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 = (T0, T1) 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:

◮ T0 is the vertex set ◮ T1 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 = (T0, T1, m)

Example

T = (T0, T1, m) 4 1 2 3 5

a b c d

T0 = {1, 2, 3, 4, 5} T1 = {a, b, c, d} m = 2 Vertex 4 is the exceptional vertex and has multiplicity 2

Example

T = (T0, T1, m) 4 1 2 3 5

a b c d

Orientation at 2 b < a and a < b

Example

T = (T0, T1, m) 4 1 2 3 5

a b c d

Orientation at 3 c < b < d < c

Example

T = (T0, T1, m) 4 1 2 3 5

a b c d

Orientation at 4 c < c

Brauer Trees − → Quivers

Let T = (T0, T1, m) be a Brauer tree.

Definition

A quiver is a finite directed graph Q = (Q0, Q1), where loops and multiple edges are allowed. Build a quiver Q = (Q0, Q1) from T. Q0 = T1, 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 = (T0, T1, m) 4 1 2 3 5

a b c d

Q = (Q0, Q1) c a b d

Special Cycles

Let v ∈ T0 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 ∈ T0 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 ∈ T0 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 ∈ T0 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 ∈ Q0 = G1, call it the special i-cycle at

v.

Example

T = (T0, T1, m) 4 1 2 3 5

a b c d

Q = (Q0, Q1) c a b d Special cycle at 2

Example

T = (T0, T1, m) 4 1 2 3 5

a b c d

Q = (Q0, Q1) c a b d Special cycle at 3

Example

T = (T0, T1, m) 4 1 2 3 5

a b c d

Q = (Q0, Q1) c a b d Special cycle at 4

Brauer Tree − → Algebra

Let T = (T0, T1, m). There are two ways of building an algebra

• ver a field k associated to T.
• 1. Get the associated quiver Q, define certain relations I, and

define ΓT = kQ/I to be the path algebra with relations.

• 2. Define an algebra ΛT over k by defining the projective

indecomposable Λ-modules via the graph T. These two methods gives the same result. That is, ΓT ∼ = ΛT.

ΓT, Path Algebra with Relations

c a b d

γ ι α β δ ǫ

Special a-cycle at 2: αβ Special b-cycle at 2: βα Special c-cycle at 3: γδǫ Special b-cycle at 3: δǫγ Special d-cycle at 3: ǫγδ Special c cycle at 4: ι

ΓT, Path Algebra with Relations

c a b d

γ ι α β δ ǫ

Relations βα = δǫγ γδǫ = ι2 αβα = βαβ = γδǫγ = δǫγδ = ǫγδǫ = ι3 = 0 αδ = γβ = ǫι = ιγ = 0

ΓT, Path Algebra with Relations

c a b d

γ ι α β δ ǫ

kQ/I is a k-vector space with allowable paths given by α, β, γ, δ, ǫ, ι αβ, βα, δǫ, ǫγ, γδ, ι2 ǫγδ Multiply by concatenating paths

Review of Brauer Trees

Brauer Tree Quiver Path Algebra

Groups

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

8

Groups

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

8 + 1

Groups

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

8 + 2

Groups

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

8 + 3

Groups

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

8 + 4

Groups

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

8 + 5

Groups

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

8 + 6

Groups

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

8 + 6 = 2 mod 12

Groups

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

2 + 0 = 2

Groups

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

2

Groups

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

2 − 1

Groups

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

2 − 2

Groups

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

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/3Z = {0, 1, 2}, addition modulo 3, 2 + 2 = 1

Examples

◮ (R − {0}, ·) ◮ (Z, +) ◮ Z/3Z = {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 = D8 = r, s | r4 = 1 = s2, srs = r−1.

Example: Dihedral Group

Consider G = D8 = r, s | r4 = 1 = s2, srs = r−1. r rotation by 90◦ −1 1

• s

reflection about y-axis −1 1

Example: Dihedral Group

Consider G = D8 = r, s | r4 = 1 = s2, srs = r−1. r rotation by 90◦ −1 1

• s

reflection about y-axis −1 1

Example: Dihedral Group

Consider G = D8 = r, s | r4 = 1 = s2, srs = r−1. r rotation by 90◦ −1 1

• s

reflection about y-axis −1 1

Example: Dihedral Group

Consider G = D8 = r, s | r4 = 1 = s2, srs = r−1. r rotation by 90◦ −1 1

• s

reflection about y-axis −1 1

• ρ : G → M2(C) defined by ρ(r) =

−1 1

• and ρ(s) =

−1 1

• defines a representation of G.

Subrepresentations and Direct Sums

Definition (Direct Sum)

V ⊕ W = V W

Subrepresentations and Direct Sums

Definition (Direct Sum)

V ⊕ W = V W

• V and W are subrepresentations of the direct sum of V and W

Subrepresentations and Direct Sums

Definition (Direct Sum)

V ⊕ W = V 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 .

