Algorithmen zur Berechnung von multiplikativen Invariantenringen

• Algorithms for computing multiplicative invariant rings

Korodi, Tamás-István; Plesken, Wilhelm (Thesis advisor); Nebe, Gabriele (Thesis advisor)

Aachen (2020)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2020

Abstract

In the most general case invariant theory considers with the construction and structure of invariant rings and their geometric interpretations. For the construction of an invariant ring we take an arbitrary ring $R$ and a group $G$ operating by ring automorphisms on $R$. The invariant ring $R^G$ consists of all elements in $R$ which are fixed by the operation of $G$. \[R^G = \{ x \in R \mid g(x) = x \text{ for all } g \in G \}.\] In this thesis we look at the situation of multiplicative invariant rings. Here the ring $R$ is a group ring $\mathbb{Z}[\Gamma]$ for a $d$-dimensional lattice $\Gamma$ and the group $G$ is a finite subgroup of the lattice point group of $\Gamma$. For elements $a \in \Gamma$ and $g \in G$ this operation is defined on the monomials by the formula $(g, X^a) \mapsto X^{g(a)}$. There already exists an extensive monograph by Prof. Dr. Martin Lorenz for the algebraic theory of multiplicative invariant rings. A yet remaining problem was the algorithmic treatment of these multiplicative invariant rings. There has already been done significant work for the cases where $G$ is a reflection group or a subgroup of a reflection group. However a general algorithmic treatment is missing just like an easily accessible database for generators of a multiplicative invariant ring. An aim of this work is to close the gap of non-reflection-groups and give algorithms for computing multiplicative invariant rings. At the beginning we describe a simple method for computing generators by using a term order on $R$ given by a positive definite $G$-invariant bilinear form. For finding relations between generators we also need to compute with ideals inside multiplicative invariant rings. For the realization of these algorithms we develop a notion of $G$-invariant Groebner bases which are a generalization of usual Groebner bases for the case of multiplicative invariant rings. It turns out that this structure of multiplicative invariant rings is essentially determined by the geometric structure of related $G$-invariant cone decompositions and polytopes. Experimental tests give good results for subgroups of $GL_d(\mathbb{Z})$ up to dimension $d = 8$. For higher dimensions the running time of the program becomes too slow due to the combinatorial explosion of higher-dimensional polytopes.