Canonical systems of basic invariants for symmetry groups of Hessian polyhedrons

O. I. Rudnitsky

Taurida Journal of Computer Science Theory and Mathematics, no. 3 (2017), pp. 73-78

Résumé

Let $G$ be a finite unitary reflection group acting on the $n$-dimensional unitary space ${{U}^{n}}$. The algebra ${{I}^{G}}$ of $G$-invariant polynomials is generated by $n$ algebraically independent homogeneous polynomials ${{f}_{1}}({x}_{1},\dots,{x}_{n}),\dots,{{f}_{n}}({x}_{1},\dots,{x}_{n})$ of degrees ${m}_{1}\leqslant{m}_{2}\leqslant \dots\leqslant{m}_{n}$ (a system of basic invariants of group $G$) [1]. According to [4] (cf. [2]) a system $\{{{f}_{1}},\dots,{{f}_{n}}\}$ of basic invariants is said to be canonical if it satisfies the following system of partial differential equations: $$\bar{f}_{i}(\partial){f}_{j}=0$$ where a differential operator $\bar{f}_{i}(\partial)$ is obtained from polynomial ${f}_{i}$ if coefficients of polynomial to substitute by the complex conjugate and variables ${x}_{i}$ to substitute by $\frac{\partial}{\partial {x}_{i}}.$ In this paper, canonical systems of basic invariants were constructed in explicit form for symmetry groups of Hessian polyhedrons –- groups $W({{L}_{3}}),$ $W({{M}_{3}})$ generated by reflections in unitary space ${{U}^{3}}$.

Détail

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