Dehn–Sommerville equations


In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their general form was established by Duncan Sommerville in 1927. The Dehn–Sommerville equations can be restated as a symmetry condition for the h-vector'' of the simplicial polytope and this has become the standard formulation in recent combinatorics literature. By duality, analogous equations hold for simple polytopes.

Statement

Let P be a d-dimensional simplicial polytope. For i = 0, 1,..., d−1, let fi denote the number of i-dimensional faces of P. The sequence
is called the f-vector of the polytope P. Additionally, set
Then for any k = −1, 0,..., d−2, the following Dehn–Sommerville equation holds:
When k = −1, it expresses the fact that Euler characteristic of a -dimensional simplicial sphere is equal to 1 + d−1.
Dehn–Sommerville equations with different k are not independent. There are several ways to choose a maximal independent subset consisting of equations. If d is even then the equations with k = 0, 2, 4,..., d−2 are independent. Another independent set consists of the equations with k = −1, 1, 3,..., d−3. If d is odd then the equations with k = −1, 1, 3,..., d−2 form one independent set and the equations with k = −1, 0, 2, 4,..., d−3 form another.

Equivalent formulations

Sommerville found a different way to state these equations:
where 0 ≤ k ≤. This can be further facilitated introducing the notion of h-vector of P. For k = 0, 1,..., d, let
The sequence
is called the h-vector of P. The f-vector and the h-vector uniquely determine each other through the relation
Then the Dehn–Sommerville equations can be restated simply as
The equations with 0 ≤ k ≤ are independent, and the others are manifestly equivalent to them.
Richard Stanley gave an interpretation of the components of the h-vector of a simplicial convex polytope P in terms of the projective toric variety X associated with P. Namely, they are the dimensions of the even intersection cohomology groups of X:
. In this language, the last form of the Dehn–Sommerville equations, the symmetry of the h-vector, is a manifestation of the Poincaré duality in the intersection cohomology of X.