Representation Theory and Holomorphic Polydifferentials Adam Wood - - PowerPoint PPT Presentation

Representation Theory and Holomorphic Polydifferentials

Adam Wood

Department of Mathematics University of Iowa

Research Seminar, St. Olaf College April 26, 2019

Outline

Introduction to (Modular) Representation Theory Representation Theory of Cyclic Groups Algebraic Geometry Space of Holomorphic Polydifferentials

Definition

A group is a set G with an operation · so that

◮ g · h ∈ G for all g, h ∈ G ◮ (a · b) · c = a · (b · c) for all a, b, c ∈ G ◮ There is an identity e ∈ G satisfying e · g = g for all g ∈ G ◮ For every g ∈ G, there is an inverse element, g−1, so that

g · g−1 = e = g−1 · g

Definition

A group is a set G with an operation · so that

◮ g · h ∈ G for all g, h ∈ G ◮ (a · b) · c = a · (b · c) for all a, b, c ∈ G ◮ There is an identity e ∈ G satisfying e · g = g for all g ∈ G ◮ For every g ∈ G, there is an inverse element, g−1, so that

g · g−1 = e = g−1 · g

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: Symmetric Group

Let G = S3. The permutation matrix (aij) associated to σ ∈ S3 satisfies aij =

• 1

if σ(i) = j

• therwise.

Example: Symmetric Group

Let G = S3. The permutation matrix (aij) associated to σ ∈ S3 satisfies aij =

• 1

if σ(i) = j

• therwise.

For example, the permutation matrix associated to (1 2) is   1 1 1   .

Example: Symmetric Group

Let G = S3. The permutation matrix (aij) associated to σ ∈ S3 satisfies aij =

• 1

if σ(i) = j

• therwise.

For example, the permutation matrix associated to (1 2) is   1 1 1   . Let Mσ denote the permutation matrix associated to σ. Then, ρ(σ) = Mσ defines a representation of G called the permutation representation

• f G.

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.

Example: Cyclic Group

Consider G = Z/3Z = {0, 1, 2}

Example: Cyclic Group

Consider G = Z/3Z = {0, 1, 2} The complex 3rd roots of unity are

◮ 1 ◮ ω = e2πi/3 ◮ ω2 = e4πi/3

Example: Cyclic Group

Consider G = Z/3Z = {0, 1, 2} The complex 3rd roots of unity are

◮ 1 ◮ ω = e2πi/3 ◮ ω2 = e4πi/3

The three maps give three distinct representations of G 0 → 1 V0 1 → 1 2 → 1 0 → 1 V1 1 → ω 2 → ω2 0 → 1 V2 1 → ω2 2 → ω

Example: Cyclic Group

Consider G = Z/3Z = {0, 1, 2} The complex 3rd roots of unity are

◮ 1 ◮ ω = e2πi/3 ◮ ω2 = e4πi/3

The three maps give three distinct representations of G 0 → 1 V0 1 → 1 2 → 1 0 → 1 V1 1 → ω 2 → ω2 0 → 1 V2 1 → ω2 2 → ω Trivial Representation

Example: Cyclic Group

0 → 1 V0 1 → 1 2 → 1 0 → 1 V1 1 → ω 2 → ω2 0 → 1 V2 1 → ω2 2 → ω

Example: Cyclic Group

0 → 1 V0 1 → 1 2 → 1 0 → 1 V1 1 → ω 2 → ω2 0 → 1 V2 1 → ω2 2 → ω Define 0 → 1 1

• W

1 → ω ω2

• 2 →

ω2 ω

Example: Cyclic Group

0 → 1 V0 1 → 1 2 → 1 0 → 1 V1 1 → ω 2 → ω2 0 → 1 V2 1 → ω2 2 → ω Define 0 → 1 1

• W

1 → ω ω2

• 2 →

ω2 ω

• W = V1 ⊕ V2

Example: Cyclic Group

0 → 1 V0 1 → 1 2 → 1 0 → 1 V1 1 → ω 2 → ω2 0 → 1 V2 1 → ω2 2 → ω Define 0 → 1 1

• W

1 → ω ω2

• 2 →

ω2 ω

• W = V1 ⊕ V2

Direct Sum

Definition

Let G be a finite group and let k be a field. Define kG =   

• g∈G

agg | ag ∈ k    . This space is called the group ring and is a finite dimensional vector space over k with basis {g | g ∈ G}.

