In this paper we explore the combinatorial automorphism group of the linear ordering polytope P n LO for each n ? 1. We establish that this group is isomorphic to Z 2 \Theta Sym(n + 1) if n ? 2 (and to Z 2 if n = 2). Doing so, we provide a simple and unified interpretation of all the automorphisms. Key words: Linear ordering polytope, automorphism group, facets 1 Introduction The linear ordering polytope is a familiar object from polyhedral combinatorics. It is defined as the convex hull of the 0/1--vectors encoding linear orders (or total orders) on a given base set. Exploiting results on the facial structure of this family of polytopes and using advanced techniques in linear programming, efficient algorithms could be designed to solve real-world instances of some hard combinatorial optimization problems. For example, the triangulation problem for input-output tables can be formulated as a linear program on the linear ordering polytope. This problem asks, given a matrix of n \The...

. The weak order polytopes are studied in Gurgel and Wakabayashi (1997), Gurgel and Wakabayashi (1996), and Fiorini and Fishburn (1999). We make use of their natural, affine projection onto the partition polytopes to determine several new families of facets for them. It turns out that not all facets of partition polytopes are lifted into facets of weak order polytopes. We settle the cases of all facet-defining inequalities established for partition polytopes by Grotschel and Wakabayashi (1990). Our method, although rather simple, allows us to establish general families of facets which contain two particular cases previously requiring long proofs. Key words. weak order polytope, partition polytope, facet of convex polytope AMS subject classifications. 52B12, 06A07, 90C57 1. Introduction. In order to solve real-life problems which require finding an optimal linear ordering, Grotschel, Junger and Reinelt [4] introduce the `linear ordering polytope'. After determining several facets, th...