1. CMB Online first
 Nakashima, Norihiro; Terao, Hiroaki; Tsujie, Shuhei

Canonical systems of basic invariants for unitary reflection groups
It has been known that there exists a canonical system for every
finite real reflection group. The first and the third authors
obtained
an explicit formula for a canonical system in the previous paper.
In this article, we first define canonical systems for the finite
unitary reflection groups, and then prove their existence.
Our proof does not depend on the classification of unitary reflection
groups.
Furthermore, we give an explicit formula for a canonical system
for every unitary reflection group.
Keywords:basic invariant, invariant theory, finite unitary reflection group Categories:13A50, 20F55 

2. CMB 2010 (vol 53 pp. 404)
 Broer, Abraham

Invariant Theory of Abelian Transvection Groups
Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called \emph{coregular} if the invariant ring is generated by algebraically independent homogeneous invariants, and the \emph{direct summand property} holds if there is a surjective $k[V]^G$linear map $\pi\colon k[V]\to k[V]^G$. The following ChevalleyShephardTodd type theorem is proved. Suppose $G$ is abelian. Then the action is coregular if and only if $G$ is generated by pseudoreflections and the direct summand property holds.
Category:13A50 

3. CMB 2009 (vol 53 pp. 77)
4. CMB 2009 (vol 52 pp. 72)
5. CMB 2002 (vol 45 pp. 272)
 Neusel, Mara D.

The Transfer in the Invariant Theory of Modular Permutation Representations II
In this note we show that the image of the transfer for permutation
representations of finite groups is generated by the transfers of
special monomials. This leads to a description of the image of the
transfer of the alternating groups. We also determine the height of
these ideals.
Keywords:polynomial invariants of finite groups, permutation representation, transfer Category:13A50 

6. CMB 1999 (vol 42 pp. 155)
 Campbell, H. E. A.; Geramita, A. V.; Hughes, I. P.; Shank, R. J.; Wehlau, D. L.

NonCohenMacaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants
This paper contains two essentially independent results in the
invariant theory of finite groups. First we prove that, for any
faithful representation of a nontrivial $p$group over a field of
characteristic $p$, the ring of vector invariants of $m$ copies of
that representation is not \comac\ for $m\geq 3$. In the second
section of the paper we use Poincar\'e series methods to produce upper
bounds for the degrees of the generators for the ring of invariants as
long as that ring is Gorenstein. We prove that, for a finite
nontrivial group $G$ and a faithful representation of dimension $n$
with $n>1$, if the ring of invariants is Gorenstein then the ring is
generated in degrees less than or equal to $n(G1)$. If the ring of
invariants is a hypersurface, the upper bound can be improved to $G$.
Category:13A50 

7. CMB 1999 (vol 42 pp. 125)
 Smith, Larry

Modular Vector Invariants of Cyclic Permutation Representations
Vector invariants of finite groups (see the introduction for an
explanation of the terminology) have often been used to illustrate the
difficulties of invariant theory in the modular case: see,
\eg., \cite{Ber}, \cite{norway}, \cite{fossum}, \cite{MmeB},
\cite{poly} and \cite{survey}. It is therefore all the more
surprising that the {\it unpleasant} properties of these invariants
may be derived from two unexpected, and remarkable, {\it nice}
properties: namely for vector permutation invariants of the cyclic
group $\mathbb{Z}/p$ of prime order in characteristic $p$ the
image of the transfer homomorphism $\Tr^{\mathbb{Z}/p} \colon
\mathbb{F}[V] \lra \mathbb{F}[V]^{\mathbb{Z}/p}$ is a prime ideal,
and the quotient algebra $\mathbb{F}[V]^{\mathbb{Z}/p}/ \Im
(\Tr^{\mathbb{Z}/p})$ is a polynomial algebra on the top Chern
classes of the action.
Keywords:polynomial invariants of finite groups Category:13A50 
