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Two Poset Polytopes Discrete Comput Geom 1:9-23 (1986) G eometrv)i.~.reh, ~ ( :*mllmlati~ml © l~fi $1~ter-Vtrlq New Yorklu¢. t¢ Two Poset Polytopes Richard P. Stanley* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 Abstract. Two convex polytopes, called the order polytope d)(P) and chain polytope <~(P), are associated with a finite poset P. There is a close interplay between the combinatorial structure of P and the geometric structure of E~(P). For instance, the order polynomial fl(P, m) of P and Ehrhart poly- nomial i(~9(P),m) of O(P) are related by f~(P,m+l)=i(d)(P),m). A "transfer map" then allows us to transfer properties of O(P) to W(P). In particular, we transfer known inequalities involving linear extensions of P to some new inequalities. I. The Order Polytope Our aim is to investigate two convex polytopes associated with a finite partially ordered set (poset) P. The first of these, which we call the "order polytope" and denote by O(P), has been the subject of considerable scrutiny, both explicit and implicit, Much of what we say about the order polytope will be essentially a review of well-known results, albeit ones scattered throughout the literature, sometimes in a rather obscure form. The second polytope, called the "chain polytope" and denoted if(P), seems never to have been previously considered per se. It is a special case of the vertex-packing polytope of a graph (see Section 2) but has many special properties not in general valid or meaningful for graphs. There is a surprising connection (Section 3) between (P(P) and (~(P) which will allow us to "transfer" properties of O(P) over to r((p). Given the poset P = { x 1..... x, } (where by standard abuse of notation we identify p with its set of points), the set R e of all functions f: P---, R is an n-dimensional real vector space with a scalar product defined by (f, g)= ~-.~ E j(x)g(x), which makes R e a Euclidean space. In particular, we can talk about convex subsets of R p and their volumes, orthogonal projections, etc. *Partially supported by NSF Grant No. 8104855-MCS and by a GuggenheimFellowship. 10 R.P. Stanley Definition 1.1. The order polytope d)(P) of the poset P is the subset of R p defined by the conditions 0 < f(x) < 1, for all x e P, (1) f(x) < f(y) ifx < y in P. (2) Note that O(P) is a convex polytope since it is defined by linear inequalities and is bounded because of (1). Clearly, because of (2), we can replace (1) by the conditions 0 < f(x), if x is a minimal element of P, f(x) < 1, if x is a maximal element of P. (1') By the transitivity of P, we can replace (2) by the equivalent conditions f(x) < f(y) if y covers x in P. (2') Let o: P ~ {1 ..... n } be a linear extension (order-preserving bijection) of P. We identify a with the permutation Yl ..... Yn of the elements x 1..... x n of P defined by o(y,) = i. All functions f ~ R e satisfying 0 < f(Yl) < "'" < f(Yn) < 1 belong to 0(P). These functions form an n-dimensional simplex, so we conclude dim tV(P)= n. It is easily seen that conditions (1') and (2') are independent, so they define the facets [(n -1)-dimensional faces] of 0(P). More precisely a facet of O(P) consists of those f ~d~(P) satisfying exactly one of the following conditions: f(x) = 0, for some minimal x ~ P, (3a) f(x) = 1, for some maximal x ~ P, (3b) f(x) = f(y), for some y covering x in P. (3c) It is convenient to state the above conditions in a more uniform way. Let be the poset obtained from P by adjoining a minimum element 0 and a maximum element i. Define a polytope ~(P) to be the set of functions g ~ R ~' satisfying g(0) = 0, g(i) = 1, g(x) <_ g(y) if x _< y in ,b. The linear map P: ~(P) -* O(P) obtained by restriction to P is clearly a bijection and hence (since P is linear) defines a combinatorial equivalence of polytopes. Thus by (3) a facet of O(P) consists of those g ~ ~(P) satisfying g(x) = g(y) for some fixed pair (x, y)~ for which y covers x in P. In particular, the number of facets of O(P) or O(P) is the number c(t') of cover relations in P, or equivalently c(P)+ a+ b, where P has a minimal elements and