On a basic invariants of the imprimitive groups $G(m,p,n), B_n^m, D_n^m$

O. I. Rudnitsky

Taurida Journal of Computer Science Theory and Mathematics, no. 4 (2023), pp. 81-91

Résumé

In $n$-dimensional unitary space $U^n$ we introduce an coordinate system with origin $O$ and the orthonormal basis vectors $\vec{e}_{i}$ ($i=\overline{1,n}$); vector $\vec{x}=\sum\limits_{i=1}^{n}{x}_{i}\vec{e}_{i}$. Let $G$ be a finite irreducible unitary group generated by reflections with respect to hyperplanes with the common point $O$. A polynomial $f={f}(\vec{x})=f({x}_{i})\in\mathbf{C}$[${x}_{1}$, \ldots,${x}_{n}$] is called a invariant ($G-$invariant) of the group $G$ if $$\sigma\cdot f=\sigma\cdot f(\vec{x})=f({\sigma}^{-1}\vec{x})=f, \ \forall \ \sigma \in G.$$ The set of all $G-$invariants forms an algebra, with is generated by $n$ algebraically independent polynomials of degrees $ {m}_{i},\ i=\overline{1,n} $, called a basic invariants of group $G$ (Shephard G.C., Todd J.A.). In this paper, we study the properties of basic invariants of the imprimitive group $G$ (the group of number 2 in the list of Shephard and Todd). These are the symmetry group $G(m,p,n)$ of the complex polytope $\frac{1}{p}{\gamma}_{n}^{m}$ and the symmetry group $G(m,1,n)=B_n^m$ of the generalized $n-$cube ${\gamma}_{n}^{m}$, as well as its subgroup $G(m,m,n)=D_n^m\subset B_n^m.$ In the paper provides an overview of known approaches to constructing in explicit form the basis invariants of these groups – of the methods of Shephard-Todd, of the Pogorelov polynomials, of the «vertex problem». Also in the paper we present a new method for constructing in explicit form the basis invariants of groups $G(m,p,n), B_n^m, D_n^m$. This method is based on the use of the differential operator for constructing in explicit form the basis invariants of the odd degrees.

Détail

Keywords: Unitary space, reflection,invariant, basic invariant, complex polytope.