Definition

Let G be a finite group and let k be a field. Define kG =   

• g∈G

agg | ag ∈ k    . This space is called the group ring and is a finite dimensional vector space over k with basis {g | g ∈ G}.

Definition

Let R be a ring. A module over R is an additive abelian group M with an action of R satisfying

Definition

Let G be a finite group and let k be a field. Define kG =   

• g∈G

agg | ag ∈ k    . This space is called the group ring and is a finite dimensional vector space over k with basis {g | g ∈ G}.

Definition

Let R be a ring. A module over R is an additive abelian group M with an action of R satisfying

◮ r.(m1 + m2) = r.m1 + r.m2

for all r1, r2 ∈ R and for all m1, m2 ∈ M.

Definition

Let G be a finite group and let k be a field. Define kG =   

• g∈G

agg | ag ∈ k    . This space is called the group ring and is a finite dimensional vector space over k with basis {g | g ∈ G}.

Definition

Let R be a ring. A module over R is an additive abelian group M with an action of R satisfying

◮ r.(m1 + m2) = r.m1 + r.m2 ◮ (r1 + r2).m = r1.m + r2.m

for all r1, r2 ∈ R and for all m1, m2 ∈ M.

Definition

Let G be a finite group and let k be a field. Define kG =   

• g∈G

agg | ag ∈ k    . This space is called the group ring and is a finite dimensional vector space over k with basis {g | g ∈ G}.

Definition

Let R be a ring. A module over R is an additive abelian group M with an action of R satisfying

◮ r.(m1 + m2) = r.m1 + r.m2 ◮ (r1 + r2).m = r1.m + r2.m ◮ (r1r2).m = r1.(r2.m)

for all r1, r2 ∈ R and for all m1, m2 ∈ M.

Alternate View of Representations

{Representations of G, ρ : G → GL(V )} ← → {kG-modules}

Alternate View of Representations

{Representations of G, ρ : G → GL(V )} ← → {kG-modules} Think of a k-vector space V that allows “multiplication” by elements of G

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

Let k be a field. The characteristic of k is the smallest nonnegative integer n so that 1k + · · · + 1k

• n times

= 0. If no such nonnegative integer exists, then char(k) = 0.

Characteristic of a Field

Let k be a field. The characteristic of k is the smallest nonnegative integer n so that 1k + · · · + 1k

• n times

= 0. If no such nonnegative integer exists, then char(k) = 0.

Example

◮ char(C) = 0 ◮ char(Fp) = p

Characteristic of a Field

Let k be a field. The characteristic of k is the smallest nonnegative integer n so that 1k + · · · + 1k

• n times

= 0. If no such nonnegative integer exists, then char(k) = 0.

Example

◮ char(C) = 0 ◮ char(Fp) = p

Fact: If k is a field, either

◮ char(k) = 0 or ◮ char(k) = p for some prime number p.

Modular Representation Theory

Study of representations of G over k when char(k) = p

Modular Representation Theory

Study of representations of G over k when char(k) = p Every representation can be written as a direct sum of indecomposable representations.

Modular Representation Theory

Study of representations of G over k when char(k) = p Every representation can be written as a direct sum of indecomposable representations. Understand the indecomposable representations

Representations of Cyclic p-groups

◮ G cyclic of order pn

Representations of Cyclic p-groups

◮ G cyclic of order pn ◮ The representation theory is “nice”

Representations of Cyclic p-groups

◮ G cyclic of order pn ◮ The representation theory is “nice” ◮ Finitely many indecomposable representations, know how to

describe them

Representations of Cyclic p-groups

◮ G cyclic of order pn ◮ The representation theory is “nice” ◮ Finitely many indecomposable representations, know how to

describe them

◮ Same is true for groups containing cyclic p-groups

Lemma

Let G be a cyclic p-group, |G| = pn, and let k be a field of characteristic p. The only irreducible representation of G is the trivial representation.

