On a basic invariants of the symmetry group of polyhedron.

In a $n$-dimensional unitary space $ \mathrm{U}_{}^{n} $ $(n > 4)$ there are three series of regular polytopes: the regular simplex $ \mathrm{\alpha}_{n}^{} $ , the generalized cross polytopes $ \mathrm{\beta}_{n}^{m} $ and the generalized $n$-cube $ \mathrm{\gamma}_{n}^{m} $. The generalized $n$-cube has $ m^{n} $ vertices:
$ (\mathrm{\Theta}_{}^{\mathrm{k}_{1}^{}}, \mathrm{\Theta}_{}^{\mathrm{k}_{2}^{}}, ..., \mathrm{\Theta}_{}^{\mathrm{k}_{n}^{}} ) $,
where $ \mathrm{k}_{1}^{}, \mathrm{k}_{2}^{}, ..., \mathrm{k}_{n}^{} $ take any integral values and $θ$ is a primitive $m$th root of unity.
For a certain divisor $p$ of the number m the vertices of $ \mathrm{\gamma}_{n}^{m} $ with
$ \sum_{i=1}^{n}\mathrm{k}_{i}^{} \equiv 0 $ (mod p)
(there are $ qm^{n-1} $ of them if $ m = pq $) determine a complex polytope $ \mathrm{\gamma}_{n}^{m} $ . The symmetry group of $ \mathrm{\gamma}_{n}^{m} $ is the imprimitive group $G(m, p, n)$ generated by reflections. It is well known that the set of polynomials invariant with respect to $G(m, p, n)$ forms an algebra generated by $n$ algebraically independent homogeneous polynomials of degrees $m, 2m, . . . ,(n − 1)m, qn$ (a system of basic invariants of group $G(m, p, n)$).
In this paper, we study the properties of basic invariants of group $G(m, p, n)$. It is given a positive solution to the "vertex problem" for the polytope $ \mathrm{\gamma}_{n}^{m} $ if $p$ and $n$ is mutually prime.
Namely, polynomials
$ \mathrm{V}_{s}^{} = \sum_{\mathrm{k}_{i}^{}}^{}(\mathrm{\Theta}_{}^{\mathrm{k}_{1}^{}}\mathrm{x}_{1}^{}, \mathrm{\Theta}_{}^{\mathrm{k}_{2}^{}}\mathrm{x}_{2}^{}, ... ,
\mathrm{\Theta}_{}^{\mathrm{k}_{n}^{}}\mathrm{x}_{n}^{} )^{ms}, \sum_{i=1}^{n}\mathrm{k}_{i}^{} \equiv 0 $ (mod p), $ s = \overline{1,n-1} $
are algebraically independent and are basic invariants of group $ G(m, p, n) $ if $ p $ and $n$ is mutually
prime.
Keywords: unitary space, reflection, basic invariant, algebra of invariants, complex polyhedron