On a basic invariants of the symmetry group of complex polyhedron $\frac{1}{p}{\gamma}_{n}^{m}$
O. I. Rudnitsky V. I. Vernadsky Crimean Federal University, Simferopol
Abstract:
In a
$n$-dimensional unitary space
${U}^{n}$ (
${n}>4$) there are three series of regular polytopes: the regular simplex
$\alpha_{n}$, the generalized cross polytopes
$\beta^{m}_{n}$ and the generalized
$n$-cube
$\gamma^{m}_{n}$. The generalized
$n$-cube has
${m}^{n}$ vertices:
$$ (\theta^{{k}_{1}},\theta^{{k}_{2}},\dots, \theta^{{k}_{n}}),$$
where
${k}_{1}, {k}_{2},\dots, {k}_{n}$ take any integral values and
$\theta$ is a primitive
$m$th root of unity. For a certain divisor
$p$ of the number
$m$ the vertices of
$\gamma^{m}_{n}$ with
$$ \sum_{i=1}^{n}{{k}_{i}}\equiv 0\pmod{p}$$
(there are
$qm^{n-1}$ of them if
$m=pq$) determine a complex polytope
$\frac{1}{p}\gamma^{m}_{n}$. The symmetry group of
$\frac{1}{p}\gamma^{m}_{n}$ 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
$m, 2m,\dots, (n-1)m, qn$ (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
$\frac{1}{p}\gamma^{m}_{n}$ if
$p$ and
$n$ is mutually prime. Namely, polynomials
$$ {V}_{s}=\sum_{{k}_{i}}(\theta^{{k}_{1}}{x}_{1}+\theta^{{k}_{2}}{x}_{2}+\dots+\theta^{{k}_{n}}{x}_{n})^{ms}, \sum_{i=1}^{n}{{k}_{i}}\equiv 0\pmod{p}, s=\overline{1,n-1} $$
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.
UDC:
514.7
MSC: 51F15,
14L24