Algorithmic and structural aspects of modular invariant rings
Peter Fleischmann
Institute for Mathematics, Statistics and Actuarial Science,
University of Kent at Canterbury
Let $G$ be a finite group, acting linearly on the polynomial ring
$A:=F[x_1,\dots,x_d]$ over the field $F$.The ring of invariants
$A^G:=\{f\in A\ |\ g(f)=f\}$ is a central object of study in
invariant theory. This theory is very well developed in the
``classical case" where the characteristic of $F$ is zero, but far
less so in the case of positive characteristic $p$, in particular the
``modular case" where $p$ divides the group order $|G|$. Unlike the
classical case, modular invariant rings are in general not
Cohen - Macaulay, but still finitely generated by a fundamental theorem of
Emmy Noether. However, in the modular situation there remain open
questions about the constructive complexity of modular invariant rings,
measured by degree bounds for generators and the structural complexity,
measured for example by the depth of $A^G$. In my talk I will report on
some recent results in that area, including a new construction
algorithm for $A^G$, developed joinly with G Kemper and C F Woodcock.