| The study of the relationship between a ring R and its subring of invariants RG under the action of a group G, invariant theory for short, is a classical algebraic theme permeating virtually all areas of pure mathematics, some areas of applied mathematics, notably coding theory, and certain parts of theoretical physics as well.; Although the field of invariant theory is over a century old, one particular branch, multiplicative invariant theory, has emerged and attracted much attention in the last 35 years. In multiplicative invariant theory one considers a free abelian group A of finite rank n (A ≅ ) on which a group G acts by automorphisms ( G → GLn()). The G-action on A extends uniquely to an action on the group algebra R = k[ A] ().; The motivating problem of this dissertation can be summed up as follows: Let A be a free abelian group of finite rank, let k be any commutative ring, and let G be a finite group acting multiplicatively on the group algebra k [A]. Construct and implement an efficient algorithm for computing generators for k[A] G, the subalgebra of multiplicative invariants. ; Finding explicit generators for invariants has always been at the heart of classical invariant theory; in recent years powerful algorithms have been developed for the computation of invariants in the classical setting. However, very little has been done for the computation of multiplicative invariants (exceptions include [Lor01] and [Rei02]).; In Chapter 3 we provide algorithms for computing generators for k[A]G when G is a reflection group (§3.3), and in chapter 4 we compute generators for k[A]G when G is a subgroup of a reflection group (§4.2). The algorithms have been implemented in the computer algebra system Magma , and the program source code along with sample output can be found in the appendices. |
