Dehn–Sommerville equations

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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

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

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

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

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

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

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

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

where 0 ≤ k ≤ ½(d−1). This can be further facilitated introducing the notion of h-vector of P. For k = 0, 1, …, d, let

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

The sequence

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

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

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

Then the Dehn–Sommerville equations can be restated simply as

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

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:

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

(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

• 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