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 pn, k algebraically closed field, char(k) = p

Motivating Example

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

Motivating Example

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

Motivating Example

Indecomposable representations

Motivating Example

Indecomposable representations Uniserial kG-modules of length j for 1 ≤ j ≤ pn, with trivial composition factors

Motivating Example

Indecomposable representations Uniserial kG-modules of length j for 1 ≤ j ≤ pn, with trivial composition factors kG is a Brauer tree algebra for the Brauer tree

• with multiplicity m = pn − 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 =   

• g∈G

agg | ag ∈ k    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 =   

• g∈G

agg | ag ∈ k    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

• ne 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

kx, 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

• ne 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

kx, 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

• nly if G has cyclic Sylow p-subgroups.

Blocks and Brauer Trees

kG = B1 ⊕ · · · ⊕ Bm Unique decomposition into indecomposable subalgebras

Blocks and Brauer Trees

kG = B1 ⊕ · · · ⊕ Bm 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 = B1 ⊕ · · · ⊕ Bm 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)

radi(M) = rad(radi−1(M))

Module Definitions

Let M be a kG-module. rad(M) =

• (maximal submodules of M)

soc(M) =

• (simple submodules of M)

radi(M) = rad(radi−1(M)) 0 = radn(M) ⊂ radn−1(M) ⊂ · · · ⊂ rad2(M) ⊂ rad(M) ⊂ M

Module Definitions

Let M be a kG-module. rad(M) =

• (maximal submodules of M)

soc(M) =

• (simple submodules of M)

radi(M) = rad(radi−1(M)) 0 = radn(M) ⊂ radn−1(M) ⊂ · · · ⊂ rad2(M) ⊂ rad(M) ⊂ M 0 ⊂ soc(M) ⊂ soc2(M) ⊂ · · · ⊂ socm−1(M) ⊂ socm(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 = (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

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

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

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

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

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

Example

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

a b c d

T0 = {1, 2, 3, 4, 5}

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

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.

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.

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.

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

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.

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.

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.

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: βα Special b-cycle at 2: αβ Special c-cycle at 3: ǫδγ Special b-cycle at 3: γǫδ Special d-cycle at 3: δγǫ Special c cycle at 4: ι

ΓT, Path Algebra with Relations

c a b d

γ ι α β δ ǫ

Relations αβ = γǫδ ǫδγ = ι2 αβα = βαβ = γǫδγ = δγǫδ = ǫδγǫ = ι3 = 0 δα = βγ = ιǫ = γι = 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 βα, αβ, ǫδ, γǫ, δγ, ι2 δγǫ Multiply by concatenating paths

ΛT, Projective Indecomposable Modules

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

a b c d

For each edge i, get a module Mi as follows Ma =

a b a

Mb =

b a d c b

Mc =

c b c d c

Md =

d c b d

Define ΛT so that the projective indecomposable Λ-modules are Ma, Mb, Mc, and Md.

Overview

G finite group kG = B1 ⊕ · · · ⊕ Bm Goal: Understand indecomposable modules for a block B

Overview

G finite group kG = B1 ⊕ · · · ⊕ Bm Goal: Understand indecomposable modules for a block B Suppose B has cyclic defect group

Overview

G finite group kG = B1 ⊕ · · · ⊕ Bm Goal: Understand indecomposable modules for a block B Suppose B has cyclic defect group B Brauer Tree Quiver with Relations String Modules

Butler and Ringel (1987)

Overview

G finite group kG = B1 ⊕ · · · ⊕ Bm Goal: Understand indecomposable modules for a block B Suppose B has cyclic defect group B Brauer Tree Quiver with Relations String Modules

Butler and Ringel (1987)

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.

Blocks

k[SL2(Fp)] = B0 ⊕ Bodd ⊕ Beven There are p simple modules, one for each dimension 1, . . . , p B0 trivial defect group Bodd cyclic defect group, contains odd dimensional simple modules Beven cyclic defect group, contains even dimensional simple modules

Brauer Tree and Quiver with Relations

Assuming p ≡ −1 mod 4 Bodd

• · · ·
• 1

p−2 3 (p−1)/2

multiplicity 2

Brauer Tree and Quiver with Relations

Assuming p ≡ −1 mod 4 Bodd

• · · ·
• 1

p−2 3 (p−1)/2

multiplicity 2 1 p − 2 3 · · · (p − 1)/2

α1 α2 β1 α3 β2 α(p−3)/2 β3 β(p−3)/2 α

Bodd for p = 7

• 1

5 3

1 5 3

α1 α2 β1 β2 α

Relations α1β1α1, β1α1β1, α2β2α2, β2α2β2 α3 α1β1 − β2α2, α2β2 − α2 α2α1, β1β2, αα2, β2α

