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A Geometrical Construction for the Polynomial Invariants of some Reflection Groups

Sarti, Alessandra (2005)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 20F55, 13F20; Secondary 14L30.We construct invariant polynomials for the reflection groups [3, 4, 3] and [3, 3, 5] by using some special sets of lines on the quadric P1 × P1 in P3. Then we give a simple proof of the well known fact that the ring of invariants are rationally generated in degree 2,6,8,12 and 2,12,20,30.

A minimal Set of Generators for the Ring of multisymmetric Functions

David Rydh (2007)

Annales de l’institut Fourier

The purpose of this article is to give, for any (commutative) ring A , an explicit minimal set of generators for the ring of multisymmetric functions T S A d ( A [ x 1 , , x r ] ) = A [ x 1 , , x r ] A d 𝔖 d as an A -algebra. In characteristic zero, i.e. when A is a -algebra, a minimal set of generators has been known since the 19th century. A rather small generating set in the general case has also recently been given by Vaccarino but it is not minimal in general. We also give a sharp degree bound on the generators, improving the degree bound previously...

A nonlinearizable action of S 3 on 4

Gene Freudenburg, Lucy Moser-Jauslin (2002)

Annales de l’institut Fourier

The main purpose of this article is to give an explicit algebraic action of the group S 3 of permutations of 3 elements on affine four-dimensional complex space which is not conjugate to a linear action.

A note on the kernels of higher derivations

Jiantao Li, Xiankun Du (2013)

Czechoslovak Mathematical Journal

Let k k ' be a field extension. We give relations between the kernels of higher derivations on k [ X ] and k ' [ X ] , where k [ X ] : = k [ x 1 , , x n ] denotes the polynomial ring in n variables over the field k . More precisely, let D = { D n } n = 0 a higher k -derivation on k [ X ] and D ' = { D n ' } n = 0 a higher k ' -derivation on k ' [ X ] such that D m ' ( x i ) = D m ( x i ) for all m 0 and i = 1 , 2 , , n . Then (1) k [ X ] D = k if and only if k ' [ X ] D ' = k ' ; (2) k [ X ] D is a finitely generated k -algebra if and only if k ' [ X ] D ' is a finitely generated k ' -algebra. Furthermore, we also show that the kernel k [ X ] D of a higher derivation D of k [ X ] can be generated by a set...

A Remark on a Paper of Crachiola and Makar-Limanov

Robert Dryło (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

A. Crachiola and L. Makar-Limanov [J. Algebra 284 (2005)] showed the following: if X is an affine curve which is not isomorphic to the affine line ¹ k , then ML(X×Y) = k[X]⊗ ML(Y) for every affine variety Y, where k is an algebraically closed field. In this note we give a simple geometric proof of a more general fact that this property holds for every affine variety X whose set of regular points is not k-uniruled.

A remarkable contraction of semisimple Lie algebras

Dmitri I. Panyushev, Oksana S. Yakimova (2012)

Annales de l’institut Fourier

Recently, E.Feigin introduced a very interesting contraction 𝔮 of a semisimple Lie algebra 𝔤 (see arXiv:1007.0646 and arXiv:1101.1898). We prove that these non-reductive Lie algebras retain good invariant-theoretic properties of 𝔤 . For instance, the algebras of invariants of both adjoint and coadjoint representations of 𝔮 are free, and also the enveloping algebra of 𝔮 is a free module over its centre.

A Survey of Counterexamples to Hilbert's Fourteenth Problem

Freudenburg, Gene (2001)

Serdica Mathematical Journal

We survey counterexamples to Hilbert’s Fourteenth Problem, beginning with those of Nagata in the late 1950s, and including recent counterexamples in low dimension constructed with locally nilpotent derivations. Historical framework and pertinent references are provided. We also include 8 important open questions.

Actions of Hopf algebras on pro-semisimple noetherian algebras and their invariants

Andrzej Tyc (2001)

Colloquium Mathematicae

Let H be a Hopf algebra over a field k such that every finite-dimensional (left) H-module is semisimple. We give a counterpart of the first fundamental theorem of the classical invariant theory for locally finite, finitely generated (commutative) H-module algebras, and for local, complete H-module algebras. Also, we prove that if H acts on the k-algebra A = k[[X₁,...,Xₙ]] in such a way that the unique maximal ideal in A is invariant, then the algebra of invariants A H is a noetherian Cohen-Macaulay...

AK-invariant, some conjectures, examples and counterexamples

L. Makar-Limanov (2001)

Annales Polonici Mathematici

In my talk I am going to remind you what is the AK-invariant and give examples of its usefulness. I shall also discuss basic conjectures about this invariant and some positive and negative results related to these conjectures.

Algebras with finitely generated invariant subalgebras

Ivan V. Arzhantsev (2003)

Annales de l’institut Fourier

We classify all finitely generated integral algebras with a rational action of a reductive group such that any invariant subalgebra is finitely generated. Some results on affine embeddings of homogeneous spaces are also given.

Calculs d'invariants primitifs de groupes finis

Ines Abdeljaouad (2010)

RAIRO - Theoretical Informatics and Applications

We introduce in this article a new method to calculate all absolute and relatif primitive invariants of finite groups. This method is inspired from K. Girstmair which calculate an absolute primitive invariant of minimal degree. Are presented two algorithms, the first one enable us to calculate all primitive invariants of minimal degree, and the second one calculate all absolute or relative primitive invariants with distincts coefficients. This work take place in Galois Theory and Invariant Theory. ...

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