The Linear Ordering Polytope via Representations

TL;DR

This paper explores the structure of the linear ordering polytope by defining equivariant projections to permutahedra and lower-dimensional polytopes, revealing symmetries and orthogonal decompositions.

Contribution

It introduces projections from the linear ordering polytope to permutahedra and lower-order polytopes that preserve symmetry and orthogonality, advancing understanding of their geometric and algebraic structure.

Findings

Projections are equivariant under the symmetric group action.

Projections map to orthogonal subspaces.

Defines an $S_n$-action on $P_{n-1}$.

Abstract

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 $\Sn$-action and they project to orthogonal subspaces. In particular the second projection defines an $S_n$-action in $P_{n-1}$.