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Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Modular Invariant Theory of

Elementary Abelian p-groups in

dimensions 2 and 3

H E A (Eddy) Campbell

University of New Brunswick

October 19, 2014

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Outline

1 Introduction

2 Dimension 2

3 Dimension 3

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory in general: ingredients

A group G represented on a vector space V over a field F of characteristic p.

A basis {x1, x2, . . . , xn} for V ∗. The action of G on V ∗ by σ(f )(v) = f (σ−1(v)).

The induced action of G by algebra automorphisms on F[V ] = F[x1, x2, . . . , xn]. The ring, F[V ]G , of polynomials fixed by the action of G .

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory in general: ingredients

A group G represented on a vector space V over a field F of characteristic p. A basis {x1, x2, . . . , xn} for V ∗.

The action of G on V ∗ by σ(f )(v) = f (σ−1(v)).

The induced action of G by algebra automorphisms on F[V ] = F[x1, x2, . . . , xn]. The ring, F[V ]G , of polynomials fixed by the action of G .

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory in general: ingredients

A group G represented on a vector space V over a field F of characteristic p. A basis {x1, x2, . . . , xn} for V ∗. The action of G on V ∗ by σ(f )(v) = f (σ−1(v)).

The induced action of G by algebra automorphisms on F[V ] = F[x1, x2, . . . , xn]. The ring, F[V ]G , of polynomials fixed by the action of G .

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory in general: ingredients

A group G represented on a vector space V over a field F of characteristic p. A basis {x1, x2, . . . , xn} for V ∗. The action of G on V ∗ by σ(f )(v) = f (σ−1(v)).

The induced action of G by algebra automorphisms on F[V ] = F[x1, x2, . . . , xn].

The ring, F[V ]G , of polynomials fixed by the action of G .

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory in general: ingredients

A group G represented on a vector space V over a field F of characteristic p. A basis {x1, x2, . . . , xn} for V ∗. The action of G on V ∗ by σ(f )(v) = f (σ−1(v)).

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Invariant theory: goal

We seek to understand F[V ]G in terms of its generators and relations or in turns of its structure such as the Cohen-Macaulay property by relating algebraic properties to the geometric properties of the representation.

We refer to the case that the order of G is divisible by p as the modular case, non-modular otherwise. Much more is known about the latter case than the former.

For example, in the non-modular case it is a famous theorem due to Coxeter, Shephard and Todd, Chevalley, Serre that F[V ]G is a polynomial algebra if and only if G is generated by (pseudo-)reflections.

The modular version of this theorem is still open.

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Invariant theory: goal

We seek to understand F[V ]G in terms of its generators and relations or in turns of its structure such as the Cohen-Macaulay property by relating algebraic properties to the geometric properties of the representation.

We refer to the case that the order of G is divisible by p as the modular case, non-modular otherwise. Much more is known about the latter case than the former.

For example, in the non-modular case it is a famous theorem due to Coxeter, Shephard and Todd, Chevalley, Serre that F[V ]G is a polynomial algebra if and only if G is generated by (pseudo-)reflections.

The modular version of this theorem is still open.

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Invariant theory: goal

We seek to understand F[V ]G in terms of its generators and relations or in turns of its structure such as the Cohen-Macaulay property by relating algebraic properties to the geometric properties of the representation.

We refer to the case that the order of G is divisible by p as the modular case, non-modular otherwise. Much more is known about the latter case than the former.

For example, in the non-modular case it is a famous theorem due to Coxeter, Shephard and Todd, Chevalley, Serre that F[V ]G is a polynomial algebra if and only if G is generated by (pseudo-)reflections.

The modular version of this theorem is still open.

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Invariant theory: goal

The modular version of this theorem is still open.

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Composition Series

In theory, one can hope to understand the invariant theory of a p-group by induction on a composition series

1G C G2 C G3 C · · ·C Gr = G with Gi+1 Gi

= Cp ,

for then F[V ]Gi+1 = (F[V ]Gi )Cp . Here Cp denotes the cyclic group of prime order p.

In practice, the representation theory of G is known to be wild unless r = 1 or r = 2 and p = 2, and F[V ]Gi is known not to be Cohen-Macaulay in “most” instances.

Wehlau proved that the invariant ring of any representation of Cp is generated by norms and traces and “rational” functions determined by the classical invariant theory of SL2(C).

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Composition Series

In theory, one can hope to understand the invariant theory of a p-group by induction on a composition series

1G C G2 C G3 C · · ·C Gr = G with Gi+1 Gi

= Cp ,

for then F[V ]Gi+1 = (F[V ]Gi )Cp . Here Cp denotes the cyclic group of prime order p.

In practice, the representation theory of G is known to be wild unless r = 1 or r = 2 and p = 2, and F[V ]Gi is known not to be Cohen-Macaulay in “most” instances.

Wehlau proved that the invariant ring of any representation of Cp is generated by norms and traces and “rational” functions determined by the classical invariant theory of SL2(C).

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Composition Series

In theory, one can hope to understand the invariant theory of a p-group by induction on a composition series

1G C G2 C G3 C · · ·C Gr = G with Gi+1 Gi

= Cp ,

for then F[V ]Gi+1 = (F[V ]Gi )Cp . Here Cp denotes the cyclic group of prime order p.

In practice, the representation theory of G is known to be wild unless r = 1 or r = 2 and p = 2, and F[V ]Gi is known not to be Cohen-Macaulay in “most” instances.

Wehlau proved that the invariant ring of any representation of Cp is generated by norms and traces and “rational” functions determined by the classical invariant theory of SL2(C).

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory of (Cp) r in dimenson 2

Suppose we have a representation ρ of (any) p-group G on a vector space V of dimension 2 over a field of characteristic p. We may assume that

G ρ ↪→ (

1 ∗ 0 1

)

Hence G is a subgroup of the additive group (F,+), and so is elementary Abelian, G = (Cp)

r for some r . It is not hard to see that

F[V ]G = F[x ,N(y)]

is a polynomial algebra on two generators of degrees 1 and |G |.

Modular Invariant Theory

H E A (Eddy) Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory of (Cp) r in dimenson 2

Suppose we have a representation ρ of (any) p-group G on a vector space V of dimension 2 over a field of characteristic p. We may assume that

G ρ ↪→