Bodd for p = 7

• 1

5 3

1 5 3

α1 α2 β1 β2 α

Relations α1β1α1, β1α1β1, α2β2α2, β2α2β2 α3 α1β1, β2α2, α2β2, α2 α2α1, β1β2, αα2, β2α

Bodd for p = 7

• 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

Q quiver, I admissible ideal

Strings

Q quiver, I admissible ideal α arrow, define α−1 formal inverse, “flipped” arrow

Strings

Q quiver, I admissible ideal α arrow, define α−1 formal inverse, “flipped” arrow

Definition

A string of length m is a finite concatenation of arrows and inverses of arrows c1c2 · · · cm−1cm so that

◮ ci+1 = c−1 i

for all 1 ≤ i ≤ m

◮ No subpath cici+1 · · · ci+t or its inverse belongs to I

Strings

Q quiver, I admissible ideal α arrow, define α−1 formal inverse, “flipped” arrow

Definition

A string of length m is a finite concatenation of arrows and inverses of arrows c1c2 · · · cm−1cm so that

◮ ci+1 = c−1 i

for all 1 ≤ i ≤ m

◮ No subpath cici+1 · · · ci+t or its inverse belongs to I

Visualize α1α2α−1

3 α4α−1 5 α−1 6 α7

as

• α1

α2 α3 α4 α5 α6 α7

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 Theory

Know indecomposable modules Goal: Understand maps between indecomposable modules Subgoal: Understand “irreducible” maps between indecomposable modules

Definition

Let f : A → B be a module homomorphism. Then, f is irreducible if f is not an isomorphism and if f = hg is a factorization of f , either g is a split monomorphism or h is a split epimorhpism. A B X

f g h

Auslander-Reiten Theory

The Auslander-Reiten quiver of Λ is a quiver defined by Vertices: indecomposable Λ-modules Arrows: irreducible maps between indecomposable modules Can be built from “almost split exact sequences”

Strings − → Auslander-Reiten Quiver

Butler and Ringel (1987) give method for getting almost split exact sequences

Definition

Let C be a string.

• 1. A string C starts on a peak if there does not exist an arrow β

so that Cβ is a string.

• 2. A string C starts in a deep if there does not exist an arrow γ

so that Cγ−1 is a string.

• 3. A string C ends on a peak if there does not exist an arrow β

so that β−1C is a string.

• 4. A string C ends in a deep if there does not exist an arrow γ so

that γC is a string. Build almost split exact sequences from strings

Example

String αβ−1

2

does not end on a peak, does not start on a peak β1α−1

2 αβ−1 2

“maximal” string to the left αβ−1

2 α1 “maximal” string to the right

β1α−1

2 αβ−1 2 α1 “maximal” string in both directions

Get almost split exact sequence 0 → αβ−1

2

→ β1α−1

2 αβ−1 2

⊕ αβ−1

2 α1 → β1α−1 2 αβ−1 2 α1 → 0

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

Connection to Current Research

X smooth projective curve over algebraically closed field k of characteristic p, G finite group acting on X For m > 1, define H0(X, Ω⊗m

X ), space of holomorphic

polydifferentials (Representation of G) Special Case:

◮ ℓ = p prime, X(ℓ) modular curve of level ℓ ◮ Xp(ℓ) reduction of X(ℓ) modulo p ◮ G = PSL(2, Fℓ) acts on Xp(ℓ) ◮ Understand H0(Xp(ℓ), Ω⊗m Xp(ℓ)) as a representation of

PSL(2, Fℓ)

Connection to Current Research

Blocks of PSL(2, Fℓ) over k look like

• where the exceptional vertex has multiplicity pn−1

2

References

J.L. Alperin. Local Representation Theory, Cambridge University Press, 1986. Maurice Auslander, Idun Reiten, and Sverre O. Smalø. Representation Theory of Artin Algebras, Cambridge University Press, 1995. M.C.R. Butler and Claus Michael Ringel. “Auslander-Reiten sequences with few middle terms and applications to string algebras”. Communications in Algebra, 15(1):145-179, 1987. 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.