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

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1. CJM 2014 (vol 67 pp. 1024)

Ashraf, Samia; Azam, Haniya; Berceanu, Barbu
 Representation Stability of Power Sets and Square Free Polynomials The symmetric group $\mathcal{S}_n$ acts on the power set $\mathcal{P}(n)$ and also on the set of square free polynomials in $n$ variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group. Keywords:symmetric group modules, square free polynomials, representation stability, Arnold algebraCategories:20C30, 13A50, 20F36, 55R80

2. CJM 2012 (vol 66 pp. 3)

Abdesselam, Abdelmalek; Chipalkatti, Jaydeep
 On Hilbert Covariants Let $F$ denote a binary form of order $d$ over the complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant $\mathcal{H}_{r,d}(F)$ vanishes exactly when $F$ is the perfect power of an order $r$ form. In geometric terms, the coefficients of $\mathcal{H}$ give defining equations for the image variety $X$ of an embedding $\mathbf{P}^r \hookrightarrow \mathbf{P}^d$. In this paper we describe a new construction of the Hilbert covariant; and simultaneously situate it into a wider class of covariants called the GÃ¶ttingen covariants, all of which vanish on $X$. We prove that the ideal generated by the coefficients of $\mathcal{H}$ defines $X$ as a scheme. Finally, we exhibit a generalisation of the GÃ¶ttingen covariants to $n$-ary forms using the classical Clebsch transfer principle. Keywords:binary forms, covariants, $SL_2$-representationsCategories:14L30, 13A50

3. CJM 2008 (vol 60 pp. 556)

Draisma, Jan; Kemper, Gregor; Wehlau, David
 Polarization of Separating Invariants We prove a characteristic free version of Weyl's theorem on polarization. Our result is an exact analogue of Weyl's theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of \emph{cheap polarization}, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups. Keywords:Jan Draisma, Gregor Kemper, David WehlauCategories:13A50, 14L24

4. CJM 1999 (vol 51 pp. 616)

Panyushev, Dmitri I.
 Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory Let $L$ be a simple algebraic group and $P$ a parabolic subgroup with Abelian unipotent radical $P^u$. Many familiar varieties (determinantal varieties, their symmetric and skew-symmetric analogues) arise as closures of $P$-orbits in $P^u$. We give a unified invariant-theoretic treatment of various properties of these orbit closures. We also describe the closures of the conormal bundles of these orbits as the irreducible components of some commuting variety and show that the polynomial algebra $k[P^u]$ is a free module over the algebra of covariants. Categories:14L30, 13A50
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