A new index for polytopes

Abstract

A new index for convex polytopes is introduced. It is a vector whose length is the dimension of the linear span of the flag vectors of polytopes. The existence of this index is equivalent to the generalized Dehn-Sommerville equations. It can be computed via a shelling of the polytope. The ranks of the middle perversity intersection homology of the associated toric variety are computed from the index. This gives a proof of a result of Kalai on the relationship between the Betti numbers of a polytope and those of its dual.

Manifold arrangements

“…See [12, Section 3] for a detailed discussion. Furthermore, Theorem 5.2 reduces to the usual notion of the cd-index [2,31].…”

“…where in the second step we used Lemma 6.3, λ(1) = 1 and λ(w (2) · a) = λ(w (2) ) · λ(a) = 0, and in the fourth step Lemma 6.3 was applied again.…”

Quasisymmetric functions and Kazhdan-Lusztig polynomials

“…Formally, the cd-index is the homogeneous noncommutative polynomial, ψ Q = ψ Q (c, d) = w [w] Q w in c and d, where the sum is over all cd-words of degree ρ(Q) − 1; see [4].…”

The Tchebyshev Transforms of the First and Second Kind

“…When P is Eulerian the ab-index of P can be written in terms of c = a + b and d = a · b + b · a, and the resulting noncommutative polynomial is called the cd-index [4]. Its importance lies in that it removes all the linear redundancies in the flag f -vector entries [3], it mirrors geometric operations on a polytope as operators on the corresponding cd-index [19,21], and it is amenable to algebraic techniques to derive inequalities on the flag vectors [6,15,17].…”