Suppose that $v=(v\_1,\ldots, v\_d)\in \mathbb{R}^d$ lies in the linear subspace $v\_1+\cdots +v\_d=0$, and moreover that the coordinates are pairwise distinct. The permutahedron \begin{equation} P(\mathcal{S}\_d;v)=Conv(\mathcal{S}\_d\cdot v) \end{equation} is the convex hull of $v$ under the symmetric group action on coordinates. It is a $(d-1)$-dimensional polytope.


Now consider the cyclic subgroup $C\_d$ of $\mathcal{S}\_d$ generated by the permutation $(123\cdots d)$ and consider the corresponding orbitope $P(C\_d; v)=Conv(C\_d\cdot v)$.


**Question:** Is $P(C\_d;v)$ necessarily a $(d-1)$-dimensional simplex?



Are cyclic orbitopes of permutahedra necessarily simplicies?
Let $M$ be the circulant matrix whose rows are given by cyclic shifts of $(v\_1,\dots v\_d)$ and let $P(x)=v\_1+v\_2x+\cdots+v\_dx^{d-1}$ be the associated polynomial. Moreover, let $s$ be the degree of $\gcd(P(x),x^{d}-1)$.Then the rank of $M$ is equal to $d-s$, so it is possible to come up with examples of vectors $v$ such that the $C\_d$ orbit is not $(d-1)$-dimensional by making $s$ large.


For example the matrix
$$\begin{bmatrix}
 2 & 1 & -2 & -1 \\ 
1 & -2 & -1 & 2 \\
 -2 & -1 & 2 & 1 \\
 -1 & 2 & 1 & -2 
\end{bmatrix}$$
has rank 2, so the associated orbit polytope has dimension $2$ rather than $3$.


