On the basic invariants of the unitary group $W({N}_{4})$

O. I. Rudnitsky

V. I. Vernadsky Crimean Federal University, Simferopol

Abstract: In this paper some properties of basis invariants of the unitary group $W({N}_{4})$ of order 7680 generated by reflections of second order with respect to the 3-planes in 4-dimensional unitary space are studied (the group of number 29 in the list of Shephard-Todd). As is well known, the set of all $W({N}_{4})$-invariant polynomials forms an algera generated by 4 algebraically independent homogeneous polynomials $J_{m_i}$ of degrees $m_i=4, 8, 12, 20$ (a system of basic invariants of group $W({N}_{4})$). In the previous works author, using the Pogorelov polynomials, obtained in explicit form all basic invariants $J_{m_i}$. The main purpose of the article is to consider another method of finding in explicit form of the basic invariants of group $W({N}_{4})$. This method is based on the following property of group $W({N}_{4})$. A. M. Cohen proved what group $W({N}_{4})\supset B_4$, where $B_4$ is the group of symmetries of the $\text{4-cube}$. Сonsequently each of polynomials $J_{m_i}$ is written as a polynomial ${\phi}_{t}({I}_{k})$ of the polynomials $I_{k}=\sum\limits_{i=1}^{4}{x_i}^{2k} (k=\overline{1,4})$ — the basic invariants of group $B_4$. In the present paper, the explicit form of polynomials ${\phi}_{t}({I}_{k}) (t=2, 4, 6, 10)$ is found, namely each of the basic invariants of group $W({N}_{4})$ is written as a polynomial of the power sum symmetric polynomials $I_{k}$.

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

UDC: 514.7

MSC: 51F15, 14L24