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.
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:
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.
Sommerville found a different way to state these equations:
where 0 ≤ k ≤ ½(d−1). This can be further facilitated introducing the notion of h-vector of P. For k = 0, 1, …, d, let
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 ≤ ½(d−1) 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 (the dual of) P. Namely, they are the dimensions of the even intersection cohomology groups of X:
(the odd intersection cohomology groups of X are all zero). 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.
- Branko Grünbaum, Convex polytopes. Second edition. Graduate Texts in Mathematics, 221, Springer, 2003 ISBN 0-387-00424-6
- Richard Stanley, Combinatorics and commutative algebra. Second edition. Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1996. x+164 pp. ISBN 0-8176-3836-9
- Duncan Sommerville (1927) The relations connecting the angle sums and volume of a polytope in space of n dimensions Proceedings of the Royal Society Series A 115:103–19, weblink from JSTOR.
- G. Ziegler, Lectures on Polytopes, Springer, 1998. ISBN 0-387-94365-X