Rings of Generalized and Stable Invariants of Pseudoreflections and Pseudoreflection Groups Original Research Article

Let varrho:Gright arrow-hookedGL(n, image) be a representation of a finite groupGover the field image and image[V] the space of polynomial functions onV=imagen. We associate toGan idealJ∞(G)subset ofimage[V] called the ideal of stable invariants ofG(or more precisely varrho). If imagesubset ofGL(n, image) is a set of pseudoreflections we associate to image an idealI(image)subset ofimage[V] called the ideal of generalized invariants of image. WhenGis a pseudoreflection group we investigateI(image) for various choices of imagesubset ofGand the relation betweenJ∞(G) andI(image). To a representation varrho:Gright arrow-hookedGL(n, image) of a finite group, respectively to a set imagesubset ofGL(n, image) of pseudoreflections, we also associate a ring[formula], respectively[formula]. We show that[formula]is always a polynomial algebra over image, and whenever varrho(G) is generated by semisimple pseudoreflections image that[formula].