Character triples and group graded equivalences Virgilius-Aurelian - - PowerPoint PPT Presentation

Character triples and group graded equivalences

Virgilius-Aurelian Minut ¸˘ a

Babes

,-Bolyai University of Cluj-Napoca

Faculty of Mathematics and Informatics

Groups, Rings and Associated Structures Spa, Belgium | June 09-15, 2019

Introduction and preliminaries

Motivation Categorical version of reduction theorems involving character triples. Assumptions and notations G is a finite group (K, O, k ) is a splitting p-modular system N G, G′ ≤ G, and N′ G′ Assume: N′ = G′∩N and G = G′N, hence ¯ G := G/N ≃ G′/N′ b ∈ Z(ON) and b′ ∈ Z(ON′) are ¯ G-invariant blocks A := bOG and A′ := b′OG′, strongly ¯ G-graded algebras with 1-components B = bON and B′ = b′ON′ CG(N) ⊆ G′ C := OCG(N)

¯ G-graded ¯ G-acted algebras

Definition An algebra C is a ¯ G-graded ¯ G-acted algebra if

1 C is ¯

G-graded, i.e. C = ⊕¯

g∈GC¯ g;

2

¯ G acts on C (always on the left in this presentation);

3 ∀¯

h ∈ ¯ G, ∀c ∈ C¯

h we have that

c

¯ g

∈ C ¯

h

¯ g

for all ¯ g ∈ ¯ G. Remark C := OCG(N) is a ¯ G-graded ¯ G-acted algebra. Moreover, there ex- ists two ¯ G-graded ¯ G-acted algebra homomorphisms ζ : C → CA(B) and ζ′ : C → CA′(B′), i.e. for any ¯ h ∈ ¯ G and c ∈ CA(B)¯

h, we

have ζ(c) ∈ CA(B)¯

h and ζ′(c) ∈ CA′(B′)¯ h and for every ¯

g ∈ ¯ G, ζ( c

¯ g ) = ζ(c) ¯ g

and ζ′( c

¯ g ) = ζ′(c) ¯ g

.

¯ G-graded bimodules over C

Definition We say that ˜ M is a ¯ G-graded (A, A′)-bimodule over C if:

1

˜ M is an (A, A′)-bimodule;

2

˜ M has a decomposition ˜ M =

¯ g∈¯ G ˜

M¯

g such that A¯ g ˜

M¯

xA′ ¯ h ⊆

˜ M¯

g¯ x¯ h, for all ¯

g, ¯ x, ¯ h ∈ ¯ G;

3

˜ m¯

g · c =

c

¯ g

· ˜ m¯

g, for all c ∈ C, ˜

m¯

g ∈ ˜

M¯

g, ¯

g ∈ ¯ G, where c · ˜ m = ζ(c) · ˜ m and ˜ m · c = ˜ m · ζ′(c), for all c ∈ C, ˜ m ∈ ˜ M. Remark Note that homomorphisms between ¯ G-graded (A, A′)-bimodules

• ver C are just homomorphism between ¯

G-graded (A, A′)-bimodules.

¯ G-graded Morita equivalences over C

Definition We say that a ¯ G-graded (A, A′)-bimodule over C, ˜ M, induces a ¯ G- graded Morita equivalence over C between A and A′, if ˜ M ⊗A′ ˜ M∗ ∼ = A as ¯ G-graded (A, A)-bimodules over C and that ˜ M∗ ⊗A ˜ M ∼ = A′ as ¯ G-graded (A′, A′)-bimodules over C, where the A-dual ˜ M∗ = HomA( ˜ M, A) of ˜ M is a ¯ G-graded (A′, A)-bimodule.

∆C

We regard A′op as a ¯ G-graded algebra with components (A′op)¯

g =

A′

¯ g−1, ∀¯

g ∈ ¯

• G. We denote by ∗ the multiplicative operations in

A′op. We also define the (¯ g, ¯ h) component of (A ⊗C A′op)(¯

g,¯ h) :=

A¯

g ⊗C A′op ¯ h . Let

δ(¯ G) := {(¯ g, ¯ g) | ¯ g ∈ ¯ G} be the diagonal subgroup of ¯ G × ¯ G, and let ∆C be the diagonal part

• f A ⊗C A′op:

∆C := ∆(A ⊗C A′op) := (A ⊗C A′op)δ(¯

G) =

• ¯

g∈¯ G

A¯

g ⊗C A′ ¯ g−1,

which clearly has the 1-component defined as follows: ∆C

1 = B ⊗C B′op.

Lemma ∆C is an O-algebra and there exists an O-algebra homomorphism from C to ∆C: ϕ : C → Z(∆C

1 ), ϕ(c) := ζ(c) ⊗C 1 = 1 ⊗C ζ′(c).

Lemma A ⊗C A′op is a right ∆C-module and a ¯ G-graded (A, A′)-bimodule

• ver C.

Lemma Let M be a ∆C-module, then A ⊗B M, M ⊗B′ A′, (A ⊗C A′op) ⊗∆C M are isomorphic as ¯ G-graded (A, A′)-bimodules over C. We shall denote them by M.

