We study the properties of the basis invariants of the symmetry groups of the complex polytope ~.ym and the generalized n-cube "yn m, as well as its subgroups D~. We 9ire art explicit construction of all the basis invariants of even degree of these groups.In n-dimensional unitary space U n we introduce an orthogonal coordinate system with origin 0 and basis vectors -ei (i = 1---,-~); G is the finite irreducible group generated by reflections with respect to (n -1)-dimensional planes containing the common point 0. We define the vertices of the generalized n-cube "~ to be the following m ~ vectors aEO here a is reflection with respect to the planes of symmetry; ~ is the unit normal vector (with origin 0) of one of them, and the vector ~ is given by x" = (x~). 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~. TO(ra,P,n)Theorem. The polynomials "rat are the generators of even degree mt (1 < t < n -1) of the algebra I G (ra,p,'O for any n, provided rn ~ 2 l ~ = 1, 2,...). In the case rn = 2 t the number n satisfies and n reflections of order 2 with respect to the planes with equations z~=xj+x=0, j=l,n-1; (5) zl -Oz2 = 0.
Продолжена работа по построению канонических систем базисных инвариантов для конечных унитарных примитивных групп, порождённых отражениями. А именно: построена в явном виде каноническая система базисных инвариантов для конечной унитарной примитивной группы W(K 5), порождённой отражениями, в пятимерном унитарном пространстве.
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