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 f_i denote the number of i-dimensional faces of P. The sequence

${\displaystyle f(P)=(f_{0},f_{1},\ldots ,f_{d-1})}$

is called the f-vector of the polytope P. Additionally, set

${\displaystyle f_{-1}=1,f_{d}=1.}$

Then for any k = −1, 0, ..., d − 2, the following Dehn–Sommerville equation holds:

${\displaystyle \sum _{j=k}^{d-1}(-1)^{j}{\binom {j+1}{k+1}}f_{j}=(-1)^{d-1}f_{k}.}$

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 ${\textstyle \left[{\frac {d+1}{2}}\right]}$ 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:

${\displaystyle \sum _{i=-1}^{k-1}(-1)^{d+i}{\binom {d-i-1}{d-k}}f_{i}=\sum _{i=-1}^{d-k-1}(-1)^{i}{\binom {d-i-1}{k}}f_{i},}$

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

${\displaystyle h_{k}=\sum _{i=0}^{k}(-1)^{k-i}{\binom {d-i}{k-i}}f_{i-1}.}$

The sequence

${\displaystyle h(P)=(h_{0},h_{1},\ldots ,h_{d})}$

is called the h-vector of P. The f-vector and the h-vector uniquely determine each other through the relation

${\displaystyle \sum _{i=0}^{d}f_{i-1}(t-1)^{d-i}=\sum _{k=0}^{d}h_{k}t^{d-k}.}$

Then the Dehn–Sommerville equations can be restated simply as

${\displaystyle h_{k}=h_{d-k}\quad {\text{ for }}0\leq k\leq d.}$

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:

${\displaystyle h_{k}=\dim _{\mathbb {Q} }\operatorname {IH} ^{2k}(X,\mathbb {Q} )}$

(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