$\langle\cdot,\cdot\rangle$ denote the inner product. We then have the reflection by $r_i$: $$\begin{aligned}
r_i x&= x-\tfrac{2}{\langle \alpha_i,\alpha_i\rangle}\langle x,\alpha_i\rangle\alpha_i
=(x_1,x_2,\dots,x_{n+1})_e-(x_i-x_{i+1})(e_i-e_{i+1})\\
&=(x_1,\dots,x_{i-1},x_{i+1},x_i,x_{i+2},\dots,x_{n+1})_e.
\end{aligned}$$ Such transpositions generate the full permutation group ${\rm S}_{n+1}$. Thus, $W(A_n)$ is isomorphic to ${\rm S}_{n+1}$, and the points of the orbit $W_\lambda({\rm S}_{n+1})$ and $W_\lambda(A_n)$ coincide.

Definitions of orbit functions {#ssec_def_orb_func}
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The notion of an orbit function in $n$ variables depends essentially on the underlying semisimple Lie group $G$ of rank $n$. In our case, $G={\rm SU}(n+1)$ (equivalently, Lie algebra $A_n$). Let the basis of the simple roots be denoted by $\alpha$, and the basis of fundamental weights by $\omega$.

The *weight lattice* $P$ is formed by all integer linear combinations of the $\omega$-basis, $$\begin{gathered}
P=\Z\omega_1+\Z\omega_2+\cdots+\Z\omega_n.\end{gathered}$$ In the weight lattice $P$, we define the *cone of dominant weights* $P^+$ and its subset of strictly dominant weights $P^{++}$ $$\begin{gathered}
P\;\supset\; P^+=\Z^{\ge 0}\omega_1+\cdots+\Z^{\ge 0}\omega_n
\;\supset\; P^{++}=\Z^{>0}\omega_1+\cdots+\Z^{>0}\omega_n.\end{gathered}$$

Hereafter, $W^e\subset W$ denotes the *even subgroup* of the Weyl group formed by an even number of reflections that generate $W$. $W_{\lambda}$ and $W^e_{\lambda}$ are the corresponding group orbits of a point $\lambda\in \R^n$.

We also introduce the notion of fundamental region $F(G)\subset \R^n$. For $A_n$ the *fundamental region* $F$ is the convex 