Trees, Strings, and Representation Theory Adam Wood Department of - PowerPoint PPT Presentation

Trees, Strings, and Representation Theory Adam Wood Department of Mathematics University of Iowa Graduate and Undergraduate Student Seminar University of Iowa October 9, 2019

Outline Motivating Example Representation Theory of Finite Groups Brauer Tree Algebras Representation Theory of Special Linear Groups

Motivating Example p prime, G cyclic group order p n , k algebraically closed field, char( k ) = p

Motivating Example p prime, G cyclic group order p n , k algebraically closed field, char( k ) = p Describe ALL the indecomposable representations

Motivating Example p prime, G cyclic group order p n , k algebraically closed field, char( k ) = p Describe ALL the indecomposable representations Indecomposable representations given by Jordan blocks   1 1 0 · · · 0 0    0 1 1 · · · 0 0       . . . .   ... ... . . . .     . . . . j     1 0        0 0 · · · 1 1       0 0 · · · 0 1  for 1 ≤ j ≤ p n (see Local Representation Theory , J.L. Alperin)

Motivating Example Indecomposable representations

Motivating Example Indecomposable representations Uniserial kG -modules of length j for 1 ≤ j ≤ p n , with trivial composition factors

Motivating Example Indecomposable representations Uniserial kG -modules of length j for 1 ≤ j ≤ p n , with trivial composition factors kG is a Brauer tree algebra for the Brauer tree ◦ • with multiplicity m = p n − 1

Representation Theory of Finite Groups Definition Let G be a finite group and let k be a field. The group ring is defined to be the set     � kG = a g g | a g ∈ k   g ∈ G with multiplication given by group multiplication. This space is a vector space of dimension | G | over k .

Representation Theory of Finite Groups Definition Let G be a finite group and let k be a field. The group ring is defined to be the set     � kG = a g g | a g ∈ k   g ∈ G with multiplication given by group multiplication. This space is a vector space of dimension | G | over k . Definition A representation of a finite group G over a field k is a kG -module.

Modular Representation Theory Representations of a group over a field of prime characteristic

Modular Representation Theory Representations of a group over a field of prime characteristic Study of kG -modules

Modular Representation Theory Representations of a group over a field of prime characteristic Study of kG -modules Theorem (Drozd, Crawley-Boevey) A finite dimensional algebra Λ over an algebraically closed field is one of the following mutually exclusive types: 1. Finite (finitely many indecomposable modules) 2. Tame (infinitely many indecomposable modules, can be parametrized) 3. Wild (A full subcategory of Λ -mod is equivalent to k � x , y � -mod)

Modular Representation Theory Representations of a group over a field of prime characteristic Study of kG -modules Theorem (Drozd, Crawley-Boevey) A finite dimensional algebra Λ over an algebraically closed field is one of the following mutually exclusive types: 1. Finite (finitely many indecomposable modules) 2. Tame (infinitely many indecomposable modules, can be parametrized) 3. Wild (A full subcategory of Λ -mod is equivalent to k � x , y � -mod) Theorem (Higman) Let G be a finite group and let k be an algebraically closed field of characteristic p. Then, kG is of finite representation type if and only if G has cyclic Sylow p-subgroups.

Blocks and Brauer Trees kG = B 1 ⊕ · · · ⊕ B m Unique decomposition into indecomposable subalgebras

Blocks and Brauer Trees kG = B 1 ⊕ · · · ⊕ B m Unique decomposition into indecomposable subalgebras Each block B has a defect group D ≤ G , measures deviation of B from being semisimple as an algebra

Blocks and Brauer Trees kG = B 1 ⊕ · · · ⊕ B m Unique decomposition into indecomposable subalgebras Each block B has a defect group D ≤ G , measures deviation of B from being semisimple as an algebra Theorem (See Chapter V, Alperin) If B is a block of kG with cyclic defect group, then B is a Brauer tree algebra

Module Definitions Let M be a kG -module.

