Dehn–Sommerville equations

Equivalent formulations

Sommerville found a different way to state these equations:

${\displaystyle \sum _{i=-1}^{k-1}(-1{)}^{d+i}(\genfrac{}{}{0ex}{}{d-i-1}{d-k})f_i=\sum _{i=-1}^{d-k-1}(-1{)}^i(\genfrac{}{}{0ex}{}{d-i-1}{k})f_i,}$

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

${\displaystyle h_k=\sum _{i=0}^k(-1{)}^{k-i}(\genfrac{}{}{0ex}{}{d-i}{k-i})f_{i-1}.}$

The sequence

${\displaystyle h(P)=(h_0,h_1,\dots ,h_d)}$

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

${\displaystyle h_k=h_{d-k}\text{ for }0\le k\le d.}$

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:

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

(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.