Reflection groups and semigroup algebras in multiplicative invariant theory

In classical invariant theory one considers the situation where a group G of n x n matrices over a field K acts on the polynomial algebra K&sqbl0;x1,&ldots;,xn&sqbr0; by linear substitutions of the variables xi. The subalgebra of all polynomials fixed (invariant) under the action of G is called the algebra of polynomial G -invariants, usually denoted by K&sqbl0;x1,&ldots;,xn&sqbr0; G . One of the most celebrated results on polynomial invariants is the Shephard-Todd-Chevalley Theorem:; Assume that G is finite of order not divisible by the characteristic of K . Then the invariant algebra K&sqbl0;x1,&ldots;,xn&sqbr0; G is again a polynomial algebra if and only if G acts as a pseudo-reflection group on the vector space V = ⊕ni=1Kxi .; Here an element g ∈ G is called a pseudo-reflection on V if G acts trivially on a hyperplane in V or, equivalently, if the n x n matrix g - 1V has rank one. The group G is said to act as a pseudo-reflection group on V if G is generated by elements that are pseudo-reflections on V .; A different kind of group action, on Laurent polynomial algebras instead of polynomial algebras, is defined as follows. Let G be a finite group acting by automorphism on a lattice A≅Z n , and hence on the group algebra KA over a field K . Fixing an isomorphism A≅Z n , we may think of the G -action on A as given by a homomorphism G→GLn Z , the group of invertible n x n-matrices over Z , and the group algebra KA can be identified with the Laurent polynomial algebra K&sqbl0;x+/-11,&ldots;,x +/-1n&sqbr0; . The subalgebra KA G of G -invariant Laurent polynomials in KA is called an algebra of multiplicative invariants. A result of Lorenz ([?]) states that if G acts as a pseudo-reflection group on the Q -vectorspace A⊗Q then KA G is a semigroup algebra over K . This is a multiplicative analogue of the "if"-part of Shephard-Todd-Chevalley theorem for polynomial invariants. However, the converse of Lorenz's theorem is open. In this thesis we will state and prove an extended version of this theorem which does indeed have a converse. Moreover, we take a different approach than [?] inasmuch as our arguments are based on the geometry of simplicial cones rather than root systems.