Canonical systems of basic invariants for unitary groups $W({{J}_{3}}(m)),$ $m=4,5$
O. I. Rudnitsky Crimea Federal University, Simferopol
Abstract:
Let
$G$ be a finite unitary reflection group acting on the
$n$-dimensional unitary space
${{U}^{n}}$. Then
$G$ acts on the polynomial ring
$R=\mathbf {C}$[
${x}_{1}$,…,
${x}_{n}$] in a natural manner and there exists
$n$-tuple ${m}_{1}\leqslant{m}_{2}\leqslant \dots\leqslant{m}_{n}$ 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
$G$).
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:
$$\bar{f}_{i}(\partial){f}_{j}=0, \ i,j=\overline{1,n} \ (i < j),$$
where a differential operator
$\bar{f}_{i}(\partial)$ 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
$\frac{{\partial}^{p}}{\partial {{x}_{i}}^{p}}$.
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.
UDC:
514.7
MSC: 51F15,
14L24