result extensively used the coalgebra techniques developed in 13]. ... ag h-vector is a non-negative linear combination of coe cients of the cd-index. In turn ... join is smaller than the cd-index of the Cartesian product of two polytopes. ... For two elements x and y in a graded poset P, de ne the interval x; y] to be the set fz 2 P :.
Products of polytopes and inequalities Richard EHRENBORG and Harold FOX Abstract The cd-index is a polynomial which encodes the ag f -vector of a convex polytope. We determine explicit recurrences for computing the cd-index of the free join U _ V of two polytopes U and V and the cd-index of the Cartesian product U V . As an application of these recurrences, we prove the inequality (U _ (V W )) ((U _ V ) W ) involving the cd-indices of three polytopes.
1 Introduction The cd-index, (V ), is a combinatorial invariant that compactly encodes the ag f -vector of a convex polytope V . Much work done recently has been to understand the cd-index of a polytope. One important issue is how to calculate the cd-index of a given polytope. Such work was begun by Ehrenborg and Readdy [13] who determined what happens to the cd-index when applying a geometric operation to the underlying polytope. The operations they considered were prism, pyramid, cutting o a vertex and Minkowski sum with a line segment in general direction. They introduced coalgebra techniques in order to obtain their results. Moreover, these results were essential in proving that the ag f -vector of zonotopes linearly span the same space as ag f -vector of polytopes [6]. Later in [7], Billera, Ehrenborg and Readdy determined how to compute cd-index of zonotopes. This result extensively used the coalgebra techniques developed in [13]. Other related work of interest on computing cd-indices can be found in [11, 14]. The usefulness of knowing the cd-index of a polytope is that one can easily expand the coecients to obtain the ag h-vector and the ag f -vector of the polytope. More importantly, an entry in the
ag h-vector is a non-negative linear combination of coecients of the cd-index. In turn the ag f -vector is a non-negative linear combination of the ag h-vector. Hence, inequalities for the cd-index implies inequalities for the ag h-vector and the ag f -vector. In this paper we will consider the cd-index under two products of polytopes: the free join, denoted by _ , and the Cartesian product, denoted by . These products were considered by Kalai [16] in his proof that the ag f -vectors of polytopes span the generalized Dehn-Sommerville relations. Our work continues the work in [13]. The authors obtained explicit expressions for the cd-index of the pyramid of a polytope and for the prism of a polytope. Both are special cases of the two products we will consider here. More generally, they showed that there exist two bilinear operators M and N on non-commuting polynomials in the variables c and d such that 1
(V _ W ) = M ( (V ); (W )); (V W ) = N ( (V ); (W )); for two polytopes V and W . However they did not give explicit recursions for these two operators. Instead their expressions were long and only gave the ab-index of the products of the polytopes which is equivalent to the ag h-vector. In this paper we develop recursions for the products which only involve the cd-indices. The underlying combinatorial structure we will be working with are graded partially ordered sets (posets); see Section 2. Coalgebra techniques, outlined in Section 3, will be an essential tool in order to express our recursions. In Section 4 we introduce the extended coalgebra which, although is a non-associative algebra, allows us to write our recursions more succinctly. The free join of polytopes is presented in Section 5. For face lattices, the free join corresponds to the Cartesian product of posets. We review the mixing operator which was introduced in [13] to understand the ab-index of Cartesian products of posets. In the next section, we develop recursions for the mixing operator. By considering the Cartesian product P (Q A1 ), where Q A1 is the poset Q with a new coatom added, we obtain expressions for the mixing operator M (u; v a) in terms of u = (P ) and v = (Q). By the coassociativity of the coproduct, it is straightforward to obtain recursions for the mixing operator only involving cd-polynomials. The recursions for the Cartesian product of polytopes is obtained similarly in Sections 7 and 8. In Section 9 we prove our main result, namely the inequality (U _ (V W )) ((U _ V ) W ); for three polytopes U , V and W . Informally this inequality means that the cd-index of the free join is smaller than the cd-index of the Cartesian product of two polytopes. For instance, when the polytope V is a point then we have (U _ W ) (Pyr(V ) W ). The proof of the main inequality relies on the extended coalgebra developed in Section 4. Namely, the extended coproduct is essential in this proof to keep the notation compact. In the concluding remarks, we give a corollary of the inequality which gives evidence for Stanley's conjecture on Gorenstein lattices.
2 Graded posets and the cd-index For terminology on partially ordered sets (posets), we refer the reader to [21]. A poset P is ranked if there is a function : P ?! Zsuch that (y ) = (x) + 1 if the element y covers the element x. A poset is graded if it is ranked, has a minimal element ^0, a maximal element ^1 and the rank function satis es (^0) = 0. The rank of a graded poset is the value (^1). All posets we will consider here are graded unless we state otherwise. For two elements x and y in a graded poset P , de ne the interval [x; y ] to be the set fz 2 P : x z yg. Moreover, let the interval [x; y] inherit the same order relations as the poset P . Observe that the interval [x; y ] is also a graded poset of rank (x; y ) = (y ) ? (x). 2
The example of graded posets the reader should keep in mind is the face lattice of a convex polytope. For a polytope V we denote the face lattice by L(V ). The minimal element of the face lattice is the empty face and the maximal element is the entire polytope. The rank of a face F is given by (F ) = dim(F ) + 1. A chain c in a poset P is a linearly ordered subset of P . A chain is maximal (saturated) if one cannot add one more element without breaking the condition that the chain is linearly ordered. In a graded poset P we will consider chains containing the minimal element ^0 and the maximal element ^1. That is, we write c = f^0 = x0 < x1 < < xk = ^1g for a chain c. Let P be a graded poset of rank n + 1, and let S be a subset of the set f1; 2; : : :; ng. De ne fS (P ) = fS to be the number of chains ( ags) of poset P whose ranks are exactly given by the set S . The 2n values given by fS constitutes the ag f -vector of a poset. The ag h-vector is de ned by the identity X hS = (?1)jS?T j fT : This is equivalent to the relation
T S
fS =
X T S
hT :
Hence the ag f -vector and the ag h-vector contain the same information. Let a and b be two non-commutative variables. For a subset S of f1; : : :; ng de ne the ab-monomial uS = u1 un by ui = a if i 62 S and ui = b if i 2 S . The ab-index (P ) of a poset of rank n + 1 is the ab-polynomial X (P ) = hS uS ; S
where S ranges over all subsets of the set f1; : : :; ng. An alternative de nition of the ab-index of a poset P is as follows. For each chain c = f^0 = x0 < x1 < < xk = ^1g give it a weight w(c) = w1 wn, where ( if i 2 f(x1); : : :; (xk?1)g; wi = a ?b b otherwise. Then the ab-index of the poset P is given by X (P ) = w(c); c
where the sum is over all chains c in the poset P . This way to view the ab-index is useful in proving identities about the ab-index. The Mobius function of a poset P is de ned by the relations (x; x) = 1 and (x; y ) = P ? xz
