identify facet-defining hyperplanes which will allow us to classify the reflexive $\lambda$-permutohedra.

Let $\lambda$ be a partition with at most $m$ parts and let $\x=(x_1,\ldots, x_m)$. Recall that the Newton polytope ${\mathrm{Newt}}(s_\lambda(\x))$ is the $\lambda$-permutohedron $\mathcal{P}^m_\lambda$, which is the convex hull of the $S_m$-orbit of $(\lambda_1,\ldots, \lambda_m)$ in $\mathbb{R}^m$. This polytope is of dimension $m-1$, and is determined by Rado’s inequalities [@Rado]: $$\sum_{i\in I} x_i \leq  \sum_{i=1}^{|I|} \lambda_i \hbox{ for all }  I\subseteq [m], \qquad\hbox{ and }\qquad \sum_{i=1}^m x_i = |\lambda|.$$

Note that whether one of Rado’s inequalities is facet-defining depends only on $|I|$ and $\lambda$, and not on the set $I$ itself.

\[prop:facets\] The facets of $\mathcal{P}_\lambda^m$ are determined by the following inequalities.

1.  For all $i=1,\ldots, m$, $x_i\leq \lambda_1$, unless $\lambda_2 = \lambda_3 = \cdots =\lambda_m$.

2.  For $2\le |I|\le m-2$, $$\sum_{i\in I} x_i \leq \sum_{i=1}^{|I|}\lambda_i,$$ unless $\lambda_1 = \lambda_2 = \cdots =\lambda_{|I|}$ or $\lambda_{|I|+1} = \lambda_{|I|+2} = \cdots =\lambda_{m}$.

3.  For $|I|=m-1$, $$\sum_{i\in I} x_i \leq \sum_{i=1}^{m-1}\lambda_i,$$ unless $\lambda_1 = \lambda_2 = \cdots =\lambda_{m-1}$.

The set of permutations $(x_1,\ldots, x_m)$ of $(\lambda_1, \ldots, \lambda_m)$ that satisfy $\sum_{i\in I} x_i \leq \sum_{i=1}^{|I|} \lambda_i$ with equality is the set of products of the permutations of $(\lambda_1, \lambda_2,  \ldots, \lambda_{|I|})$ 