Lemma

Let G be a cyclic p-group, |G| = pn, and let k be a field of characteristic p. The only irreducible representation of G is the trivial representation. #( irreducible representations) = #( conjugacy classes of elements g ∈ G with p not dividing order of g)

Lemma

Let G be a cyclic p-group, |G| = pn, and let k be a field of characteristic p. The only irreducible representation of G is the trivial representation. #( irreducible representations) = #( conjugacy classes of elements g ∈ G with p not dividing order of g) kG decomposes into a sum of the projective indecomposable representations kG ∼ = k[x]/(xpn − 1) ∼ = k[x]/(x − 1)pn ((a − b)p = ap − bp since char(k) = p)

• ne projective indecomposable representation ⇒ one irreducible

representation

Indecomposable Representations

|G| = pn, G = σ Trivial Representation g → 1 for all g ∈ G, matrix (1) Dimension 2 Representation σ → 1 1 1

• Dimension 3

Representation σ →   1 1 1 1 1   . . . . . . Dimension pn Representation σ →        1 1 · · · 1 . . . ... ... . . . · · · 1 1 · · · 1       

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  

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   Compare with....

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   Compare with.... char(k) = 0, Irreducible Representations σ → 1 σ → ω σ → ω2

Example: Cyclic Group and “Other” Group

Consider G = Z/3Z ⋊ Z/4Z, char(k) = 3

Example: Cyclic Group and “Other” Group

Consider G = Z/3Z ⋊ Z/4Z, char(k) = 3 Let ω be a 4th root of unity in k. We get 3 · 4 = 12 indecomposable representations visualized as follows:

Example: Cyclic Group and “Other” Group

Consider G = Z/3Z ⋊ Z/4Z, char(k) = 3 Let ω be a 4th root of unity in k. We get 3 · 4 = 12 indecomposable representations visualized as follows: (1) 1 1 1

• 

 1 1 1 1 1   (ω) ω 1 ω

• 

 ω 1 ω 1 ω   (ω2) ω2 1 ω2

• 

 ω2 1 ω2 1 ω2   (ω3) ω3 1 ω3

• 

 ω3 1 ω3 1 ω3  

The previous example generalizes. Let G = P ⋊ C, where P is cyclic of order pn and C is cyclic of order c, with p ∤ c. There are c · pn indecomposable representations of G.

1-dim (1) (ω) · · · ωc−1 2-dim

• 1

1 1

• ω

1 ω

• · · ·
• ωc−1

1 ωc−1

• .

. . pn-dim       1 1 · · · ... . . . . . . ... 1 · · · 1             ω 1 · · · ... . . . . . . ... 1 · · · ω       · · ·       ωc−1 1 . . . . . . 1 · · · ωc−1      

Algebraic Geometry

Study of algebraic curves

Algebraic Geometry

Study of algebraic curves Goal: Define a representation of a group using geometry

Algebraic Geometry

Study of algebraic curves Goal: Define a representation of a group using geometry

◮ Define an algebraic curve

Algebraic Geometry

Study of algebraic curves Goal: Define a representation of a group using geometry

◮ Define an algebraic curve ◮ Define the module of relative differentials

Algebraic Geometry

Study of algebraic curves Goal: Define a representation of a group using geometry

◮ Define an algebraic curve ◮ Define the module of relative differentials ◮ Group actions on curves

Algebraic Geometry

Study of algebraic curves Goal: Define a representation of a group using geometry

◮ Define an algebraic curve ◮ Define the module of relative differentials ◮ Group actions on curves ◮ Define a representation using geometry

Example: Affine Plane Curve

Let f (x, y) ∈ k[x, y] be an irreducible polynomial. The affine plane curve defined by f is Xf = {(a, b) ∈ k × k | f (a, b) = 0}.

Example: Affine Plane Curve

Let f (x, y) ∈ k[x, y] be an irreducible polynomial. The affine plane curve defined by f is Xf = {(a, b) ∈ k × k | f (a, b) = 0}. The function field of Xf is k(Xf ) = k(x)[y]/(f (x, y)).

Example: Affine Plane Curve

Let f (x, y) ∈ k[x, y] be an irreducible polynomial. The affine plane curve defined by f is Xf = {(a, b) ∈ k × k | f (a, b) = 0}. The function field of Xf is k(Xf ) = k(x)[y]/(f (x, y)). We say that the curve Xf corresponds to the ring k[x, y]/(f (x, y)).

