morita equivalence classes of blocks with defect group...

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.)

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

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 .

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 .

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.

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.

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.

Defect group: D

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.

Morita equivalence

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.

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)

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.

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.

More details and classifications available (or soon available) onhttps://wiki.manchester.ac.uk/blocks/

morita equivalence classes of blocks with defect group...

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