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
- [math]\displaystyle{ f(P)=(f_0,f_1,\ldots,f_{d-1}) }[/math]
is called the f-vector of the polytope P. Additionally, set
- [math]\displaystyle{ f_{-1}=1, f_d=1. }[/math]
Then for any k = −1, 0, ..., d − 2, the following Dehn–Sommerville equation holds:
- [math]\displaystyle{ \sum_{j=k}^{d-1} (-1)^j \binom{j+1}{k+1} f_j = (-1)^{d-1}f_k. }[/math]
When k = −1, it expresses the fact that Euler characteristic of a (d − 1)-dimensional simplicial sphere is equal to 1 + (−1)d − 1.
Dehn–Sommerville equations with different k are not independent. There are several ways to choose a maximal independent subset consisting of [math]\displaystyle{ \left[\frac{d+1}{2}\right] }[/math] 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:
- [math]\displaystyle{ \sum_{i=-1}^{k-1}(-1)^{d+i}\binom{d-i-1}{d-k} f_i = \sum_{i=-1}^{d-k-1}(-1)^i \binom{d-i-1}{k} f_i, }[/math]
where 0 ≤ k ≤ 1⁄2(d−1). This can be further facilitated introducing the notion of h-vector of P. For k = 0, 1, ..., d, let
- [math]\displaystyle{ h_k = \sum_{i=0}^k (-1)^{k-i}\binom{d-i}{k-i}f_{i-1}. }[/math]
The sequence
- [math]\displaystyle{ h(P)=(h_0,h_1,\ldots,h_d) }[/math]
is called the h-vector of P. The f-vector and the h-vector uniquely determine each other through the relation
- [math]\displaystyle{ \sum_{i=0}^d f_{i-1}(t-1)^{d-i}=\sum_{k=0}^d h_k t^{d-k}. }[/math]
Then the Dehn–Sommerville equations can be restated simply as
- [math]\displaystyle{ h_k = h_{d-k} \quad\text{ for } 0\leq k\leq d. }[/math]
The equations with 0 ≤ k ≤ 1⁄2(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:
- [math]\displaystyle{ h_k=\dim_{\mathbb{Q}}\operatorname{IH}^{2k}(X,\mathbb{Q}) }[/math]
(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.
References
- Branko Grünbaum, Convex Polytopes. Second edition. Graduate Texts in Mathematics, Vol. 221, Springer, 2003 ISBN:0-387-00424-6
- Richard P. Stanley, Combinatorics and Commutative Algebra. Second edition. Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1996. ISBN:0-8176-3836-9
- D. M. Y. 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.
- Günter M. Ziegler, Lectures on Polytopes. Springer, 1998. ISBN:0-387-94365-X
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