¯ G-graded bimodules over C

Lemma

1 Let M be a ∆(A⊗C A′op)-module and M′ be a ∆(A′ ⊗C A′′op)-

• module. Then M ⊗B′ M′ is a ∆(A ⊗C A′′op)-module with the

multiplication operation defined as follows: (a¯

g⊗Ca′′op ¯ g−1)(m⊗B′m′) := (a¯ g⊗C(u′−1 ¯ g

)op)m⊗B′(u′

¯ g⊗Ca′′op ¯ g−1)m′

for all ¯ g ∈ ¯ G, a¯

g ∈ A¯ g, a′′op ¯ g−1 ∈ A′′op ¯ g−1, m ∈ M, m′ ∈ M′.

Moreover, we have the isomorphism

• M ⊗B′ M′ ≃

M ⊗A′ M′

• f ¯

G-graded (A, A′′)-bimodules over C.

¯ G-graded bimodules over C

Lemma

2 Let M be a ∆(A′ ⊗C Aop)-module and M′ be a ∆(A′ ⊗C A′′op)-

• module. Then HomB′ (M, M′) is a ∆(A ⊗C A′′op)-module with

the following operation: (a¯

gfa′′ ¯ g−1)(m) := (u¯ g ⊗C (a′′ ¯ g−1)op)f ((u−1 ¯ g

⊗C aop

¯ g )m)

for all ¯ g ∈ ¯ G and for all a¯

g ∈ A¯ g, a′′ ¯ g−1 ∈ A′′ ¯ g−1, m ∈ M,

f ∈ HomB′ (M, M′). Moreover, we have the isomorphism

• HomB′ (M, M′) ≃ HomA′
• M,

M′

• f ¯

G-graded (A, A′′)-bimodules over C.

Theorem Let M

B B′ and

M∗

B′ B := HomB (M, B) (the B-dual of M) be two

bimodules that induce a Morita equivalence between B and B′: B

M∗⊗B−

B′

M⊗B′−

• If M extends to a ∆C-module, then we have the following:

1 M∗ becomes a ∆(A′ ⊗C Aop)-module; 2

• M := (A ⊗C A′op) ⊗∆C M and

M∗ := (A′ ⊗C Aop) ⊗∆(A′⊗CAop) M∗ are ¯ G-graded (A, A′)-bimodules over C and they induce a ¯ G-graded Morita equivalence over C between A and A′: A

∼

A′.

Module triples and character triples

In this section, we attempt to give a version with Morita equivalences for the relationship ≤c given in [2, Definition 2.7.]. Proposition Let A and A′ be two strongly ¯ G-graded algebras over C. Assume that ˜ M is a ¯ G-graded (A, A′)-bimodule over C, which induces a Morita equivalence between A and A′. Let U be a (left) B-module and let U′ be a (left) B′-module corresponding to U under the given

• equivalence. Then there is a commutative diagram:

E(U)

∼

E(U′)

CA(B)

∼

• CA′(B′)
• C
• idC

C.

Relation ≥c for module triples

Definition Let V be a G-invariant simple KB-module, V ′ a G′-invariant simple KB′-module. We say that (A, B, V ) ≥c (A′, B′, V ′) if

1 G = G′N, N′ = N ∩ G′ 2 CG(N) ⊆ G′ 3 we have the following commutative diagram of ¯

G-graded K- algebras: E(V )

∼

E(V ′)

KC

• idC

KC.

• where KC = KCG(N) is regarded as a ¯

G-graded ¯ G-acted K- algebra, with 1-component KZ(N).

Relation ≥c for module triples

Proposition Assume that ˜ M induces a ¯ G-graded Morita equivalence over C := OCG(N) between A and A′. Let V be a simple KB-module and V ′ be a simple KB′-module corresponding to V ′ via the given corre-

• spondence. Then we have that (A, B, V ) ≥c (A′, B′, V ′).

Proposition Let θ be the character associated to V and θ′ the character as- sociated to V ′. If (A, B, V ) ≥c (A′, B′, V ′), then (G, N, θ) ≥c (G′, N′, θ′).

Butterfly theorem Let ˆ G be another group with normal subgroup N. Assume that:

1 CG(N) ⊆ G′, 2

˜ M induces a ¯ G-graded Morita equiv. over C between A and A′;

3 the conjugation maps ε : G → Aut(N) and ˆ

ε : ˆ G → Aut(N) satisfy ε(G) = ˆ ε(ˆ G). Denote ˆ G′ = ˆ ε−1(ε(G′)). Then there is a ˆ G/N-graded Morita equiv- alence over ˆ C := OCˆ

G(N) between ˆ

A := bO ˆ G and ˆ A′ := b′O ˆ G′. ˆ A := bO ˆ G A := bOG

˜ M ∼ A′ := b′OG′

ˆ A′ := b′O ˆ G′ bONCˆ

G(N)

bONCG(N)

∼ b′ON′CG(N)

b′ON′Cˆ

G(N)

B := ONb

M ∼ B′ := ON′b′.

Bibliography

[1]

Marcus, A. and Minut

,˘

a, V.A., Group graded endomorphism algebras and Morita equivalences, preprint 2019;

[2]

Sp¨ ath, B., Reduction theorems for some global-local conjec-

• tures. In: Local Representations Theory and Simple Groups.

EMS Ser. Congr. Rep, Eur. Math. Soc., Z¨ urich (2018), 23– 62.