Modular Invariant Theory of Elementary Abelian p-groups in

43 Pages·2014·0.54 MB·English

ModularInvariant
Theory
HEA(Eddy)
Campbell Modular Invariant Theory of
Introduction Elementary Abelian p-groups in
Dimension2
Dimension3 dimensions 2 and 3
H E A (Eddy) Campbell
UniversityofNewBrunswick
October 19, 2014
Outline
ModularInvariant
Theory
HEA(Eddy)
Campbell
Introduction 1 Introduction
Dimension2
Dimension3
2 Dimension 2
3 Dimension 3
A basis {x ,x ,...,x } for V∗.
1 2 n
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[x ,x ,...,x ].
1 2 n
The ring, F[V]G, of polynomials fixed by the action
of G.
Invariant Theory in general: ingredients
ModularInvariant
Theory
HEA(Eddy)
Campbell
A group G represented on a vector space V over a
Introduction field F of characteristic p.
Dimension2
Dimension3
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[x ,x ,...,x ].
1 2 n
The ring, F[V]G, of polynomials fixed by the action
of G.
Invariant Theory in general: ingredients
ModularInvariant
Theory
HEA(Eddy)
Campbell
A group G represented on a vector space V over a
Introduction field F of characteristic p.
Dimension2
A basis {x ,x ,...,x } for V∗.
Dimension3 1 2 n
The induced action of G by algebra automorphisms
on F[V] = F[x ,x ,...,x ].
1 2 n
The ring, F[V]G, of polynomials fixed by the action
of G.
Invariant Theory in general: ingredients
ModularInvariant
Theory
HEA(Eddy)
Campbell
A group G represented on a vector space V over a
Introduction field F of characteristic p.
Dimension2
A basis {x ,x ,...,x } for V∗.
Dimension3 1 2 n
The action of G on V∗ by σ(f)(v) = f(σ−1(v)).
The ring, F[V]G, of polynomials fixed by the action
of G.
Invariant Theory in general: ingredients
ModularInvariant
Theory
HEA(Eddy)
Campbell
A group G represented on a vector space V over a
Introduction field F of characteristic p.
Dimension2
A basis {x ,x ,...,x } for V∗.
Dimension3 1 2 n
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[x ,x ,...,x ].
1 2 n
Invariant Theory in general: ingredients
ModularInvariant
Theory
HEA(Eddy)
Campbell
A group G represented on a vector space V over a
Introduction field F of characteristic p.
Dimension2
A basis {x ,x ,...,x } for V∗.
Dimension3 1 2 n
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[x ,x ,...,x ].
1 2 n
The ring, F[V]G, of polynomials fixed by the action
of G.
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.
Invariant theory: goal
ModuTlahreIonrvyariant We seek to understand F[V]G in terms of its generators
HEA(Eddy) and relations or in turns of its structure such as the
Campbell
Cohen-Macaulay property by relating algebraic properties
Introduction to the geometric properties of the representation.
Dimension2
Dimension3
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.
Invariant theory: goal
ModuTlahreIonrvyariant We seek to understand F[V]G in terms of its generators
HEA(Eddy) and relations or in turns of its structure such as the
Campbell
Cohen-Macaulay property by relating algebraic properties
Introduction to the geometric properties of the representation.
Dimension2
Dimension3 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.
The modular version of this theorem is still open.
Invariant theory: goal
ModuTlahreIonrvyariant We seek to understand F[V]G in terms of its generators
HEA(Eddy) and relations or in turns of its structure such as the
Campbell
Cohen-Macaulay property by relating algebraic properties
Introduction to the geometric properties of the representation.
Dimension2
Dimension3 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.