over Characteristic 2 R. A. Spencer DPMMS, University of Cambridge - PowerPoint PPT Presentation

Tensor Products of Restricted Simples of SL 4 over Characteristic 2 R. A. Spencer DPMMS, University of Cambridge

The Question If { L ( λ ) : λ ∈ Λ } is the set of all simple SL 4 modules over an algebraically closed field k of characteristic 2, what is the structure of L ( λ ) ⊗ k L ( µ )? 1

A Generalised Form of Alperin Diagram Quasi-Hereditary Algebras Tensor Products of Simples of SL 4 Application 2

A Generalised Form of Alperin Diagram

Alperin Diagrams: An Overview • Diagram for conveying submodule structure • Defined in the 1980s, but often used loosely • Only describes a small class of modules Definition (Often) A D An Alperin Diagram for module M is a quiver Q with a lattice bijection δ from the lattice of arrow B C closed subsets of Q to the lattice of submodules of M . A • Vertices of quiver labeled with simple module isomorphism classes • Edges correspond to non-split extensions as subquotients 3

Alperin Diagrams: An Overview • Diagram for conveying submodule structure • Defined in the 1980s, but often used loosely • Only describes a small class of modules Definition (Often) A D An Alperin Diagram for module M is a quiver Q with a lattice bijection δ from the lattice of arrow B C closed subsets of Q to the lattice of submodules of M . A • Vertices of quiver labeled with simple module isomorphism classes • Edges correspond to non-split extensions as subquotients 3

Alperin Diagrams: An Overview • Diagram for conveying submodule structure • Defined in the 1980s, but often used loosely • Only describes a small class of modules Definition (Often) A D An Alperin Diagram for module M is a quiver Q with a lattice bijection δ from the lattice of arrow B C closed subsets of Q to the lattice of submodules of M . A • Vertices of quiver labeled with simple module isomorphism classes • Edges correspond to non-split extensions as subquotients 3

Alperin Diagrams: An Overview • Diagram for conveying submodule structure • Defined in the 1980s, but often used loosely • Only describes a small class of modules Definition (Often) A D An Alperin Diagram for module M is a quiver Q with a lattice bijection δ from the lattice of arrow B C closed subsets of Q to the lattice of submodules of M . A • Vertices of quiver labeled with simple module isomorphism classes • Edges correspond to non-split extensions as subquotients 3

Alperin Diagrams: An Overview • Diagram for conveying submodule structure • Defined in the 1980s, but often used loosely • Only describes a small class of modules Definition (Often) A D An Alperin Diagram for module M is a quiver Q with a lattice bijection δ from the lattice of arrow B C closed subsets of Q to the lattice of submodules of M . A • Vertices of quiver labeled with simple module isomorphism classes • Edges correspond to non-split extensions as subquotients 3

Alperin Diagrams: An Overview • Diagram for conveying submodule structure • Defined in the 1980s, but often used loosely • Only describes a small class of modules Definition (Often) A D An Alperin Diagram for module M is a quiver Q with a lattice bijection δ from the lattice of arrow B C closed subsets of Q to the lattice of submodules of M . A • Vertices of quiver labeled with simple module isomorphism classes • Edges correspond to non-split extensions as subquotients 3

Alperin Diagrams: When They Fail and Alternatives Problems: • The requirement δ is a bijection is very strong • Requires infinite quivers or only finitely many submodules • Infinitely many submodules occur frequently (e.g. R ⊕ R over R ) Possible Solutions: • Drop surjectivity requirement on δ • Generalise diagrams based on certain classes of filtrations (e.g. radical, socle, socle-isotypic, etc.) • Require socle and radical series to be read off 4

Alperin Diagrams: Our Alternative • An injective diagram, based on generated submodules , annotated to give the socle and radical series • Procedure for module M : • Find n vectors { v i } where n is the composition length of M such that, • � v 1 � = M • � v i � = � v j � ⇐ ⇒ i = j • � v i � / rad � v i � is simple • Draw a line v i → v j if v j ∈ rad � v i �\ rad 2 � v i � and � v j � / rad 2 � v i � ֒ → � v i � / rad 2 � v i � is not split • Construct δ to take the arrow-closure of v i to � v i � , be lattice and top preserving. • Decorate with more vectors to highlight socle and radical series and other submodule structure . • Examples to come in the context of quasi-hereditary algebras 5

Quasi-Hereditary Algebras

Quasi-hereditary Algebras • Really a class of categories of modules • Simple modules L ( λ ) labeled by poset (Λ , ≤ ) • Standard and costandard modules ∆( λ ) and ∇ ( λ ) for each λ ∈ Λ • Simple head (resp. socle) of L ( λ ) • All other factors L ( µ ) for µ < λ • Maximal such quotient of projective cover (resp. submodule of injective hull) of L ( λ ) • Indecomposable tilting modules (both ∆- and ∇ -filtrations) T ( λ ) 6

Rational (co)Modules of Algebraic Groups • Λ is the set of dominant weights • Tuples of naturals • ≤ not lexicographical: depends on certain coroots • Each L ( λ ), ∆( λ ), ∇ ( λ ) and T ( λ ) have highest weight λ . • Contravariant dual • Tilting modules contravariantly self-dual 7

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Example of Alternative Alperin Diagram The module ∆(3 , 0) of type G 2 over characteristic 2 v 0 30 v 7 10 v 8 00 ⊕ v 1 ⊕ v 2 20 20 ⊕ v 6 v 5 v 3 00 00 01 ⊕ v 4 10 8

Tensor Products of Simples of SL 4

Return to the question If { L ( λ ) : λ ∈ Λ } is the set of all simple SL 4 modules over an algebraically closed field k of characteristic 2, what is the structure of L ( λ ) ⊗ k L ( µ )? 9

Philosophy • “Twisting” by the Frobenius automorphism of G allows us to reduce to finitely many cases sometimes • Write “base p ” � p j λ j � p j µ j λ = , µ = j ≥ 0 j ≥ 0 for p -restricted weights λ j and µ j • E.g. (3 , 14 , 5) = (1 , 0 , 1)+2 × (1 , 1 , 0)+2 2 × (0 , 1 , 1)+2 3 × (0 , 1 , 0) • By the Steinburg tensor product theorem L ( λ ) ⊗ L ( µ ) ∼ ( L ( λ j ) ⊗ L ( µ j )) [ j ] � = j ∈ N 0 10

Philosophy • “Twisting” by the Frobenius automorphism of G allows us to reduce to finitely many cases sometimes • Write “base p ” � p j λ j � p j µ j λ = , µ = j ≥ 0 j ≥ 0 for p -restricted weights λ j and µ j • E.g. (3 , 14 , 5) = (1 , 0 , 1)+2 × (1 , 1 , 0)+2 2 × (0 , 1 , 1)+2 3 × (0 , 1 , 0) • By the Steinburg tensor product theorem L ( λ ) ⊗ L ( µ ) ∼ ( L ( λ j ) ⊗ L ( µ j )) [ j ] � = j ∈ N 0 10