Inequalities for cd -Indices of Joins and Products of Polytopes

Richard Ehrenborg, Harold Fox
2003 Combinatorica  
The cd-index is a polynomial which encodes the flag f -vector of a convex polytope. For polytopes U and V , we determine explicit recurrences for computing the cd-index of the free join U ∨ V and the cd-index of the Cartesian product U × V . As an application of these recurrences, we prove the inequality Ψ involving the cd-indices of three polytopes. 1991 Mathematics Subject Classification: Primary 06A07, Secondary 52B05. * Combinatorica 23 (2003), 427-452. than or equal to the cd-index of the
more » ... -dimensional simplex. In the case of polytopes this conjecture was proved by Billera and Ehrenborg [5] . A second reason to consider the cd-index is that it is a powerful invariant for computations. There are explicit expressions for how the cd-index changes after applying a geometric operation to the underlying polytope. The operations which have been studied so far are prism, pyramid, cutting off a face and Minkowski sum with a line segment in general direction [11, 13] . Also the cd-index of zonotopes, a special class of polytopes, is fairly well-understood [6, 7] . In this paper we continue this line of work. We first consider the problem of computing the cdindex of the free join of two polytopes knowing the cd-indices of each polytope. We then study the related question of the Cartesian product of polytopes. These problems were studied in [13], but without a satisfying answer in the following sense. The authors constructed two bilinear operators M and N such that where V ∨ W denotes the free join of the polytopes V and W , V × W denotes the Cartesian product, and Ψ denotes the cd-index. The expressions given for these two bilinear operators were cumbersome since rather than expressing all the computations in Z c, d they involved the auxiliary variables a and b. In this paper we develop recursions for the operators M and N that only involve the variables c and d. That is, with these recursions one can compute the cd-index of a free join of polytopes completely in terms of cd-polynomials. The main technique that we use is to work with the underlying coalgebra structure of posets and the cd-index. This method was introduced in [13] and has revealed a rich variety of results; for example, see [2, 5, 7] . In order to obtain more succinct expressions for the recursions, we were forced to include a new element of degree minus one. Moreover, since this extended algebra does not satisfy the associative law, care must be taken when working with it. In Section 8 we prove our main inequality which relates the cd-indices between the free join of polytopes and the Cartesian product. It is: for U , V and W three polytopes; see Theorem 8.2. The proof of this theorem relies on the coalgebra techniques and the recursions which we develop in the earlier sections of this paper. In the concluding remarks, we give a corollary of the inequality which provides evidence for Stanley's conjecture on Gorenstein * lattices. Posets and the cd-index For terminology on partially ordered sets (posets), we refer the reader to [22] . Unless otherwise stated, throughout this paper we assume that all posets are graded with unique minimal element0, unique
doi:10.1007/s00493-003-0026-z fatcat:e4iqzdg3pnblzb7ioigyhcjkji