kG with defect group D, and bD is a block of kCG (D)corresponding to B, then E = NG (D, bD)/DCG (D) is the inertial quotient of B.
Important: E is a subgroup of Out(D) with odd order.
Cesare Giulio Ardito University of Manchester 5 / 10
Studying blocks: important concepts
Defect group: D
A defect group of a block B of kG is a minimal p-subgroup D of OG thatcontains a vertex of every indecomposable module in B.
It measures the complexity of a block, or how “far” it is from being projective.
If D = 1 the block contains a single simple projective module.
If B contains the trivial module (the “principal” block) then D = Sylp(G ).
Inertial quotient: E
If B is a block of kG with defect group D, and bD is a block of kCG (D)corresponding to B, then E = NG (D, bD)/DCG (D) is the inertial quotient of B.
Important: E is a subgroup of Out(D) with odd order.
Cesare Giulio Ardito University of Manchester 5 / 10
Studying blocks: important concepts
Defect group: D
A defect group of a block B of kG is a minimal p-subgroup D of OG thatcontains a vertex of every indecomposable module in B.
It measures the complexity of a block, or how “far” it is from being projective.
If D = 1 the block contains a single simple projective module.
If B contains the trivial module (the “principal” block) then D = Sylp(G ).
Inertial quotient: E
If B is a block of kG with defect group D, and bD is a block of kCG (D)corresponding to B, then E = NG (D, bD)/DCG (D) is the inertial quotient of B.
Important: E is a subgroup of Out(D) with odd order.
Cesare Giulio Ardito University of Manchester 5 / 10
Studying blocks: the right kind of equivalence
Morita equivalence
Two algebras A and B are said to be Morita equivalent if the category ofleft A-modules is equivalent to the category of left B-modules.
Two Morita equivalent algebras can be very different. For instance, kis Morita equivalent to Mn(k) for any n.
The defect group and the inertial quotient are not known, in general,to be invariant under Morita equivalence between two blocks.
A Morita equivalence preseves the number of ordinary and Brauercharacters in a block, and the order, the exponent and the p-rank ofthe defect group.
Cesare Giulio Ardito University of Manchester 6 / 10
Studying blocks: the right kind of equivalence
Morita equivalence
Two algebras A and B are said to be Morita equivalent if the category ofleft A-modules is equivalent to the category of left B-modules.
Two Morita equivalent algebras can be very different. For instance, kis Morita equivalent to Mn(k) for any n.
The defect group and the inertial quotient are not known, in general,to be invariant under Morita equivalence between two blocks.
A Morita equivalence preseves the number of ordinary and Brauercharacters in a block, and the order, the exponent and the p-rank ofthe defect group.
Cesare Giulio Ardito University of Manchester 6 / 10
Studying blocks: the right kind of equivalence
Morita equivalence
Two algebras A and B are said to be Morita equivalent if the category ofleft A-modules is equivalent to the category of left B-modules.
Two Morita equivalent algebras can be very different. For instance, kis Morita equivalent to Mn(k) for any n.
The defect group and the inertial quotient are not known, in general,to be invariant under Morita equivalence between two blocks.
A Morita equivalence preseves the number of ordinary and Brauercharacters in a block, and the order, the exponent and the p-rank ofthe defect group.
Cesare Giulio Ardito University of Manchester 6 / 10
Studying blocks: the right kind of equivalence
Morita equivalence
Two algebras A and B are said to be Morita equivalent if the category ofleft A-modules is equivalent to the category of left B-modules.
Two Morita equivalent algebras can be very different. For instance, kis Morita equivalent to Mn(k) for any n.
The defect group and the inertial quotient are not known, in general,to be invariant under Morita equivalence between two blocks.
A Morita equivalence preseves the number of ordinary and Brauercharacters in a block, and the order, the exponent and the p-rank ofthe defect group.
Cesare Giulio Ardito University of Manchester 6 / 10
Studying blocks: Donovan’s conjecture
Donovan’s Conjecture
For any fixed p-group D, there are only finitely many Morita equivalenceclasses of blocks of kG with defect group D for all finite groups G .
Proved for any p, cyclic D. (Dade, Janusz, Kupisch, 1960s).
Proved for p = 2, abelian D, blocks of quasisimple groups (Eaton,Kessar, Kulshammer, Linckelmann, 2013).
Proved for p = 2, any abelian D. (Eaton, Livesey, 2018).
Proved in every case above for blocks of OG as well.
...and many other groups!
but the proof usually does not give an explicit list for each given D.
Cesare Giulio Ardito University of Manchester 7 / 10
Studying blocks: Donovan’s conjecture
Donovan’s Conjecture
For any fixed p-group D, there are only finitely many Morita equivalenceclasses of blocks of kG with defect group D for all finite groups G .
Proved for any p, cyclic D. (Dade, Janusz, Kupisch, 1960s).
Proved for p = 2, abelian D, blocks of quasisimple groups (Eaton,Kessar, Kulshammer, Linckelmann, 2013).
Proved for p = 2, any abelian D. (Eaton, Livesey, 2018).
Proved in every case above for blocks of OG as well.
...and many other groups!
but the proof usually does not give an explicit list for each given D.
