we present some standard definitions and properties of orbit functions.

Permutation group ${\rm S}_{n+1}$
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The group ${\rm S}_{n+1}$ of order $(n+1)!$ transforms the ordered number set $[l_1,l_2,\dots,l_n,l_{n+1}]$ by permuting the numbers.

We introduce an orthonormal basis in the real Euclidean space $\R^{n+1}$, $${e_i}\in\R^{n+1}\,,\qquad
\l e_i , e_j\r=\delta_{ij}\,,\qquad    1\leq i,j\leq n+1\,,$$ and use the $l_k$’s as the coordinates of a point $\mu$ in the $e$-basis: $$\mu=\sum_{k=1}^{n+1}l_ke_k\,,\qquad l_k\in\R\,.$$

The group ${\rm S}_{n+1}$ permutes the coordinates $l_k$ of $\mu$, thus generating other points from it. The set of all distinct points, obtained by application of ${\rm S}_{n+1}$ to $\mu$, is called the orbit of ${\rm S}_{n+1}$. We denote an orbit by $W_\lambda$, where $\lambda$ is a unique point of the orbit, such that $$l_1\geq l_2\geq\cdots\geq l_n\geq l_{n+1}\,.$$ If there is no pair of equal $l_k$’s in $\lambda$, the orbit $W_\lambda$ consists of $(n+1)!$ points.

Further on, we will only consider points $\mu$ from the $n$-dimensional subspace ${\mathcal H}\subset\R^{n+1}$ defined by the equation $$\begin{gathered}
\label{plane H}
\sum_{k=1}^{n+1}l_k=0.\end{gathered}$$

Lie algebra $A_n$
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Let us recall basic properties of the simple Lie algebra $A_n$ of the compact Lie group ${\rm SU}(n+1)$. Consider the general value $(1\leq n<\infty)$ of the rank. The Coxeter-Dynkin diagram, Cartan matrix 