About one property of the basic invariants of the unitary group $EW({N}_{4})$

O. I. Rudnitsky

V. I. Vernadsky Crimean Federal University, Simferopol

Abstract: In this paper we are considering the finite unitary primitive group $EW(N_4)$ of order $64\cdot6!$ generated by reflections of second order with respect to the $3$-planes in the $4$-dimensional unitary space (the group of number $31$ in the list of Shephard-Todd). This group is neat extension by A. M. Cohen of group $W({N}_{4})$(the group of number $29$ in the list of Shephard-Todd). The group $EW(N_4)$ acts on the polynomial ring in $4$ variables over the field of complex numbers in a natural manner and, as is well known (Shephard and Todd), there are four algebraically independent homogeneous polynomials $J_{m_i}$ of degrees $m_i=8, 12, 20, 24$, called basic invariants, such that the algebra of all $EW({N}_{4})$-invariant polynomials is generated by these polynomials. Note that there are infinitely many possible choices of a basic invariants of the given group, but their degrees are well known and typical. In the previous works, author, using the Pogorelov polynomials, obtained in explicit form all basic invariants $J_{m_i}$ of the group $EW(N_4)$. The main purpose of the article is to consider another method of finding in explicit form of the basic invariants of group $EW({N}_{4})$. This method is based on the following property of group $EW({N}_{4})$. Let $G(4, 4, 4)$ be the finite imprimitive unitary group generated by reflections. Since $G(4, 4, 4) \subset EW(N_4)$, each of the basic invariants of group $EW({N}_{4})$ is written as a polynomial ${\phi}_{t}({I}_{k})$ of the polynomials $I_{k}=\sum\limits_{i=1}^{4}{x_i}^{4k}, (k=1,2,3)$ and $I_{4}=x_1x_2x_3x_4$ - the basic invariants of the group $G(4, 4, 4)$. In the present paper, the explicit form of polynomials ${\phi}_{t}({I}_{k}) (t=2, 3, 5, 6)$ is found, namely each of the basic invariants of group $EW({N}_{4})$ is written as a polynomial of the power sum symmetric polynomials $I_{k} \ (k=1,2,3)$ and the elementary symmetric polynomial $I_{4}=x_1x_2x_3x_4$. Note that previously a similar problem was solved in the paper by M. Oura and J. Sekiguchi for the group $W(K_6)$ and in the author’s papers for the groups $W(K_5), W(N_4)$.

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

UDC: 514.7

MSC: 51F15, 14L24