# Are cyclic orbitopes of permutahedra necessarily simplicies?

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?

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 $$3$$.
• Thank you! Given the gcd interpretation (thank you for altering me to that), It doesn't seem possible to find linear conditions so that the circulant matrix has rank $d-1$ provided $(v_1,\ldots, v_d)$ lies outside the resulting linear subspaces (that's really what I needed for my purposes) – Bob Jul 12 at 14:52