Invariant Theory
Given an action of a group on some set, invariant theory studies the subset of
elements which remain fixed under this action. Most of the time, the set has
additional
structure properties (e.g. a ring structure)
some of which the set of invariants will inherit. For a typical example,
consider the action of a group on a
polynomial ring in n variables induced by an n-dimensional linear representation.
My own research activity in this area is mostly restricted to the case of the
action of a finite group on a polynomial algebra as above, or on the tensor
product of a polynomial algebra and an exterior algebra on the same number of
generators on which the group acts simultaneously. Such rings of invariants
occur for example in group cohomology.
The behaviour of such rings of invariants is particularly nice and
well understood in the case when the characteristic of the field over which the
representation is given does not divide the order of the
group.
Much less is known in the remaining case - which makes it all the more interesting.
In projects with Anne
Shepler,
we study the Jacobian determinant of a group with a polynomial ring of invariants over
arbitrary fields, as well as the module of invariant differential forms.
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