Canonical system of basic invariants for unitary group $W(K_5)$

O. I. Rudnitskii

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 58 (2019), pp. 32-40

Résumé

For a finite group $G$ generated by reflections in the $n$-dimensional unitary space $U^n$, the algebra $I^G$ of all $G$-invariant polynomials $f(x_1,\dots,x_n)$ is generated by $n$ algebraically independent homogeneous polynomials $f_i\in I^G$ with $\mathrm{deg}\,f_i=m_i$ ($i=\overline{1,n}$); $m_1\leqslant m_2\leqslant\dots\leqslant m_n$ (Shephard G. C., Todd J. A.). According to Nakashima N., Terao H., and Tsujie S., system $\{f_1,\dots, f_n\}$ of basic invariants of the group $G$ is said to be canonical if it satisfies the following system of partial differential equations: $$ \overline{f}_i(\partial) f_j=0,\quad i,j=\overline{1,n}\ (i), $$ where the differential operator $\overline{f}_i(\partial)$ is obtained from polynomial $f_i$ if each its coefficient is replaced by the complex conjugate and each variable $x_k$ is replaced by $\frac\partial{\partial x_k}$. In the previous works, the author obtained in an explicit form canonical systems of basic invariants for all finite primitive unitary groups $G$ generated by reflections in unitary spaces of dimensional $2$, $3$, and $4$. In this paper, canonical systems of basic invariants were constructed in an explicit form for unitary groups $W(K_5)$ generated by reflections in space $U^5$.

Détail

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