Role of the officer in Air Force Public Relations

Thesis (M.S.)--Boston Universit

Degree bounds for modular covariants

Let V,W be representations of a cyclic group G of prime order p over a field k of characteristic p. The module of covariants k[V,W]^G is the set of G-equivariant polynomial maps from V to W, and is a module over the algebra of invariants k[V]^G. We give a formula for the Noether bound of k[V,W]^G over k[V]^G, i.e. the minimal degree d such that k[V,W]^G is generated over k[V]^G by elements of degree at most d

Zero-separating invariants for linear algebraic groups

Let G be linear algebraic group over an algebraically closed field k acting rationally on a G-module V , and N(G,V) its nullcone. Let δ(G, V ) and σ(G, V ) denote the minimal number d, such that for any v ∈ V^G \ N(G,V) and v ∈ V \ N(G,V) respectively, there exists a homogeneous invariant f of positive degree at most d such that f (v) = 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V . For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL 2 (k) which contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G 0 is unipotent. Our results also lead to a more elementary proof that β_sep(G) is finite if and only if G is finite

On separating a fixed point from zero by invariants

Assume a fixed point v in V^G can be separated from zero by a homogeneous invariant f ∈ k[V]^G of degree p^r d where p > 0 is the characteristic of the ground field k and p, d are coprime. We show that then v can also be separated from zero by an invariant of degree p^r , which we obtain explicitly from f . It follows that the minimal degree of a homogeneous invariant separating v from zero is a p-power

Zero-separating invariants for finite groups

We fix a field k of characteristic p. For a finite group G denote by δ(G) and σ(G) respectively the minimal number d, such that for every finite dimensional representation V of G over k and every v ∈ V^G \ {0} or v ∈ V \ {0} respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) = 0. Let P be a Sylow-p-subgroup of G (which we take to be trivial if the group order is not divisble by p). We show that δ(G) = |P|. If N_G(P)/P is cyclic, we show σ(G) ≥ |N_G(P)|. If G is p-nilpotent and P is not normal in G, we show σ(G) ≤ |G|/l , where l is the smallest prime divisor of |G|. These results extend known results in the non-modular case to the modular case

Separating Invariants for the Basic G_a actions

Abstract. We explicitly construct a finite set of separating invariants for the basic G_a -actions. These are the finite dimensional indecomposable rational linear representations of the additive group G_a of a field of characteristic zero, and their invariants are the kernel of the Weitzenbock derivation

Locally finite derivations and modular coinvariants

We consider a finite dimensional kG-module V of a p-group G over a field k of characteristic p. We describe a generating set for the corresponding Hilbert Ideal. In case G is cyclic this yields that the algebra k[V]_G of coinvari-ants is a free module over its subalgebra generated by kG-module generators of V^∗ . This subalgebra is a quotient of a polynomial ring by pure powers of its variables. The coinvariant ring was known to have this property only when G was cyclic of prime order. In addition, we show that if G is the Klein 4-group and V does not contain an indecomposable summand isomorphic to the regular module, then the Hilbert Ideal is a complete intersection, extending a result of the second author and R. J. Shank