Geometric theory of invariants of groups generated by reflections

Invariants of finite and infinite groups generated by reflections

“…Still, a long time ago there arose the necessity of investigating and classifying group representations with a finite number of generators. In particular, infinite groups generated by oblique reflections respectively to surfaces [29] belong to these groups. In the paper of V. P. Platonov [54], a review of results obtained on this topic is given, and a number of interesting conjectures and questions are proposed.…”

“…be defined in the space Era, if the contrary is not specified, F. is a noncylindrical surface of the order n invariant with respect to the infinite group G generated by oblique (in particular, orthogonal) reflections with respect to planes; /z i are planes of H ~j (j = 0~) spanned by G(u), the orbits of the symmetry directions u. Then H ~j = H 'i ~ H "ri , where 71 are surfaces of II "yj possessing the following properties: symmetry planes F, conjugated to vectors II ~j are parallel to H 7j [29]. The mutual arrangement of H ~j is determined by the 7j-surfaces of H "r~ .…”

“…The structure of a degenerated diametrical quadric of the surface Fn was studied in [38]. O. I. Roudnitski showed that a degenerated diametrical surface of a arbitrary order is decomposed into real, and, possibly, imaginary surfaces; in this way, he solved, in particular, one of the problems of the paper [33]. We have obtained a new (synthetic) proof of this result; moreover, it is shown that the decomposition of this surface into planes is explained not only by its degeneration but by the existence of other surfaces Fn, as well, whose nondegenerated diametrical surfaces are decomposed into planes.…”

“…The mutual arrangement of H ~j is determined by the 7j-surfaces of H "r~ . General canonical equations of surfaces F, with groups G are given in [29] (with the corresponding historical references). In the same paper, the structure of complete symmetry groups G of the surface Fn is pointed out.…”

Some properties of basis invariants of the symmetry groupsG(m,p,n), B n m , D n m

“…The Pogorelov polynomials J~ belong to the algebra I c of polynomials that are invariant with respect to the group G [3]. In the present note we establish relations for the numbers m and n for which .the basis m (the generators of the algebra I ~ are representable invariants of the groups G = G (m,p, n), B~, and D n in the form (2); we construct all the generators of even degree of the algebras I ~ G = G (m,p, n), Bm~, D~.…”