b maximal elements. We now wish to determine the entire facial structure of t~(P), or equivalently of 0(P). Since every face is an intersection of facets, it follows that a face F,~ of ~(P) corresponds to certain partitions ~r= {B 1..... Bk} of P into nonempty pairwise disjoint blocks, viz., F,~ = ( g ~ ~ (P): g is constant on the blocks B i of Ir }. (4) Two Poset Polytopes 11 b O Fig. 1 It remains to determine for which rr F,~ is a face, and which are the distinct faces F., Call ira face partition if F,, is a face of P. It is clear that if ~r is a face partition, then ~r is connected, i.e., every block B of ~r is connected as an (induced) subposet of P. Call a partition ~r = { B 1..... B k } closed if for any i 4= j there is g ~ F~ such that g(Bi) ~ g(Bj). Every partition ~r has a unique coarsen- ing ~ for which ~ is closed and F,=F~. Moreover, if ~r~[B 1..... Bk} is a closed face partition then dim F~ = k - 2 [since if (~ ~ B, and 1 ~ By then g ~ F~ satisfies g(Bi)= 0 and g(Bj)= 1]. Hence it remains to describe the closed face partitions. This description was apparently first explicitly observed by Geissinger [6]. We will state Geissinger's result below (Theorem 1.2) but will omit the rather straightforward proof. Define a binary relation < ~ on ~r by setting B, < Bj if x _< y for some x ~ B i and y ~ By. Call ~r compatible if the transitive closure of < ~ is a partial order (i.e., is antisymmetric). If ~r is compatible then every block B of sv is convex; i.e., if x, z ~ B and x < y < z.. then y ~ B. The converse is false; e.g., let I' be given by Fig. 1. The partition into blocks 0, ad, bc, 1 is connected and convex, but not compatible. Theorem 1.2. A partition of P is a closed face partition if and on~ if it is connected and compatible. (In particular, the partition ~r into a single block P yields the empty set F, = 0, which we regard as a face.) Thus the lattice of faces 0(P) [or (~(P)] is isomorphic to the lattice of connected compatible partitions of P, ordered by reverse refinement. For in- stance, if P = { a, b } is a two-element antichain, then d~(P) is a square and Fig. 2 depicts its face lattice (with (~ and ] written 0 and 1). Define a filter (or dual order ideal, up-set, or increasing subset) of P to be a subset I of P such that if x ~ I and y > x, then y ~ I. Let XI: P ~ R denote the ~O-o-b-I Oo-b-I~ O-o-bl Ob-a-! Oob-i ~ Ob-ol Fig. 2 12 R.P. Stanley characteristic function of I; i.e., 1, x~l x~(x) = O, x~ I. The following corollary is immediate from Theorem 1.2 and can also be easily proved directly. Corollary 1.3. The vertices of tP( P) are the characteristic functions XI of filters I of P. In particular, the number of vertices of t~(P) is the number of filters of P. 2. The Chain Polytope Let us define a second polytope associated with a poset P = { x 1..... x, }. Definition 2.1. The chain polytope ~(P) of the poset P is the subset of R e defined by the conditions 0 < g(x), for all x ~ P, (5) g(Yl)+'''+g(Yk) --<1, for every chain yl < "'" < Yk °fP" (6) Again it is clear that ~(P) is a convex polytope. Since f~(P) contains the n-dimensional simplex {g~RP: g(x) >0 for all x ~ P and g(xl)+ ... + g(x,) < 1}, we have dim~(P) = n. In view of (5) we can replace (6) by g(Yl) +''" + g(Yk) < 1, for every maximal chain Yl < " " " < Yk of P. (6') Conditions (5) and (6') are easily seen to be independent and thus define the facets of ~(P). In particular, the number of facets of ~(P) is equal to n + m(P), where re(P) is the number of maximal chains of P. A description of the faces of ~(P) analogous to Theorem 1.2 seems messy and will not be pursued here. However, we do have a simple description of the vertices analogous to Corollary 1.3. Define an antichain of P to be a subset A of pairwise incomparable elements of P. Theorem 2.2. The vertices of ~¢(P) are the characteristic functions XA of anti- chains of P. In particular, the number of vertices of T ( P ) is equal to the number of antichains of P. Proof. Clearly each XA ~ ~¢(P)" Since 0 < g(x) < 1 for all g ~ ~(P) and x ~ P, it follows that XA is a vertex of if(P).
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