Brauer Trees and Brauer Tree Algebras Adam Wood Department of - PowerPoint PPT Presentation

Brauer Trees and Brauer Tree Algebras Adam Wood Department of Mathematics University of Iowa Bradley University Math Colloquium November 14, 2019

Outline Overview of Representation Theory Brauer Trees Representations of Finite Groups Connection Between Brauer Trees and Representation Theory

Representation Theory Goal: Understand representations of certain algebraic objects ◮ Different approaches and types of representation theory

Representation Theory Goal: Understand representations of certain algebraic objects ◮ Different approaches and types of representation theory ◮ Representations of finite dimensional algebras

Representation Theory Goal: Understand representations of certain algebraic objects ◮ Different approaches and types of representation theory ◮ Representations of finite dimensional algebras ◮ Special case: representations of finite groups

Representation Theory Goal: Understand representations of certain algebraic objects ◮ Different approaches and types of representation theory ◮ Representations of finite dimensional algebras ◮ Special case: representations of finite groups ◮ Indecomposable representations

Graph The main objects in this talk are graphs A graph is a a collection of vertices and edges. If the edges have a specified direction, the graph is directed . Otherwise, the graph is undirected . Examples: • • • • • • • • • Undirected Graph Directed Graph

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 ◮ We view the graph in the plane and assume a counterclockwise orientation of the edges ◮ Notation for a Brauer tree: T = ( T 0 , T 1 , m )

Example T = ( T 0 , T 1 , m ) 4 c a b 1 2 3 d 5 T 0 = { 1 , 2 , 3 , 4 , 5 } T 1 = { a , b , c , d } m = 2 Vertex 4 is the exceptional vertex and has multiplicity 2

Example T = ( T 0 , T 1 , m ) 4 c a b 1 2 3 d 5 Orientation at 2 b < a and a < b

Example T = ( T 0 , T 1 , m ) 4 c a b 1 2 3 d 5 Orientation at 3 c < b < d < c

Example T = ( T 0 , T 1 , m ) 4 c a b 1 2 3 d 5 Orientation at 4 c < c

Brauer Trees − → Quivers Let T = ( T 0 , T 1 , m ) be a Brauer tree. Definition A quiver is a finite directed graph Q = ( Q 0 , Q 1 ), where loops and multiple edges are allowed. Build a quiver Q = ( Q 0 , Q 1 ) from T . Q 0 = T 1 , the vertices of Q are the edges of T There is an arrow a : i → j if i < j and j is the “next ” edge after i . In this case, a is said to be given by the successor relation ( i , j ).

Example Recall T = ( T 0 , T 1 , m ) 4 c a b 1 2 3 d 5 Q = ( Q 0 , Q 1 ) c a b d

Special Cycles Let v ∈ T 0 be a vertex. ◮ If v is not exceptional and #(edges adjacent to v ) ≥ 2, then there is an oriented cycle in Q , unique up to cyclic permutation.

Special Cycles Let v ∈ T 0 be a vertex. ◮ If v is not exceptional and #(edges adjacent to v ) ≥ 2, then there is an oriented cycle in Q , unique up to cyclic permutation. ◮ If v is exceptional and #(edges adjacent to v ) · m ≥ 2, then there is an oriented cycle in Q , unique up to cyclic permutation.

Special Cycles Let v ∈ T 0 be a vertex. ◮ If v is not exceptional and #(edges adjacent to v ) ≥ 2, then there is an oriented cycle in Q , unique up to cyclic permutation. ◮ If v is exceptional and #(edges adjacent to v ) · m ≥ 2, then there is an oriented cycle in Q , unique up to cyclic permutation. ◮ Call this cycle the special cycle at v .

Special Cycles Let v ∈ T 0 be a vertex. ◮ If v is not exceptional and #(edges adjacent to v ) ≥ 2, then there is an oriented cycle in Q , unique up to cyclic permutation. ◮ If v is exceptional and #(edges adjacent to v ) · m ≥ 2, then there is an oriented cycle in Q , unique up to cyclic permutation. ◮ Call this cycle the special cycle at v . ◮ If the cycle starts at i ∈ Q 0 = G 1 , call it the special i -cycle at v .

Example T = ( T 0 , T 1 , m ) 4 c a b 1 2 3 d 5 Q = ( Q 0 , Q 1 ) c Special cycle at 2 a b d

Example T = ( T 0 , T 1 , m ) 4 c a b 1 2 3 d 5 Q = ( Q 0 , Q 1 ) c Special cycle at 3 a b d

Example T = ( T 0 , T 1 , m ) 4 c a b 1 2 3 d 5 Q = ( Q 0 , Q 1 ) c Special cycle at 4 a b d

Brauer Tree − → Algebra Let T = ( T 0 , T 1 , m ). There are two ways of building an algebra over a field k associated to T . 1. Get the associated quiver Q , define certain relations I , and define Γ T = kQ / I to be the path algebra with relations. 2. Define an algebra Λ T over k by defining the projective indecomposable Λ-modules via the graph T . These two methods gives the same result. That is, Γ T ∼ = Λ T .

