Pattern Recognition on Oriented Matroids

3Dehn–Sommerville Type Relations

In this chapter we continue to consider face systems that satisfy the Dehn–Sommerville type relations discussed in Section 2.3.

Throughout the chapter, m denotes an integer at least 2. Let Δ ⊆ 2[m] be an abstract simplicial complex of size d(Δ) > 0. We denote by U(d(Δ)) the backward identity matrix of order d(Δ) + 1 whose rows and columns are indexed starting with zero, and whose (i, j)th entry is the Kronecker delta δi+j,d(Δ).

Recall that the h-vector of the complex Δ is said to satisfy the Dehn–Sommerville relations if h(Δ) is a left eigenvector of the matrix U(d(Δ)) corresponding to its eigenvalue 1, that is,

$h_l\left(\Delta \right)=h_{d\left(\Delta \right)-t}\left(\Delta \right),0\le l\le d\left(\Delta \right),\left(3.1\right)$

or, equivalently, ...