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1. CMB 2010 (vol 53 pp. 404)
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 Chevalley--Shephard--Todd type theorem is proved. Suppose $G$ is abelian. Then the action is coregular if and only if $G$ is generated by pseudo-reflections and the direct summand property holds.
Category:13A50 |
2. CMB 2009 (vol 53 pp. 77)
Constructing (Almost) Rigid Rings and a UFD Having Infinitely Generated Derksen and Makar-Limanov Invariants |
Constructing (Almost) Rigid Rings and a UFD Having Infinitely Generated Derksen and Makar-Limanov Invariants An example is given of a UFD which has an infinitely generated Derksen invariant. The ring is "almost rigid" meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem.
Categories:14R20, 13A50, 13N15 |
3. CMB 2009 (vol 52 pp. 72)
A SAGBI Basis For $\mathbb F[V_2\oplus V_2\oplus V_3]^{C_p}$ Let $C_p$ denote the cyclic group of order $p$, where $p \geq 3$ is
prime. We denote by $V_n$ the indecomposable $n$ dimensional
representation of $C_p$ over a field $\mathbb F$ of characteristic
$p$. We compute a set of generators, in fact a SAGBI basis, for
the ring of invariants $\mathbb F[V_2 \oplus V_2 \oplus V_3]^{C_p}$.
Category:13A50 |
4. CMB 2002 (vol 45 pp. 272)
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 |
5. CMB 1999 (vol 42 pp. 155)
Non--Cohen-Macaulay 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 non-trivial $p$-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
non-trivial 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 $n(|G|-1)$. If the ring of
invariants is a hypersurface, the upper bound can be improved to $|G|$.
Category:13A50 |
6. CMB 1999 (vol 42 pp. 125)
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 |