Module Definitions Let M be a kG -module. � rad( M ) = (maximal submodules of M )

Module Definitions Let M be a kG -module. � rad( M ) = (maximal submodules of M ) � soc( M ) = (simple submodules of M ) rad i ( M ) = rad(rad i − 1 ( M ))

Module Definitions Let M be a kG -module. � rad( M ) = (maximal submodules of M ) � soc( M ) = (simple submodules of M ) rad i ( M ) = rad(rad i − 1 ( M )) 0 = rad n ( M ) ⊂ rad n − 1 ( M ) ⊂ · · · ⊂ rad 2 ( M ) ⊂ rad( M ) ⊂ M

Module Definitions Let M be a kG -module. � rad( M ) = (maximal submodules of M ) � soc( M ) = (simple submodules of M ) rad i ( M ) = rad(rad i − 1 ( M )) 0 = rad n ( M ) ⊂ rad n − 1 ( M ) ⊂ · · · ⊂ rad 2 ( M ) ⊂ rad( M ) ⊂ M 0 ⊂ soc( M ) ⊂ soc 2 ( M ) ⊂ · · · ⊂ soc m − 1 ( M ) ⊂ soc m ( M ) = M

Module Definitions Definition A module M is called uniserial if it satisfies one of the following equivalent conditions. ◮ M has a unique composition series

Module Definitions Definition A module M is called uniserial if it satisfies one of the following equivalent conditions. ◮ M has a unique composition series ◮ The quotients of the radical series of M are simple

Module Definitions Definition A module M is called uniserial if it satisfies one of the following equivalent conditions. ◮ M has a unique composition series ◮ The quotients of the radical series of M are simple ◮ The quotients of the socle series of M are simple

Simple and Indecomposable Representations Definition A representation V of G is called simple if the only subrepresentations of V are 0 and V .

Simple and Indecomposable Representations Definition A representation V of G is called simple if the only subrepresentations of V are 0 and V . Definition A representation V is called indecomposable if whenever V = U ⊕ W is a direct sum decomposition of V , either U or W is zero.

Simple and Indecomposable Representations Definition A representation V of G is called simple if the only subrepresentations of V are 0 and V . Definition A representation V is called indecomposable if whenever V = U ⊕ W is a direct sum decomposition of V , either U or W is zero. Note that simple = ⇒ indecomposable but

Simple and Indecomposable Representations Definition A representation V of G is called simple if the only subrepresentations of V are 0 and V . Definition A representation V is called indecomposable if whenever V = U ⊕ W is a direct sum decomposition of V , either U or W is zero. Note that simple = ⇒ indecomposable but indecomposable � = ⇒ simple .

Representation Theory of Finite Groups Theorem (Maschke) Let G be a finite group and let k be a field of characteristic p. The group algebra kG is semisimple if and only if p ∤ | G | .

Representation Theory of Finite Groups Theorem (Maschke) Let G be a finite group and let k be a field of characteristic p. The group algebra kG is semisimple if and only if p ∤ | G | . So,....

Representation Theory of Finite Groups Theorem (Maschke) Let G be a finite group and let k be a field of characteristic p. The group algebra kG is semisimple if and only if p ∤ | G | . So,.... ◮ If p ∤ | G | (or char( k ) = 0), study the simple representations

Representation Theory of Finite Groups Theorem (Maschke) Let G be a finite group and let k be a field of characteristic p. The group algebra kG is semisimple if and only if p ∤ | G | . So,.... ◮ If p ∤ | G | (or char( k ) = 0), study the simple representations ◮ If p | | G | , study the indecomposable representations

Brauer Trees Definition A Brauer tree is a finite unoriented connected graph T = ( T 0 , T 1 ) with no loops or cycles satisfying the additional properties: 1. There is an exceptional vertex with a multiplicity m ≥ 1 2. For each vertex v , there is a cyclic ordering of edges incident with v

Brauer Trees Definition A Brauer tree is a finite unoriented connected graph T = ( T 0 , T 1 ) with no loops or cycles satisfying the additional properties: 1. There is an exceptional vertex with a multiplicity m ≥ 1 2. For each vertex v , there is a cyclic ordering of edges incident with v Notation and conventions: ◮ T 0 is the vertex set

Brauer Trees Definition A Brauer tree is a finite unoriented connected graph T = ( T 0 , T 1 ) with no loops or cycles satisfying the additional properties: 1. There is an exceptional vertex with a multiplicity m ≥ 1 2. For each vertex v , there is a cyclic ordering of edges incident with v Notation and conventions: ◮ T 0 is the vertex set ◮ T 1 is the edge set

Brauer Trees Definition A Brauer tree is a finite unoriented connected graph T = ( T 0 , T 1 ) with no loops or cycles satisfying the additional properties: 1. There is an exceptional vertex with a multiplicity m ≥ 1 2. For each vertex v , there is a cyclic ordering of edges incident with v Notation and conventions: ◮ T 0 is the vertex set ◮ T 1 is the edge set ◮ The exceptional vertex will be solid or bold; the other vertices will not be filled in or plain text