Cesare Giulio Ardito University of Manchester 7 / 10
Studying blocks: Donovan’s conjecture
Donovan’s Conjecture
For any fixed p-group D, there are only finitely many Morita equivalenceclasses of blocks of kG with defect group D for all finite groups G .
Proved for any p, cyclic D. (Dade, Janusz, Kupisch, 1960s).
Proved for p = 2, abelian D, blocks of quasisimple groups (Eaton,Kessar, Kulshammer, Linckelmann, 2013).
Proved for p = 2, any abelian D. (Eaton, Livesey, 2018).
Proved in every case above for blocks of OG as well.
...and many other groups!
but the proof usually does not give an explicit list for each given D.
Cesare Giulio Ardito University of Manchester 7 / 10
Studying blocks: Donovan’s conjecture
Donovan’s Conjecture
For any fixed p-group D, there are only finitely many Morita equivalenceclasses of blocks of kG with defect group D for all finite groups G .
Proved for any p, cyclic D. (Dade, Janusz, Kupisch, 1960s).
Proved for p = 2, abelian D, blocks of quasisimple groups (Eaton,Kessar, Kulshammer, Linckelmann, 2013).
Proved for p = 2, any abelian D. (Eaton, Livesey, 2018).
Proved in every case above for blocks of OG as well.
...and many other groups!
but the proof usually does not give an explicit list for each given D.
Cesare Giulio Ardito University of Manchester 7 / 10
Studying blocks: Donovan’s conjecture
Donovan’s Conjecture
For any fixed p-group D, there are only finitely many Morita equivalenceclasses of blocks of kG with defect group D for all finite groups G .
Proved for any p, cyclic D. (Dade, Janusz, Kupisch, 1960s).
Proved for p = 2, abelian D, blocks of quasisimple groups (Eaton,Kessar, Kulshammer, Linckelmann, 2013).
Proved for p = 2, any abelian D. (Eaton, Livesey, 2018).
Proved in every case above for blocks of OG as well.
...and many other groups!
but the proof usually does not give an explicit list for each given D.
Cesare Giulio Ardito University of Manchester 7 / 10
Studying blocks: Donovan’s conjecture
Donovan’s Conjecture
For any fixed p-group D, there are only finitely many Morita equivalenceclasses of blocks of kG with defect group D for all finite groups G .
Proved for any p, cyclic D. (Dade, Janusz, Kupisch, 1960s).
Proved for p = 2, abelian D, blocks of quasisimple groups (Eaton,Kessar, Kulshammer, Linckelmann, 2013).
Proved for p = 2, any abelian D. (Eaton, Livesey, 2018).
Proved in every case above for blocks of OG as well.
...and many other groups!
but the proof usually does not give an explicit list for each given D.
Cesare Giulio Ardito University of Manchester 7 / 10
Studying blocks: Donovan’s conjecture
Donovan’s Conjecture
For any fixed p-group D, there are only finitely many Morita equivalenceclasses of blocks of kG with defect group D for all finite groups G .
Proved for any p, cyclic D. (Dade, Janusz, Kupisch, 1960s).
Proved for p = 2, abelian D, blocks of quasisimple groups (Eaton,Kessar, Kulshammer, Linckelmann, 2013).
Proved for p = 2, any abelian D. (Eaton, Livesey, 2018).
Proved in every case above for blocks of OG as well.
...and many other groups!
but the proof usually does not give an explicit list for each given D.
Cesare Giulio Ardito University of Manchester 7 / 10
The main result
Theorem (A., 2019)
A 2-block B of OG (or kG ) of any finite group G , with defect group D = (C2)5
is in one of 34 Morita equivalence classes:
The principal block of:
D o E where E is an odd-order subgroup of GL5(2) (15)
SL2(32) or Aut(SL2(32)) (2)
SL2(16)× (C2) (1)
L× (C2)2 or L× A4 where L = SL2(8),Aut(SL2(8)) or J1 (6)
L× A5 where L is as above (3)
A5 × ((C2)3 o F ) where F is an odd-order subgroup of GL3(2) (3)
A5 × A5 × C2 (1)
A nonprincipal block of:
((C2)4 o 31+2+ )× C2, (C2)5 o (C7 o 31+2
+ ), (SL2(8)× (C2)2) o 31+2+ (3)
Cesare Giulio Ardito University of Manchester 8 / 10
Further information
CorollaryThe following conjectures hold for blocks with defect group (C2)5:
Brauer’s k(B) conjecture, other counting conjectures (already known)
Harada’s conjecture
Broue’s abelian defect group conjecture
A block B with abelian defect group D is derived equivalent to itsBrauer correspondent, hence to a twisted group algebra kα(D o E )where E is the inertial quotient of B.
Cesare Giulio Ardito University of Manchester 9 / 10
Further information
CorollaryThe following conjectures hold for blocks with defect group (C2)5:
Brauer’s k(B) conjecture, other counting conjectures (already known)
Harada’s conjecture
Broue’s abelian defect group conjecture...for 32 of the 34 classes
A block B with abelian defect group D is derived equivalent to itsBrauer correspondent, hence to a twisted group algebra kα(D o E )where E is the inertial quotient of B.
Cesare Giulio Ardito University of Manchester 9 / 10
More details and classifications available (or soon available) onhttps://wiki.manchester.ac.uk/blocks/
Cesare Giulio Ardito University of Manchester 10 / 10