Γ T , Path Algebra with Relations c ι γ α a b ǫ β δ d Special a -cycle at 2: αβ Special b -cycle at 3: δǫγ Special b -cycle at 2: βα Special d -cycle at 3: ǫγδ Special c -cycle at 3: γδǫ Special c cycle at 4: ι

Γ T , Path Algebra with Relations c ι γ α a b ǫ β δ d Relations βα = δǫγ γδǫ = ι 2 αβα = βαβ = γδǫγ = δǫγδ = ǫγδǫ = ι 3 = 0 αδ = γβ = ǫι = ιγ = 0

Γ T , Path Algebra with Relations c ι γ α a b ǫ β δ d kQ / I is a k -vector space with allowable paths given by α , β , γ , δ , ǫ , ι αβ , βα , δǫ , ǫγ , γδ , ι 2 ǫγδ Multiply by concatenating paths

Review of Brauer Trees Brauer Tree Quiver Path Algebra

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 1

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 2

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 3

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 4

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 5

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 6

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 6 = 2 mod 12

Groups 12 11 1 10 2 9 3 8 4 7 5 6 2 + 0 = 2

Groups 12 11 1 10 2 9 3 8 4 7 5 6 2

Groups 12 11 1 10 2 9 3 8 4 7 5 6 2 − 1

Groups 12 11 1 10 2 9 3 8 4 7 5 6 2 − 2

Groups 12 11 1 10 2 9 3 8 4 7 5 6 2 − 2 = 0

Definition of a Group Definition A group is a set G with an operation · so that ◮ g · h ∈ G for all g , h ∈ G (closure) ◮ ( a · b ) · c = a · ( b · c ) for all a , b , c ∈ G (associativity) ◮ There is an identity e ∈ G satisfying e · g = g for all g ∈ G (identity) ◮ For every g ∈ G , there is an inverse element, g − 1 , so that g · g − 1 = e = g − 1 · g (inverse)

Examples ◮ ( R − { 0 } , · )

Examples ◮ ( R − { 0 } , · ) ◮ ( Z , +)

Examples ◮ ( R − { 0 } , · ) ◮ ( Z , +) ◮ Z / 3 Z = { 0 , 1 , 2 } , addition modulo 3, 2 + 2 = 1

Examples ◮ ( R − { 0 } , · ) ◮ ( Z , +) ◮ Z / 3 Z = { 0 , 1 , 2 } , addition modulo 3, 2 + 2 = 1 ◮ ( Z , · ) is NOT a group

A Representation of a Group Definition Let G be a finite group and let k be a field. A representation of G is a group homomorphism ρ : G → GL ( V ) , where V is a vector space over k . For g ∈ G , we think of ρ ( g ) as an n × n matrix, where n is the dimension of V over k .

Example: Dihedral Group Consider G = D 8 = � r , s | r 4 = 1 = s 2 , srs = r − 1 � .

Example: Dihedral Group Consider G = D 8 = � r , s | r 4 = 1 = s 2 , srs = r − 1 � . � 0 � rotation by − 1 r 90 ◦ 1 0 � − 1 � reflection 0 s about y -axis 0 1

Example: Dihedral Group Consider G = D 8 = � r , s | r 4 = 1 = s 2 , srs = r − 1 � . � 0 � rotation by − 1 r 90 ◦ 1 0 � − 1 � reflection 0 s about y -axis 0 1

Example: Dihedral Group Consider G = D 8 = � r , s | r 4 = 1 = s 2 , srs = r − 1 � . � 0 � rotation by − 1 r 90 ◦ 1 0 � − 1 � reflection 0 s about y -axis 0 1

Example: Dihedral Group Consider G = D 8 = � r , s | r 4 = 1 = s 2 , srs = r − 1 � . � 0 � rotation by − 1 r 90 ◦ 1 0 � − 1 � reflection 0 s about y -axis 0 1 � 0 � � − 1 � − 1 0 ρ : G → M 2 ( C ) defined by ρ ( r ) = and ρ ( s ) = 1 0 0 1 defines a representation of G .

Subrepresentations and Direct Sums Definition (Direct Sum) � V � 0 V ⊕ W = 0 W

Subrepresentations and Direct Sums Definition (Direct Sum) � V � 0 V ⊕ W = 0 W V and W are subrepresentations of the direct sum of V and W

Subrepresentations and Direct Sums Definition (Direct Sum) � V � 0 V ⊕ W = 0 W V and W are subrepresentations of the direct sum of V and W BUT, there are more subrepresentations

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