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Search: MSC category 13A50 ( Actions of groups on commutative rings; invariant theory [See also 14L24] )

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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 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.


3. CMB 2009 (vol 53 pp. 77)

Finston, David; Maubach, Stefan
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

4. CMB 2009 (vol 52 pp. 72)

Duncan, Alexander; LeBlanc, Michael; Wehlau, David L.
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 $\mathbb F[V_2 \oplus V_2 \oplus V_3]^{C_p}$.


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

6. CMB 1999 (vol 42 pp. 155)

Campbell, H. E. A.; Geramita, A. V.; Hughes, I. P.; Shank, R. J.; Wehlau, D. L.
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 $|G|$.


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

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