Transcript
Page 1: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

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

Page 2: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Outline

1 Introduction

2 Dimension 2

3 Dimension 3

Page 3: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

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 afield 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 automorphismson F[V ] = F[x1, x2, . . . , xn].

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

Page 4: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

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 afield 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 automorphismson F[V ] = F[x1, x2, . . . , xn].

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

Page 5: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

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 afield 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 automorphismson F[V ] = F[x1, x2, . . . , xn].

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

Page 6: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

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 afield 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 automorphismson F[V ] = F[x1, x2, . . . , xn].

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

Page 7: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

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 afield 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 automorphismson F[V ] = F[x1, x2, . . . , xn].

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

Page 8: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Invariant theory: goal

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

We refer to the case that the order of G is divisible by pas the modular case, non-modular otherwise. Muchmore is known about the latter case than the former.

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

The modular version of this theorem is still open.

Page 9: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Invariant theory: goal

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

We refer to the case that the order of G is divisible by pas the modular case, non-modular otherwise. Muchmore is known about the latter case than the former.

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

The modular version of this theorem is still open.

Page 10: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Invariant theory: goal

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

We refer to the case that the order of G is divisible by pas the modular case, non-modular otherwise. Muchmore is known about the latter case than the former.

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

The modular version of this theorem is still open.

Page 11: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Invariant theory: goal

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

We refer to the case that the order of G is divisible by pas the modular case, non-modular otherwise. Muchmore is known about the latter case than the former.

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

The modular version of this theorem is still open.

Page 12: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Composition Series

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

1G C G2 C G3 C · · ·C Gr = G withGi+1

Gi= Cp ,

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

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

Wehlau proved that the invariant ring of anyrepresentation of Cp is generated by norms and tracesand “rational” functions determined by the classicalinvariant theory of SL2(C).

Page 13: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Composition Series

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

1G C G2 C G3 C · · ·C Gr = G withGi+1

Gi= Cp ,

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

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

Wehlau proved that the invariant ring of anyrepresentation of Cp is generated by norms and tracesand “rational” functions determined by the classicalinvariant theory of SL2(C).

Page 14: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Composition Series

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

1G C G2 C G3 C · · ·C Gr = G withGi+1

Gi= Cp ,

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

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

Wehlau proved that the invariant ring of anyrepresentation of Cp is generated by norms and tracesand “rational” functions determined by the classicalinvariant theory of SL2(C).

Page 15: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

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 Gon a vector space V of dimension 2 over a field ofcharacteristic p. We may assume that

Gρ↪→(

1 ∗0 1

)

Hence G is a subgroup of the additive group (F,+), andso is elementary Abelian, G = (Cp)r for some r . It is nothard to see that

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

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

Page 16: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

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 Gon a vector space V of dimension 2 over a field ofcharacteristic p. We may assume that

Gρ↪→(

1 ∗0 1

)

Hence G is a subgroup of the additive group (F,+), andso is elementary Abelian, G = (Cp)r for some r . It is nothard to see that

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

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

Page 17: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Moduli space for (Cp)r , dimension 2

Given {gi}, generators for G we define ci ∈ F by

ρ(gi) =

(1 ci0 1

),

for some ci ∈ F. That is, G is determined by a vectorc = (c1, c2, . . . , cr ) ∈ Fr .

Let C denote the Fp-span of {c1, c2, . . . , cr} ⊂ F. Therepresentation is faithful if

dimFp(C ) = r .

If αb = c , for α ∈ F∗, b, c ∈ Fr then b and c determinethe same representation of G : representations of G areparametrized by P(Fr ).

Page 18: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Moduli space for (Cp)r , dimension 2

Given {gi}, generators for G we define ci ∈ F by

ρ(gi) =

(1 ci0 1

),

for some ci ∈ F. That is, G is determined by a vectorc = (c1, c2, . . . , cr ) ∈ Fr .

Let C denote the Fp-span of {c1, c2, . . . , cr} ⊂ F. Therepresentation is faithful if

dimFp(C ) = r .

If αb = c , for α ∈ F∗, b, c ∈ Fr then b and c determinethe same representation of G : representations of G areparametrized by P(Fr ).

Page 19: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Moduli space for (Cp)r , dimension 2

Given {gi}, generators for G we define ci ∈ F by

ρ(gi) =

(1 ci0 1

),

for some ci ∈ F. That is, G is determined by a vectorc = (c1, c2, . . . , cr ) ∈ Fr .

Let C denote the Fp-span of {c1, c2, . . . , cr} ⊂ F. Therepresentation is faithful if

dimFp(C ) = r .

If αb = c , for α ∈ F∗, b, c ∈ Fr then b and c determinethe same representation of G : representations of G areparametrized by P(Fr ).

Page 20: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

On the action of Aut(G ) = GLr(Fp)

Elements of Aut(G ) act as permutations on the set ofequivalence classes of representations, but preserve thering of invariants. Therefore, the collection of invariantrings is parametrized by P(Fr )//GLr (Fp).

