Canonical systems of basic invariants for unitary groups W(J3(m)), m = 4, 5

Let $G$ be a finite unitary reflection group acting on the $n$-dimensional unitary space $U^n.$ Then $G$ acts on the polynomial ring $R = C[x_1,\dots,x_n]$ in a natural manner and there exists $n$-tuple $m_1 \ge m_2 \ge \dots \ge m_n$ of positive integers, such that the algebra $I^G$ of all $G$-invariant polynomials is generated by $n$ algebraically independent homogeneous polynomials $f_1(x_1,\dots, x_n),\dots, f_n(x_1,\dots,x_n) \in I^G$ with deg $f_i = m_i$ (a system of basic invariants of group $G$) [1].
A system ${f_1,\dots, f_n}$ of basic invariants of group $G$ is said to be canonical if it satisfies the following system of partial differential equations:
$$
\overline{f}_i(\partial)f_j=0,\quad i,j = \overline{1, n} \quad (i < j),
$$
where a differential operator $\overline{f}_i(\partial)$ is obtained from polynomial $f_i$ if each coefficient of polynomial to replace by the complex conjugate and each variable $x_i^p$ to replace by $\frac{\partial^p}{\partial x_i^p}$ [2, 3].
In this paper, canonical systems of basic invariants were constructed in explicit form for unitary groups $W(J_3(m)), m = 4, 5,$ generated by reflections in space $U^3.$

Keywords: Unitary space, reflection, reflection groups, algebra of invariants, basic invariant, canonical system of basic invariants.