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MATHEMATICS
Canonical system of basic invariants for unitary group $W(K_5)$
O. I. Rudnitskii Vernadsky Crimean
Federal University, Simferopol, Russian Federation
Abstract:
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:
$$
\overline{f}_i(\partial) f_j=0,\quad i,j=\overline{1,n}\ (i<j),
$$
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 $U^5$.
Keywords:
Unitary space, reflection, reflection groups, algebra of invariants, basic invariant, canonical system of basic invariants.
Received: 04.12.2018
Citation:
O. I. Rudnitskii, “Canonical system of basic invariants for unitary group $W(K_5)$”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2019, no. 58, 32–40
Linking options:
https://www.mathnet.ru/eng/vtgu697 https://www.mathnet.ru/eng/vtgu/y2019/i58/p32
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Abstract page: | 98 | Full-text PDF : | 35 | References: | 25 |
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