Closedness of basis surfaces invariant with respect to the group $H_3$

V. A. Ternovskiy

Taurida Journal of Computer Science Theory and Mathematics, no. 2 (2022), pp. 96-102

Résumé

In the real $m$-dimensional Euclidean space $E^m$, the group $G$ is generated by orthogonal reflections with respect to $N(G)$ hyperplanes with common point $O$. In a rectangular coordinate system, we assign the algebraic hypersurface of order $n$ to $f(x)=0$, where $f(x)$ is a polynomial degrees $n$ with respect to the coordinates of the vector $\overrightarrow{x} =(x_1,\ldots,x_m)$. The set of all hypersurfaces invariant with respect to the same group $G$ corresponds to the set $f(x)$, which forms the algebra $I^G$. The description of the algebra $I^G$ is a fundamental problem in the theory of invariants. Each method for obtaining such polynomials refers to their important properties, but in practice it is not always possible to explicitly write down all the basic polynomials. In this connection, it is of interest to find the basis invariants of even degrees that belong to the algebra $\theta^G$ of polynomials \begin{equation*} {{\theta }_{2r}}\left( \overrightarrow{x} \right)=\sum\limits_{j=1}^{N(G)}{\eta_{j}^{2r}}\left( \overrightarrow{x} \right), \end{equation*} where $\eta_j (\overrightarrow{x})=0$ – normalized equations of all hyperplanes whose reflections belong to $G$. For any natural $r$, the invariant $\theta_{2r}$ has a simple geometric interpretation: the surface given by the equation $\theta_{2r}=c$ is the set of all such points in the space $E^n$, the sum $2r$-th degrees of distance of an arbitrary point of which from the planes $\eta_j (\overrightarrow{x})=0$ is equal to the constant $c$. Finding explicitly the basic invariants of finite groups generated by orthogonal reflections in Euclidean spaces, V. F. Ignatenko set the task in his work: to obtain the equations of all basic closed surfaces other than a sphere. For groups $A_3$ and $B_3$ this problem is solved by the author. In this article, we study the structure of the basis surfaces invariant with respect to finite groups generated by reflections in real space. Sufficient conditions for the closure of all basis surfaces of the algebras of invariants for the symmetry group of the icosahedron $H_3$ in three-dimensional space are obtained.

Détail

Mots-clés : invariant
Keywords: symmetry groups, basic invariant, algebra of invariants, groups generated by reflections.