Example: Affine Plane Curve

Let f (x, y) ∈ k[x, y] be an irreducible polynomial. The affine plane curve defined by f is Xf = {(a, b) ∈ k × k | f (a, b) = 0}. The function field of Xf is k(Xf ) = k(x)[y]/(f (x, y)). We say that the curve Xf corresponds to the ring k[x, y]/(f (x, y)). For example, let f (x, y) = y − x2. If k = R, then visualize the curve as

Example: Affine Plane Curve

f (x, y) = y − x2 x y

Projective Plane Curve

The projective plane over k, denoted P2(k), is defined to consist of points [x0, x1, x2], where xi ∈ k, and we declare two points to be equal if one is a nonzero scalar multiple of the other. A polynomial f (x, y, z) ∈ k[x, y, z] is homogeneous of degree d if f (λx, λy, λz) = λdf (x, y, z) for all λ ∈ k. Let f (x, y, z) ∈ k[x, y, z] be a homogeneous irreducible

• polynomial. The projective plane curve defined by f is

Xf = {[x0, x1, x2] ∈ P2(k) | f (x0, x1, x2) = 0}.

Projective Plane Curve

The projective plane over k, denoted P2(k), is defined to consist of points [x0, x1, x2], where xi ∈ k, and we declare two points to be equal if one is a nonzero scalar multiple of the other. A polynomial f (x, y, z) ∈ k[x, y, z] is homogeneous of degree d if f (λx, λy, λz) = λdf (x, y, z) for all λ ∈ k.

Projective Plane Curve

The projective plane over k, denoted P2(k), is defined to consist of points [x0, x1, x2], where xi ∈ k, and we declare two points to be equal if one is a nonzero scalar multiple of the other. A polynomial f (x, y, z) ∈ k[x, y, z] is homogeneous of degree d if f (λx, λy, λz) = λdf (x, y, z) for all λ ∈ k. Let f (x, y, z) ∈ k[x, y, z] be a homogeneous irreducible

• polynomial. The projective plane curve defined by f is

Xf = {[x0, x1, x2] ∈ P2(k) | f (x0, x1, x2) = 0}.

Projective Curve

Generalize plane curves to an arbitrary curve

Projective Curve

Generalize plane curves to an arbitrary curve Fact: Bijective correspondence between smooth projective curves X over k and function fields F over k

Projective Curve

Generalize plane curves to an arbitrary curve Fact: Bijective correspondence between smooth projective curves X over k and function fields F over k Allow points at infinity

Projective Curve

Generalize plane curves to an arbitrary curve Fact: Bijective correspondence between smooth projective curves X over k and function fields F over k Allow points at infinity Think of a plane curve

Group Actions

Definition

Let G be a group with identity e and let X be a set. A group action of G on X is a map G × X → X (g, x) → g.x satisfying

◮ e.x = x for all x ∈ X ◮ g.(h.x) = (gh).x for all g, h ∈ G, for all x ∈ X

Example

X, affine plane curve over R defined by f (x, y) = y − x2 G = Z/2Z = {0, 1}.

Example

X, affine plane curve over R defined by f (x, y) = y − x2 G = Z/2Z = {0, 1}.

◮ By definition, 0.(a, b) = (a, b)

Example

X, affine plane curve over R defined by f (x, y) = y − x2 G = Z/2Z = {0, 1}.

◮ By definition, 0.(a, b) = (a, b) ◮ Define 1.(a, b) = (−a, b)

Example

X, affine plane curve over R defined by f (x, y) = y − x2 G = Z/2Z = {0, 1}.

◮ By definition, 0.(a, b) = (a, b) ◮ Define 1.(a, b) = (−a, b)

− →

Example

X, affine plane curve over R defined by f (x, y) = y − x2 G = Z/2Z = {0, 1}.

◮ By definition, 0.(a, b) = (a, b) ◮ Define 1.(a, b) = (−a, b)

− →

Module of Relative Differentials

Let A be an algebra over a field k.

Definition

A derivation of A over k onto some space B is a map d : A → B so that

• 1. d(x + y) = d(x) + d(y)
• 2. d(xy) = xd(y) + d(x)y
• 3. d(λ) = 0

for x, y ∈ A, λ ∈ k.

Module of Relative Differentials

Let A be an algebra over a field k.

Definition

A derivation of A over k onto some space B is a map d : A → B so that

• 1. d(x + y) = d(x) + d(y)
• 2. d(xy) = xd(y) + d(x)y
• 3. d(λ) = 0

for x, y ∈ A, λ ∈ k.

