The Linear Ordering Polytope via Representations

Let $P_n$ denote the $n$-th linear ordering polytope. We define projections from $P_n$ to the $n$-th permutahedron and to the $(n-1)$-st linear ordering polytope. Both projections are equivariant with respect to the natural $\text{Sn}$-action and they project to orthogonal subspaces. In particular the second projection defines an $S_n$-action in $P_{n-1}$.