Groups, Representation Theory, and Curves Adam Wood Department of - - PowerPoint PPT Presentation

Groups, Representation Theory, and Curves

Adam Wood

Department of Mathematics University of Iowa

MAA MathFest Great Talks for a General Audience August 3, 2019

Outline

Introduction to (Modular) Representation Theory Representation Theory of Cyclic Groups Curves and Group Actions Space of Holomorphic Polydifferentials

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

Groups

Z = {. . . , −2, −1, 0, 1, 2, . . . }

Groups

Z = {. . . , −2, −1, 0, 1, 2, . . . } 4 + 3 = 7

Groups

Z = {. . . , −2, −1, 0, 1, 2, . . . } 4 + 3 = 7 4 + 0 = 4

Groups

Z = {. . . , −2, −1, 0, 1, 2, . . . } 4 + 3 = 7 4 + 0 = 4 4 + (−4) = 0

Groups

Z = {. . . , −2, −1, 0, 1, 2, . . . } 4 + 3 = 7 4 + 0 = 4 4 + (−4) = 0 Z is a set with operation +

Groups

Z = {. . . , −2, −1, 0, 1, 2, . . . } 4 + 3 = 7 4 + 0 = 4 4 + (−4) = 0 Z is a set with operation + Closure

Groups

Z = {. . . , −2, −1, 0, 1, 2, . . . } 4 + 3 = 7 4 + 0 = 4 4 + (−4) = 0 Z is a set with operation + Closure Identity

Groups

Z = {. . . , −2, −1, 0, 1, 2, . . . } 4 + 3 = 7 4 + 0 = 4 4 + (−4) = 0 Z is a set with operation + Closure Identity Inverse

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: 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.

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

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

◮ Can be extended to other groups

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   Trivial Representation Dimension 2 Representation Dimension 3 Representation

Visualization of Representations of the Cyclic Group

Let G be a cyclic group of order pn and let k be a field of characteristic p. Brauer Tree:

• pn − 1

Quiver:

• α

αpn = 0

Curves and Group Actions

Goal: Define a representation of a group using geometry

Curves and Group Actions

Goal: Define a representation of a group using geometry

◮ Define an algebraic curve

Curves and Group Actions

Goal: Define a representation of a group using geometry

◮ Define an algebraic curve ◮ Group actions on curves

Curves and Group Actions

Goal: Define a representation of a group using geometry

◮ Define an algebraic curve ◮ Group actions on curves ◮ Define a representation using geometry

Algebraic Curves

An algebraic curve over a field k is the vanishing set of some polynomial with solutions in the field k.

Example

Consider f (x, y) = y − x2. If k = R, then visualize the curve as x y

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)

− →

Space of Holomorphic Polydifferentials

X curve, k field of characteristic p, G group acting on X

Space of Holomorphic Polydifferentials

X curve, k field of characteristic p, G group acting on X H0(X, Ω⊗m

X ) space of holomorphic polydifferentials (if m = 1,

space of holomorphic differentials)

Space of Holomorphic Polydifferentials

X curve, k field of characteristic p, G group acting on X H0(X, Ω⊗m

X ) space of holomorphic polydifferentials (if m = 1,

space of holomorphic differentials) Initially, this space is a vector space over k

Space of Holomorphic Polydifferentials

X curve, k field of characteristic p, G group acting on X H0(X, Ω⊗m

X ) space of holomorphic polydifferentials (if m = 1,

space of holomorphic differentials) Initially, this space is a vector space over k Since G acts on X, the space of holomorphic polydifferentials becomes a representation of G

Research on the Space of Holomorphic Polydifferentials

Problem: Decompose H0(X, Ω⊗m

X ) into indecomposable

representations

Research on the Space of Holomorphic Polydifferentials

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

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 for groups of the form G ⋊ H

Motivation and Applications

◮ Geometric technique can possibly be used for other

representations

Motivation and Applications

◮ Geometric technique can possibly be used for other

representations

◮ For m = 2, can be used to understand the tangent space of

the deformation functor of curves

Motivation and Applications

◮ Geometric technique can possibly be used for other

representations

◮ For m = 2, can be used to understand the tangent space of

the deformation functor of curves

◮ Application to number theory and cusp forms using modular

curves

Motivation and Applications

◮ Geometric technique can possibly be used for other

representations

◮ For m = 2, can be used to understand the tangent space of

the deformation functor of curves

◮ Application to number theory and cusp forms using modular

curves

◮ Spaces like H0(X, Ω⊗m X ) appear in problems in coding theory

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.