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 .

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.

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 .

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

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 .

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.

Characteristic of a Field

Z/pZ = {0, 1, . . . , p − 1} Two types of fields: Characteristic zero Prime characteristic Think of: C, Q Fp

Characteristic of a Field

Fp = {0, 1, . . . , p − 1}

Characteristic of a Field

Fp = {0, 1, . . . , p − 1} Two types of fields:

• Characteristic zero
• Prime characteristic

Characteristic of a Field

Fp = {0, 1, . . . , p − 1} Two types of fields:

• Characteristic zero
• Prime characteristic

Think of:

• C, Q
• Fp

Modular Representation Theory

Study of representations of G over k, field of prime characteristic

Modular Representation Theory

Study of representations of G over k, field of prime characteristic Every representation can be written as a direct sum of indecomposable representations.

Modular Representation Theory

Study of representations of G over k, field of prime characteristic Every representation can be written as a direct sum of indecomposable representations. Understand the indecomposable representations

Example: Cyclic Group

Consider G = Z/3Z = {0, 1, 2}, σ = 1, char(k) = 3

Example: Cyclic Group

Consider G = Z/3Z = {0, 1, 2}, σ = 1, char(k) = 3 Indecomposable Representations σ → (1) σ → 1 1 1

• σ →

  1 1 1 1 1  

Brauer Tree and Quiver for a Cyclic Group

G = Z/3Z T = (T0, T1, m)

• where m = 2

Q = (Q0, Q1)

• α

where α3 = 0

Block Theory

Let G be a finite group and let k be a field. The group ring kG decomposes into pieces called blocks. kG = B1 ⊕ · · · ⊕ Bm Goal: Understand these blocks Each simple representation “belongs” to some block 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

Special Linear Groups

Let Fp denote the finite field with p elements Define SL2(Fp) = a b c d

• ∈ Mat2×2(Fp) | a, b, c, d ∈ Fp, ad − bc = 1
• .

|SL2(Fp)| = 1 2p(p − 1)(p + 1) Goal: Understand the indecomposable representations of SL2(Fp)

• ver a field of characteristic p.

Example: Special Linear Groups

Consider the group SL2(F7) Two blocks, odd block and even block Brauer Tree for odd block

• 1

5 3

Quiver for odd block 1 5 3

α1 α2 β1 β2 α

Brauer Tree Algebra

• 1

5 3

1 5 3

α1 α2 β1 β2 α

Relations I = β1α1β1, α1β1, β2α2, α2β2, α2, α2α1, β1β2, αα2, β2α Only allowable loops in kQ/I are β1α1 and α For rest of talk, let Λ = kQ/I

Strings in Λ

Butler and Ringel, “Auslander-Reiten Sequences with Few Middle Terms and Applications to String Algebras,” Communications in Algebra (1987)

Strings in Λ

Butler and Ringel, “Auslander-Reiten Sequences with Few Middle Terms and Applications to String Algebras,” Communications in Algebra (1987) {Indecomposable Λ-modules} ← → {Strings in Λ}

Strings in Λ

Butler and Ringel, “Auslander-Reiten Sequences with Few Middle Terms and Applications to String Algebras,” Communications in Algebra (1987) {Indecomposable Λ-modules} ← → {Strings in Λ} 1 5 3

α1 α2 β1 β2 α

Strings in Λ

Butler and Ringel, “Auslander-Reiten Sequences with Few Middle Terms and Applications to String Algebras,” Communications in Algebra (1987) {Indecomposable Λ-modules} ← → {Strings in Λ} 1 5 3

α1 α2 β1 β2 α

Length 0 e1 e5 e3 Length 1 α1 α2 β1 β2 α Length 2 β1α1 β−1

2 α1

α2β−1

1

αβ−1

2

α−1

2 α

Length 3 αβ−1

2 α1

α−1

2 αβ−1 2

β1α−1

2 α

Length 4 and 5 α−1

2 αβ−1 2 α1

β1α−1

2 αβ−1 2

β1α−1

2 αβ−1 2 α1

Auslander-Reiten Quiver

β1 α1 α β1α−1

2 α

e5 αβ−1

2 α1

β1α−1

2 αβ−1 2 α1

α−1

2 α

αβ−1

2

β1α−1

2 αβ−1 2

α−1

2 αβ−1 2 α1

e3 α2β−1

1

α−1

2 αβ−1 2

β−1

2 α1

e1 α2 β2

References

J.L. Alperin. Local Representation Theory, Cambridge University Press, 1986. Jean-Pierre Serre. Linear Representations of Finite Groups, Springer-Verlag, 1977. Sotiris Karanikolopoulos. “On holomorphic polydifferentials in positive characteristic”. Mathematische Nachrichten, 285(7):852-877, 2012. Frauke M. Bleher, Ted Chinburg, and Artistides Kontogeorgis. “Galois structure of the holomorphic differentials of curves”. In progress. 2017.