Invariant Theory and Graphs A computational approach

Specialeforsvar: Magnus Rahbek Dalgaard Hansen

Titel: Invariant Theory and Graphs - A computational approach

Abstract: In this thesis, we provide an exposition of the invariant theory of finite groups, with a focus on algorithms and the Hilbert series. We apply the built-up theory to the algebra of invariants of multigraphs, as well as s-graphs, which are graphs
weighted in {0, 1, ..., s}. Utilizing computer exploration on the invariant algebra of s-graphs, we derive a formula for the Hilbert series of any permutation group acting on a special discrete variety, V |s. We conjecture that this formula can be generalized to any finite group. Furthermore, we present a version of King’s algorithm for computing a (minimal) generating set for the algebra of invariants on simple graphs. We conjecture the correctness of this algorithm and its potential generalization to any finite group acting on V |s. Finally,  we recreate Thiery’s disproof of Pouzet’s conjecture, from [Thi00].

Vejledere: Søren Eilers, Henrik G. Holm
Censor:  Niels Lauritzen, Aarhus Universitet