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 ( O N ) and b ′ ∈ Z ( O N ′ ) are ¯ G -invariant blocks A := b O G and A ′ := b ′ O G ′ , strongly ¯ G -graded algebras with 1-components B = b O N and B ′ = b ′ O N ′ C G ( N ) ⊆ G ′ C := O C G ( 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 ; g ∈ G C ¯ ¯ G acts on C (always on the left in this presentation); 2 3 ∀ ¯ h ∈ ¯ g ∈ ¯ ¯ g G , ∀ c ∈ C ¯ h we have that ∈ C ¯ for all ¯ G . c ¯ g h Remark C := O C G ( N ) is a ¯ G -graded ¯ G -acted algebra. Moreover, there ex- ists two ¯ G -graded ¯ G -acted algebra homomorphisms ζ : C → C A ( B ) and ζ ′ : C → C A ′ ( B ′ ), i.e. for any ¯ h ∈ ¯ G and c ∈ C A ( B ) ¯ h , we g ∈ ¯ have ζ ( c ) ∈ C A ( B ) ¯ h and ζ ′ ( c ) ∈ C A ′ ( B ′ ) ¯ h and for every ¯ G , g ) = ζ ( c ) g ) = ζ ′ ( c ) ¯ ¯ ¯ ¯ g and ζ ′ ( c g ζ ( c .

¯ G -graded bimodules over C Definition We say that ˜ M is a ¯ G-graded ( A , A ′ ) -bimodule over C if: ˜ M is an ( A , A ′ )-bimodule; 1 M = � M has a decomposition ˜ ˜ G ˜ g ˜ x A ′ M ¯ g such that A ¯ M ¯ h ⊆ 2 g ∈ ¯ ¯ ¯ ˜ x , ¯ h ∈ ¯ M ¯ h , for all ¯ g , ¯ G ; x ¯ g ¯ g ∈ ˜ g ∈ ¯ ¯ g m ¯ ˜ g · c = c · ˜ m ¯ g , for all c ∈ C , ˜ m ¯ M ¯ g , ¯ G , where 3 m ∈ ˜ c · ˜ m = ζ ( c ) · ˜ m and ˜ m · c = ˜ m · ζ ′ ( c ), for all c ∈ C , ˜ M . Remark Note that homomorphisms between ¯ G -graded ( A , A ′ )-bimodules over 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 ∗ ∼ M ∗ ⊗ A ˜ = A as ¯ G -graded ( A , A )-bimodules over C and that ˜ = A ′ as ¯ M ∼ ˜ G -graded ( A ′ , A ′ )-bimodules over C , where the A -dual M ∗ = Hom A ( ˜ ˜ 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 = g ∈ ¯ A ′ g − 1 , ∀ ¯ G . We denote by ∗ the multiplicative operations in ¯ g , ¯ A ′ op . We also define the (¯ h ) component of ( A ⊗ C A ′ op ) (¯ h ) := g , ¯ g ⊗ C A ′ op h . Let A ¯ ¯ δ (¯ g ∈ ¯ G ) := { (¯ g , ¯ g ) | ¯ G } G , and let ∆ C be the diagonal part be the diagonal subgroup of ¯ G × ¯ of A ⊗ C A ′ op : � ∆ C := ∆( A ⊗ C A ′ op ) := ( A ⊗ C A ′ op ) δ (¯ g ⊗ C A ′ G ) = g − 1 , A ¯ ¯ g ∈ ¯ ¯ G 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 over 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: g ⊗ C a ′′ op g ⊗ C ( u ′− 1 g ⊗ C a ′′ op g − 1 )( m ⊗ B ′ m ′ ) := ( a ¯ ) op ) m ⊗ B ′ ( u ′ g − 1 ) m ′ ( a ¯ ¯ g ¯ ¯ ¯ g − 1 , m ∈ M , m ′ ∈ M ′ . g ∈ ¯ g , a ′′ op g − 1 ∈ A ′′ op for all ¯ G , a ¯ g ∈ A ¯ ¯ ¯ Moreover, we have the isomorphism � M ⊗ B ′ M ′ ≃ � M ⊗ A ′ � M ′ of ¯ G -graded ( A , A ′′ )-bimodules over C .

