On basis invariants of unitary group W(J_3(4)).

In this paper, some properties of basis invariants of the unitary group $W (J_3(4))$ of order 336 generated by reflections in 3-dimensional unitary space are studied. There is developed a new method of finding in explicit form the basic invariants of group $W(J_3(4))$. This method is based on the following property of group $W(J_3(4))$ – group $W(J_3(4))$ contains group $B_3$ of symmetries of the cube, and Pogorelov polynomials of the form $$
J_{m_i}(G) = \sum\limits_{\sigma \in G}(\overrightarrow{x}, \sigma\ \overrightarrow{s})^{m_i},
$$ where $G$ is a reflection group, $\sigma$ is reflection with respect to planes of symmetry, $\overrightarrow{s}$ is the unit normal vector (with origin $O$) of one of them, vector $\overrightarrow{x}$ is given by $\overrightarrow{x} = (x_i)$, $m_i$ are degrees of the basic invariants of group $G$. In the present paper, using that method, the basis invariants of group $W(J_3(4))$ in explicit form were constructed.

Keywords: Unitary space, reflection, reflection group, invariant, algebra of invariants.