About one property of basic invariants of unitary group $W(K_5)$

O. I. Rudnitsky

V. I. Vernadsky Crimean Federal University, Simferopol

Abstract: In this paper, we are considering the finite unitary primitive group $W(K_5)$ of order $72\cdot6!$ generated by reflections of second order with respect to the hyperplanes in 5-dimensional unitary space (the group of number 33 in the list of Shephard and Todd). As is well known, the algebra of all $W(K_5)$-invariant polynomials is generated by 5 algebraically independent homogeneous polynomials $J_{m_i}$ of degrees $m_i=4, 6, 10, 12, 18.$ In the previous works, author obtained in explicit form basic invariants $J_{m_i}$. The main purpose of the article is to consider another method of finding in explicit form the basic invariants of group $W(K_5)$. This method is based on the following property of group $W(K_5)$. Let $G(3, 3, 4)$ be the imprimitive unitary reflection group generated by reflections. Since $G(3, 3, 4)$ is a subgroup of $W(K_5)$, each of polynomials $J_{m_i}$ is written as a polynomial ${\phi}_{t}({I}_{k})$ of the polynomials $I_{k}=\sum\limits_{i=1}^{4}{x_i}^{3k}, (k=1,2,3)$, $I_{4}=x_1x_2x_3x_4$ and $I_{5}=x_5$ – the basic invariants of $G(3, 3, 4)$. In the present paper, the explicit form of polynomials ${\phi}_{t}({I}_{k}), t=4, 6, 10, 12, 18,$ is found and the basis invariants of group $W(K_5)$ were constructed in explicit form.

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

UDC: 514.7

MSC: 51F15, 14L24