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 primarily deals with 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 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. Even for reflection groups, not much is known, due to the presence
of nondiagonalizable reflections (transvections).
In a project with Anne
Shepler,
we study invariant rings of reflection groups and explore the role of the transvections. We are also interested in invariant differential forms in
positive characteristic; in particular, in freeness questions and twisted
algebra structures.
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