¯ G -graded bimodules over C Lemma 2 Let M be a ∆( A ′ ⊗ C A op )-module and M ′ be a ∆( A ′ ⊗ C A ′′ op )- module. Then Hom B ′ ( M , M ′ ) is a ∆( A ⊗ C A ′′ op )-module with the following operation: g − 1 ) op ) f (( u − 1 ⊗ C a op g fa ′′ g ⊗ C ( a ′′ ( a ¯ g − 1 )( m ) := ( u ¯ g ) m ) ¯ ¯ g ¯ ¯ g ∈ ¯ g , a ′′ g − 1 ∈ A ′′ for all ¯ G and for all a ¯ g ∈ A ¯ g − 1 , m ∈ M , ¯ ¯ f ∈ Hom B ′ ( M , M ′ ). Moreover, we have the isomorphism � � � � M , � Hom B ′ ( M , M ′ ) ≃ Hom A ′ M ′ of ¯ G -graded ( A , A ′′ )-bimodules over C .

� � Theorem M ∗ Let M B ′ and B := Hom B ( M , B ) (the B -dual of M ) be two B B ′ bimodules that induce a Morita equivalence between B and B ′ : M ∗ ⊗ B − � B ′ B M ⊗ B ′ − If M extends to a ∆ C -module, then we have the following: 1 M ∗ becomes a ∆( A ′ ⊗ C A op )-module; M ∗ := ( A ′ ⊗ C A op ) ⊗ ∆( A ′ ⊗ C A op ) � M := ( A ⊗ C A ′ op ) ⊗ ∆ C M and � 2 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 ) ∼ C A ( B ) C A ′ ( B ′ ) id C C . C

� � Relation ≥ c for module triples Definition Let V be a G -invariant simple K B -module, V ′ a G ′ -invariant simple K B ′ -module. We say that ( A , B , V ) ≥ c ( A ′ , B ′ , V ′ ) if 1 G = G ′ N , N ′ = N ∩ G ′ 2 C G ( N ) ⊆ G ′ 3 we have the following commutative diagram of ¯ G -graded K - algebras: ∼ � E ( V ′ ) E ( V ) id C K C K C . where K C = K C G ( N ) is regarded as a ¯ G -graded ¯ G -acted K - algebra, with 1-component K Z ( N ).

Relation ≥ c for module triples Proposition Assume that ˜ M induces a ¯ G -graded Morita equivalence over C := O C G ( N ) between A and A ′ . Let V be a simple K B -module and V ′ be a simple K B ′ -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 C G ( N ) ⊆ G ′ , M induces a ¯ ˜ G -graded Morita equiv. over C between A and A ′ ; 2 ε : ˆ 3 the conjugation maps ε : G → Aut( N ) and ˆ G → Aut( N ) ε (ˆ satisfy ε ( G ) = ˆ G ). G ′ = ˆ Denote ˆ ε − 1 ( ε ( G ′ )). Then there is a ˆ G / N -graded Morita equiv- A ′ := b ′ O ˆ alence over ˆ G ( N ) between ˆ A := b O ˆ G and ˆ C := O C ˆ G ′ . ˜ ∼ A ′ := b ′ O G ′ A ′ := b ′ O ˆ A := b O ˆ ˆ M ˆ G ′ G A := b O G ∼ b ′ O N ′ C G ( N ) b ′ O N ′ C ˆ b O NC ˆ G ( N ) b O NC G ( N ) G ( N ) ∼ B ′ := O N ′ b ′ . M B := O Nb

Bibliography Marcus, A. and Minut , ˘ a, V.A., Group graded endomorphism [1] algebras and Morita equivalences , preprint 2019; Sp¨ ath, B., Reduction theorems for some global-local conjec- [2] tures . In: Local Representations Theory and Simple Groups. EMS Ser. Congr. Rep, Eur. Math. Soc., Z¨ urich (2018), 23– 62.