The coordinate ring is therefore given by the Dicksoninvariants

F[c1, c2, . . . , cr ]GLr (Fp) = F[d1, d2, . . . , dr ]

with |di | = pr − pr−i , for 1 ≤ i ≤ r .

The representation is faithful if dr 6= 0.

Page 21: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

On the action of Aut(G ) = GLr(Fp)

Elements of Aut(G ) act as permutations on the set ofequivalence classes of representations, but preserve thering of invariants. Therefore, the collection of invariantrings is parametrized by P(Fr )//GLr (Fp).

The coordinate ring is therefore given by the Dicksoninvariants

F[c1, c2, . . . , cr ]GLr (Fp) = F[d1, d2, . . . , dr ]

with |di | = pr − pr−i , for 1 ≤ i ≤ r .

The representation is faithful if dr 6= 0.

Page 22: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

On the action of Aut(G ) = GLr(Fp)

Elements of Aut(G ) act as permutations on the set ofequivalence classes of representations, but preserve thering of invariants. Therefore, the collection of invariantrings is parametrized by P(Fr )//GLr (Fp).

The coordinate ring is therefore given by the Dicksoninvariants

F[c1, c2, . . . , cr ]GLr (Fp) = F[d1, d2, . . . , dr ]

with |di | = pr − pr−i , for 1 ≤ i ≤ r .

The representation is faithful if dr 6= 0.

Page 23: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

We assume p > 2.

Page 24: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory of (Cp)r in dimension 3

Given a representation ρ of a p-group G in dimension 3we must have

Gρ↪→

1 a b0 1 c0 0 1

for a, b, c ∈ F.

These can be classified by means of their socles.

Type (2,1): dimFp(V G ) = 2, dimFp((V /V G )G ) = 1.

Type (1,2): dimFp(V G ) = 1, dimFp((V /V G )G ) = 2.

Type (1,1): dimFp(V G ) = 1, dimFp((V /V G )G ) = 1.

Page 25: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory of (Cp)r in dimension 3

Given a representation ρ of a p-group G in dimension 3we must have

Gρ↪→

1 a b0 1 c0 0 1

for a, b, c ∈ F.

These can be classified by means of their socles.

Type (2,1): dimFp(V G ) = 2, dimFp((V /V G )G ) = 1.

Type (1,2): dimFp(V G ) = 1, dimFp((V /V G )G ) = 2.

Type (1,1): dimFp(V G ) = 1, dimFp((V /V G )G ) = 1.

Page 26: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory of (Cp)r in dimension 3

Given a representation ρ of a p-group G in dimension 3we must have

Gρ↪→

1 a b0 1 c0 0 1

for a, b, c ∈ F.

These can be classified by means of their socles.

Type (2,1): dimFp(V G ) = 2, dimFp((V /V G )G ) = 1.

Type (1,2): dimFp(V G ) = 1, dimFp((V /V G )G ) = 2.

Type (1,1): dimFp(V G ) = 1, dimFp((V /V G )G ) = 1.

Page 27: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory of (Cp)r in dimension 3

Given a representation ρ of a p-group G in dimension 3we must have

Gρ↪→

1 a b0 1 c0 0 1

for a, b, c ∈ F.

These can be classified by means of their socles.

Type (2,1): dimFp(V G ) = 2, dimFp((V /V G )G ) = 1.

Type (1,2): dimFp(V G ) = 1, dimFp((V /V G )G ) = 2.

Type (1,1): dimFp(V G ) = 1, dimFp((V /V G )G ) = 1.

Page 28: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Invariant Theory of (Cp)r in dimension 3

Given a representation ρ of a p-group G in dimension 3we must have

Gρ↪→

1 a b0 1 c0 0 1

for a, b, c ∈ F.

These can be classified by means of their socles.

Type (2,1): dimFp(V G ) = 2, dimFp((V /V G )G ) = 1.

Type (1,2): dimFp(V G ) = 1, dimFp((V /V G )G ) = 2.

Type (1,1): dimFp(V G ) = 1, dimFp((V /V G )G ) = 1.

Page 29: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Type(2,1)

In this case, G has the form

Gρ↪→

1 0 a0 1 b0 0 1

.

for some finite subgroup A ⊂ F2. The ring of invariantsis a polynomial algebra on {x , y ,N(z)} for {x , y , z} abasis for V ∗3 .

Page 30: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Type(1,2)

In this case, G has the form

Gρ↪→

1 a b0 1 00 0 1

.

for some finite subgroup A ⊂ F2. The ring of invariantsis a polynomial algebra on {x ,N(y),N(z)} for {x , y , z}a basis for V ∗3 .

Page 31: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Type(1,1)

In this case, G has at least one element G whose Jordanform consists of a single block. By choice of basis wemay assume that

g i =

1 i(i2

)0 1 i0 0 1

.

Assuming that G is Abelian, we have that

Gρ↪→

1 a b0 1 a0 0 1

,

for a, b ∈ F.

