Ratio of varieties by actions of reductive groups


  • Nélida Medina García




Varieties, small groups


We consider the ring of polynomials R = K[x1, dots, xn] in the variables x1, dots, xn and complex coefficients. The permutation group of 1, dots, n acts sore R by permuting the variables. The set of invariants by this action forms a ring generated by elementary symmetric polynomials. Emmy Noether proves that if a finite group of inverse matrices G subsetGL(n; k) acts on R, then the ring of invariants is generated by a finite number of invariant homogeneous and defines an operator in G to obtain invariant polynomials. There are algebraic relationships between the generators of the invariant ring and the orbits of Cn/G. In 1963, Masayoshi Nagata demonstrated that the ring of the invariants of geomagically reductive groups is finitely generated. We analice the existence of a quotient variety X/G where G is an algebraic group acting on an algebraic variety X.


Cox David, Little Jhon & O'shea Donald. Ideals, Varieties, and Algorithms, Tercera edición, Springer Science-Business Media, USA.2007.

Fléischmann Peter. On Invariant Theory of Finite Groups, Institut of Mathematics and Statistics. University of Kent at Canterbury. 2006.

Reynoso Claudia. Introducción a la Teoría de Invariantes Geométricos, Departamento de Guanajuato, Guanajuato, México. 2010.



How to Cite

Medina García, N. (2017). Ratio of varieties by actions of reductive groups. Selecciones Matemáticas, 4(01), 25-29. https://doi.org/10.17268/sel.mat.2017.01.03