orbit functions to form analogs of Chebyshev polynomials in $n$ variables in their non-trigonometric form. It is shown that, in the case of 2 variables, our polynomials coincide with those of Koorwider [@Koornwinder1-4](III), although the approach and terminology could not be more different, ours being purely algebraic, having originated in Lie theory.

In Section 5, we present the orbit functions of $A_n$ disguised as polynomials built from multivariate orbit functions of the symmetric group. In Section 2, such a possibility is described in terms of related bases, one orthonormal (symmetric group), the other non-orthogonal (simple roots of $A_n$ and their dual $\omega$-basis). Both forms of the same polynomials appear rather different but may prove useful in different situations.

The last section contains a few comments and some questions related to the subject of this paper that we find intriguing.

Preliminaries
=============

This section is intended to fix notation and terminology. We also briefly recall some facts about ${\rm S}_{n+1}$ and $A_{n}$, dwelling particularly on various bases in $\R^{n+1}$ and $\R^n$. In Section \[ssec\_Weyl\_group\], we identify elementary reflections that generate the $A_n$ Weyl group $W$, with the permutation of two adjacent objects in an ordered set of $n+1$ objects. And, finally, 