Abstract
The basis polynomial invariants with even degrees relatively to the symmetries group were described in cited literature. Here, the polynomial invariants with odd degrees are constructed. We give an explicit construction of all the basic polynomial invariants as algebra generators of odd degrees relatively to the symmetries group. All calculations are presented in detail.
Highlights
Using [1], we show the terms in Euclid n-dimensional space En and picture them considering the symmetries in a three-dimensional cube
We studied the properties of m the polynomial invariants’ algebra I Bn relatively to the generalized n-cube γnm symmetry group Bnm
We introduced the polynomial invariants of the symmetry group generated by reflections we pictured in the 3-dimensional real space
Summary
Using [1], we show the terms in Euclid n-dimensional space En and picture them considering the symmetries in a three-dimensional cube. The generalized n-cube γnm in n-dimensional unitary space U n was first described by Coxeter in [3]. We investigated the available literature searching for other methods giving results considering basic reflections’ algebra invariants. More general unitary reflection groups known as G (m, p, n) were described in [5,6], but no method giving results similar to ours was presented. There is no particular reason for us to mention them This was found to be the case by the authors of several other references that we previously mentioned: it was the lack of available literature that prompted us to submit this result together with pictorial representations of the basic concepts
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