Are cyclic orbitopes of permutahedra necessarily simplicies?

Question

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?

Answer 1

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 $3$.