Example

A = k[x], define d : A → A by d(f (x)) = f ′(x)

Module of Relative Differentials

Define the module of relative differentials of A over k to be an

• bject ΩA/k together with a derivation of A over k, d : A → ΩA/k.

Module of Relative Differentials

Define the module of relative differentials of A over k to be an

• bject ΩA/k together with a derivation of A over k, d : A → ΩA/k.

This space is generated by {d(a) | a ∈ A}. There could be relations!

Module of Relative Differentials

Define the module of relative differentials of A over k to be an

• bject ΩA/k together with a derivation of A over k, d : A → ΩA/k.

This space is generated by {d(a) | a ∈ A}. There could be relations!

Example

A = k[x] ΩA/k = k[x]dx

Space of Holomorphic Differentials

Idea: Make the module of relative differentials geometric

Space of Holomorphic Differentials

Idea: Make the module of relative differentials geometric Let X be a smooth projective curve over a field k.

Space of Holomorphic Differentials

Idea: Make the module of relative differentials geometric Let X be a smooth projective curve over a field k. We can cover X with two affine curves given by rings A1 and A2

Space of Holomorphic Differentials

Idea: Make the module of relative differentials geometric Let X be a smooth projective curve over a field k. We can cover X with two affine curves given by rings A1 and A2 Define the space of holomorphic differentials to be H0(X, ΩX) = ΩA1/k ∩ ΩA2/k

Space of Holomorphic Differentials

Idea: Make the module of relative differentials geometric Let X be a smooth projective curve over a field k. We can cover X with two affine curves given by rings A1 and A2 Define the space of holomorphic differentials to be H0(X, ΩX) = ΩA1/k ∩ ΩA2/k This space is a k-vector space. If a group G acts on X, then G acts on H0(X, ΩX) = ⇒ we get a representation of G

Space of Holomorphic Polydifferentials

We defined H0(X, ΩX) = ΩA1/k ∩ ΩA2/k

Space of Holomorphic Polydifferentials

We defined H0(X, ΩX) = ΩA1/k ∩ ΩA2/k Let m > 1. Define the space of holomorphic polydifferentials to be H0(X, Ω⊗m

X ) = ΩA1/k ⊗A1 · · · ⊗A1 ΩA1/k

• m times

∩ ΩA2/k ⊗A2 · · · ⊗A2 ΩA2/k

• m times

.

Space of Holomorphic Polydifferentials

We defined H0(X, ΩX) = ΩA1/k ∩ ΩA2/k Let m > 1. Define the space of holomorphic polydifferentials to be H0(X, Ω⊗m

X ) = ΩA1/k ⊗A1 · · · ⊗A1 ΩA1/k

• m times

∩ ΩA2/k ⊗A2 · · · ⊗A2 ΩA2/k

• m times

. As above, if a group G acts on X, we get a representation of G

Research on the Space of Holomorphic Polydifferentials

Let k be a field, let X be a smooth projective curve over k, and let G be a group acting on X. Problem: Decompose H0(X, Ω⊗m

X ) into indecomposable

representations

Research on the Space of Holomorphic Polydifferentials

Let k be a field, let X be a smooth projective curve over k, and let G be a group acting on X. Problem: Decompose H0(X, Ω⊗m

X ) into indecomposable

representations Variations:

◮ char(k) = 0 or char(k) = p ◮ Type of group G ◮ Ramification of the cover π : X → X/G ◮ Value of m

Previous Results

◮ char(k) = 0, Chevalley and Weil, 1936 ◮ char(k) = p, m = 1

◮ Unramified cover, Tamagawa, 1951 ◮ Tamely ramified cover, Nakajima, 1986 ◮ Cyclic p-group, Valentini and Madan, 1981 ◮ Arbitrary p-group, Karanikolopoulos and Kontogeorgis, 2013 ◮ G has cyclic Sylow p-subgroups (includes the case when

G = P ⋊ C), m = 1, Bleher, Chinburg, and Kontogeorgis, 2017

◮ char(k) = p, m > 1, cyclic p-group, Karanikolopoulos, 2012

Example, computation of H0(X, Ω⊗m

X ) f (t) = t9 − t F = k(t)[y]/(y 2 − f (t)) ∼ = k(t)( √ f ) X smooth projective curve over k with function field F, char(k) = 3 Two important affine curves B1 = k[t][ √ f ], B2 = k[t−1][t−5√ f ] ΩB1/k =

