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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.

Contents

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 , … , 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:

∑ j = k d − 1 ( − 1 ) j ( 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 [ d + 1 2 ] 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:

∑ i = − 1 k − 1 ( − 1 ) d + i ( d − i − 1 d − k ) f i = ∑ i = − 1 d − k − 1 ( − 1 ) i ( 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 = ∑ i = 0 k ( − 1 ) k − i ( d − i k − i ) f i − 1 .

The sequence

h ( P ) = ( h 0 , h 1 , … , h d )

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

∑ i = 0 d f i − 1 ( t − 1 ) d − i = ∑ k = 0 d h k t d − k .

Then the Dehn–Sommerville equations can be restated simply as

h k = h d − k for 0 ≤ k ≤ 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 = dim Q ⁡ IH 2 k ⁡ ( X , 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

Dehn–Sommerville equations Wikipedia
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