Page 32: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Given {gi}, generators for G we define bi , ci ∈ F by

ρ(gi) =

1 ai bi0 1 ai0 1 1

,

for ai , bi ∈ F. Therefore, 3-dimensional representationsof G are determined by matrices

M =

(a1 a2 . . . arb1 b2 . . . br

)in F2×r and are of type (1,1) if at least one ai 6= 0.

Page 33: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

A moduli space for dimension 3

Two matrices

M =

(a1 a2 . . . arb1 b2 . . . br

)and M ′ =

(a′1 a′2 . . . a′rb′1 b′2 . . . b′r

)give equivalent representations if and only if there areα, β in F∗ n F such that(

α 0αβ α2

)M = M ′

Page 34: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

A moduli space for dimension 3, continued

That is, 3-dimensional representations of G areparameterized by the orbits of F2×r under the action ofF n F∗. Here F∗ acts on F by multiplication. We alsohave a right action of GLr (Fp) on G by change of basis,preserving the ring of invariants, and hence an action onF2×r .

Thus the rings of invariants for dimension 3 areparameterized by the F n F∗-orbits acting onF2×r//GLr (Fp).

Page 35: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

A moduli space for dimension 3, continued

That is, 3-dimensional representations of G areparameterized by the orbits of F2×r under the action ofF n F∗. Here F∗ acts on F by multiplication. We alsohave a right action of GLr (Fp) on G by change of basis,preserving the ring of invariants, and hence an action onF2×r .

Thus the rings of invariants for dimension 3 areparameterized by the F n F∗-orbits acting onF2×r//GLr (Fp).

Page 36: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Module spaces for dimension 3, continued

We used elements of F[F 2×r ]GLr (Fp) when r = 2, 3 tostratify F2×r and subsequently provide generators andrelations for the corresponding rings of invariants in thiscases.

Pierron and Shank have extended this work to r = 4, atechnical tour-de-force.

All these rings are complete intersections.

Page 37: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Module spaces for dimension 3, continued

We used elements of F[F 2×r ]GLr (Fp) when r = 2, 3 tostratify F2×r and subsequently provide generators andrelations for the corresponding rings of invariants in thiscases.

Pierron and Shank have extended this work to r = 4, atechnical tour-de-force.

All these rings are complete intersections.

Page 38: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Module spaces for dimension 3, continued

We used elements of F[F 2×r ]GLr (Fp) when r = 2, 3 tostratify F2×r and subsequently provide generators andrelations for the corresponding rings of invariants in thiscases.

Pierron and Shank have extended this work to r = 4, atechnical tour-de-force.

All these rings are complete intersections.

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Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Theorems in dimension 3

Theorem

Let V be a 3-dimensional representation of anelementary Abelian p-group G = (Cp)r . SettingF[V ] = F[x , y , z ], we have

F[V ]G = F[x , f1, f2, . . . , fs ,N(z)]

where LT (f ) = ydi for some {di} ∈ N.

Corollary

There is an efficient algorithm, SAGBI, divide by x , forcomputing generators and relations for F[V ]G .

Page 40: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Conjectures

Conjecture 1

Any modular 3-dimensional representation of anelementary Abelian p-group has a complete intersectionas its ring of invariants of embedding dimensions ≤ dr/2e+ 3.

Page 41: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

The generic conjectures

Conjecture 2, r = 2s even

If the representation above is generic then the ring ofinvariants is a complete intersection of embeddingdimension s + 3, on generators {x , f1, f2, . . . , fs+1,N(z)}of degrees as follows:

1 The case r = 2s:

ps ps + 2ps−1 ps+1 + ps−2 . . . pr−1 + 2 ,

that is, |fi | = ps+i−2 + 2ps−i+1 for 2 ≤ i ≤ s + 1,with relations determined by (f p2 , f

p+21 ), and

(f pi , fi−1f(p2−1)pi−3

1 )

Page 42: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

Conjecture 2, r = 2s − 1 odd

If the representation above is generic then the ring ofinvariants is a complete intersection of embeddingdimension s + 3, on generators

1 The case r = 2s − 1

2ps−1 ps ps + 2ps−2 ps+1 + 2ps−3 pr−1 + 2 ,

that is, |fi | = ps+i−3 + 2ps−i+1, with relationsdetermined by (f p1 , f

22 ), (f p3 , f1f

p2 ), and

(f pi , fi−1f(p2−1)pi−4

2 ).

Page 43: Modular Invariant Theory of Elementary Abelian p-groups in … · Modular Invariant Theory of Elementary Abelian p-groups in dimensions 2 and 3 H E A (Eddy) Campbell University of

Modular InvariantTheory

H E A (Eddy)Campbell

Introduction

Dimension 2

Dimension 3

The generic cases r = 4, 5, 6, 7, 8, s = 2, 3, 4

|f1| |f2| |f3| |f4| |f5|r = 1 2 pr = 2 p p + 2r = 3 2p p2 p2 + 2pr = 4 p2 p2 + 2p p3 + 2r = 5 2p2 p3 p3 + 2p p4 + 2r = 6 p3 p3 + 2p2 p4 + 2p p5 + 2r = 7 2p3 p4 p4 + 2p2 p5 + 2p p6 + 2


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