• k[t] + k[t]

√ f

• dt

ΩB2/k =

• t−2k[t−1] + t3k[t−1]

√ f

• dt

H0(X, ΩX) = ΩB1/k ∩ ΩB2/k = (k + kt + kt2 + kt3) dt √ f H0(X, Ω⊗2

X ) = (ΩB1/k ⊗B1 ΩB1/k) ∩ (ΩB2/k ⊗B2 ΩB2/k)

=

• (k + kt) 1

√ f + (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

Example, group action on F

F = k(t)( √ f ) G = Z/3Z ⋊ Z/4Z = σ ⋊ ρ Define action of G on F by σ.t = t + 1 σ. √ f = √ f ρ.t = −t ρ. √ f = ω−1√ f ω primitive fourth root of unity in k, extend multiplicatively to all

• f F

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

ρ. 1 √ f = 1 ρ. √ f = 1 ω−1√ f = ω √ f σ.(dt) = d(σ.t) = d(t + 1) = dt ρ.1 f = 1 ρ. √ f

2 =

1 ω−2√ f = ω2 √ f = − 1 √ f ρ.(dt) = −dt −t + 1 −1

σ−1 σ−1

t5 t2 −t4 + t3 + t2 − t + 1 −t + 1 −t3 + t −1 t6 + t4 + t2

σ−1 σ−1 σ−1 σ−1 σ−1 σ−1 σ−1

Recall

Consider G = Z/3Z ⋊ Z/4Z, char(k) = 3 Let ω be a 4th root of unity in k. We get 3 · 4 = 12 indecomposable representations visualized as follows: (1) 1 1 1

• 

 1 1 1 1 1   (ω) ω 1 ω

• 

 ω 1 ω 1 ω   (ω2) ω2 1 ω2

• 

 ω2 1 ω2 1 ω2   (ω3) ω3 1 ω3

• 

 ω3 1 ω3 1 ω3  

Recall

Consider G = Z/3Z ⋊ Z/4Z, char(k) = 3 Let ω be a 4th root of unity in k. We get 3 · 4 = 12 indecomposable representations visualized as follows: (1) U0,1 1 1 1

• U0,2

  1 1 1 1 1   U0,3 (ω) U1,1 ω 1 ω

• U1,2

  ω 1 ω 1 ω   U1,3 (ω2) U2,1 ω2 1 ω2

• U2,2

  ω2 1 ω2 1 ω2   U2,3 (ω3) U3,1 ω3 1 ω3

• U3,2

  ω3 1 ω3 1 ω3   U3,3

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

ρ. 1 √ f = 1 ρ. √ f = 1 ω−1√ f = ω √ f σ.(dt) = d(σ.t) = d(t + 1) = dt ρ.1 f = 1 ρ. √ f

2 =

1 ω−2√ f = ω2 √ f = − 1 √ f ρ.(dt) = −dt −t + 1 −1

σ−1 σ−1

t5 t2 −t4 + t3 + t2 − t + 1 −t + 1 −t3 + t −1 t6 + t4 + t2

σ−1 σ−1 σ−1 σ−1 σ−1 σ−1 σ−1

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

k √ f ∼ = U1,1 σ.(dt) = d(σ.t) = d(t + 1) = dt ρ.1 f = 1 ρ. √ f

2 =

1 ω−2√ f = ω2 √ f = − 1 √ f ρ.(dt) = −dt −t + 1 −1

σ−1 σ−1

t5 t2 −t4 + t3 + t2 − t + 1 −t + 1 −t3 + t −1 t6 + t4 + t2

σ−1 σ−1 σ−1 σ−1 σ−1 σ−1 σ−1

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

k √ f ∼ = U1,1 σ.(dt) = d(σ.t) = d(t + 1) = dt k f ∼ = U2,1 ρ.(dt) = −dt −t + 1 −1

