Canonical systems of basic invariants for unitary groups $W({{J}_{3}}(m)),$ $m=4,5$

O. I. Rudnitsky

Taurida Journal of Computer Science Theory and Mathematics, no. 1 (2018), pp. 89-96

Résumé

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 ${{U}^{3}}$.

Détail

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