On a basic invariants of the symmetry group of complex polyhedron $\frac{1}{p}{\gamma}_{n}^{m}$

O. I. Rudnitsky

Taurida Journal of Computer Science Theory and Mathematics, no. 3 (2021), pp. 72-78

Résumé

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.

Détail

Keywords: Unitary space, reflection, basic invariant, algebra of invariants, complex polyhedron.