σ−1 σ−1

t5 t2 −t4 + t3 + t2 − t + 1 −t + 1 −t3 + t −1 t6 + t4 + t2

σ−1 σ−1 σ−1 σ−1 σ−1 σ−1 σ−1

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

k √ f ∼ = U1,1 kdt ∼ = U2,1 k f ∼ = U2,1 −t + 1 −1

σ−1 σ−1

t5 t2 −t4 + t3 + t2 − t + 1 −t + 1 −t3 + t −1 t6 + t4 + t2

σ−1 σ−1 σ−1 σ−1 σ−1 σ−1 σ−1

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

k √ f ∼ = U1,1 kdt ∼ = U2,1 kdt ⊗ kdt ∼ = U0,1 k f ∼ = U2,1 −t + 1 −1

σ−1 σ−1

t5 t2 −t4 + t3 + t2 − t + 1 −t + 1 −t3 + t −1 t6 + t4 + t2

σ−1 σ−1 σ−1 σ−1 σ−1 σ−1 σ−1

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

k √ f ∼ = U1,1 kdt ∼ = U2,1 kdt ⊗ kdt ∼ = U0,1 k f ∼ = U2,1 k + kt ∼ = U0,2 t5 t2 −t4 + t3 + t2 − t + 1 −t + 1 −t3 + t −1 t6 + t4 + t2

σ−1 σ−1 σ−1 σ−1 σ−1 σ−1 σ−1

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

k √ f ∼ = U1,1 kdt ∼ = U2,1 kdt ⊗ kdt ∼ = U0,1 k f ∼ = U2,1 k + kt ∼ = U0,2 t2 U2,3 −t + 1 −1 t6 + t4 + t2

σ−1 σ−1 σ−1 σ−1

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

k √ f ∼ = U1,1 kdt ∼ = U2,1 kdt ⊗ kdt ∼ = U0,1 k f ∼ = U2,1 k + kt ∼ = U0,2 U2,3 U0,3 t6 + t4 + t2

σ−1

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

k √ f ∼ = U1,1 kdt ∼ = U2,1 kdt ⊗ kdt ∼ = U0,1 k f ∼ = U2,1 k + kt ∼ = U0,2 U2,3 U0,3 U0,1

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

k √ f ∼ = U1,1 kdt ∼ = U2,1 kdt ⊗ kdt ∼ = U0,1 k f ∼ = U2,1 k + kt ∼ = U0,2 k + kt + · · · + kt6 ∼ = U2,3 ⊕ U0,3 ⊕ U0,1

Example, “nice” bases

H0(X, Ω⊗2

X ) =

• (k + kt) 1

√ f ⊕ (k + kt + · · · + kt6)1 f

• (dt ⊗ dt)

k √ f ∼ = U1,1 kdt ∼ = U2,1 kdt ⊗ kdt ∼ = U0,1 k f ∼ = U2,1 k + kt ∼ = U0,2 k + kt + · · · + kt6 ∼ = U2,3 ⊕ U0,3 ⊕ U0,1

= ⇒ H0(X, Ω⊗2

X ) = ((U0,2) ⊗ U1,1 + (U2,3 ⊕ U0,3 ⊕ U0,1) ⊗ U2,1)⊗U0,1

Example

H0(X, Ω⊗2

X ) = ((U0,2) ⊗ U1,1 + (U2,3 ⊕ U0,3 ⊕ U0,1) ⊗ U2,1) ⊗ U0,1

= (U1,2 ⊕ (U0,3 ⊕ U2,3 ⊕ U2,1)) ⊗ U0,1 = U1,2 ⊕ U0,3 ⊕ U2,3 ⊕ U2,1

Example

H0(X, Ω⊗2

X ) = U1,2 ⊕ U0,3 ⊕ U2,3 ⊕ U2,1

H0(X, Ω⊗2

X ) =

              ω 1 ω 1 1 1 1 1 ω2 1 ω2 1 ω2 ω2              

Research

Decompose H0(X, Ω⊗m

X ) into indecomposable representations

Bleher, Chinburg, and Kontogeorgis (2017), m = 1, dealt with groups of the form G ⋊ H, G cyclic p-group, exploited the geometry and took different approach compared with previous authors Karanikolopoulos (2012), m > 1, G cyclic p-group Idea: Use the geometric structure and reasoning in the m > 1 case to answer the question when m > 1 and groups of the form G ⋊ H

References

J.L. Alperin. Local Representation Theory, Cambridge University Press, 1986. Jean-Pierre Serre. Linear Representations of Finite Groups, Springer-Verlag, 1977. Peter Webb. A Course in Finite Group Representation Theory, Cambridge University Press, 2016. Robin Hartshorne. Algebraic Geometry. Springer-Verlag, 1977. Dino Lorenzini. An Invitation to Arithmetic Geometry. American Mathematical Society, 1996. 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.