# Canonical systems of basic invariants for symmetry groups of Hessian polyhedrons.

Let $G$ be a ﬁnite unitary reﬂection group acting on the $n$-dimensional unitary

space $U^n$. The algebra $I^G$ of $G$-invariant polynomials is generated by n algebraically independent

homogeneous polynomials $f_1(x_1, . . . , x_n), . . . , f_n(x_1, . . . , x_n)$ of degrees $m_1 \leqslant m_2 \leqslant · · · \leqslant m_n$ (a

system of basic invariants of group $G$) [1]. According to [4] (cf. [2]) a system ${f_1, . . . , f_n}$ of

basic invariants is said to be canonical if it satisﬁes the following system of partial diﬀerential

equations:

$$ \bar{f_i}(\partial)f_j = 0$$

where a diﬀerential operator $\bar {f_i}(\partial)$ is obtained from polynomial $f_i$ if coeﬃcients of polynomial to

substitute by the complex conjugate and variables $x_i$ to substitute by $\frac{\partial}{\partial x_i}$ .

In this paper, canonical systems of basic invariants were constructed in explicit form for

symmetry groups of Hessian polyhedrons — groups $W(L_3), W(M_3)$ generated by reﬂections in

unitary space $U^3$.

**Keywords:** unitary space, reﬂection, reﬂection groups, algebra of invariants, basic invariant,

canonical system of basic invariants.