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 &minus; 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 = &minus;1, 0, ..., d &minus; 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 = &minus;1, it expresses the fact that Euler characteristic of a (d &minus; 1)-dimensional simplicial sphere is equal to 1 + (&minus;1)d &minus; 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 &minus; 2 are independent. Another independent set consists of the equations with k = &minus;1, 1, 3, ..., d &minus; 3. If d is odd then the equations with k = &minus;1, 1, 3, ..., d &minus; 2 form one independent set and the equations with k = &minus;1, 0, 2, 4, ..., d &minus; 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 &le; k &le; $1/2$(d&minus;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


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

The equations with 0 &le; k &le; $1/2$(d&minus;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:


 * $$ h_k=\dim_{\mathbb{Q}}\operatorname{IH}^{2k}(X,\mathbb{Q}) $$

(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.