Modular Invariant Theory of Elementary Abelian p-groups in PDF
Preview Modular Invariant Theory of Elementary Abelian p-groups in
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.
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