Zigzag structure of thin chamber complexes (Q1704897)

Let \(\Delta\) be a finite simplicial complex of rank \(n\). Assume that \(\Delta\) is thin, meaning that any codimension 1 face is contained in exactly two maximal faces. Given a flag \(F = (X_0, X_1, \dots, X_{n-1})\), where each \(X_i\) is a \(i\)-face of \(\Delta\), for each \(k\) there is a unique \(k\)-face \(X'_k\) which is adjacent to \(X_k\) and incident to all other \(X_j\). One defines \(\sigma_k(F)\) as the flag obtained by replacing \(X_k\) by \(X'_k\), and set \(T(F) = \sigma_{n-1} \dots \sigma_1 \sigma_0(F)\). A zig-zag in \(\Delta\) is an orbit for the action of the operator \(T\) on the set of flags of \(\Delta\). A description of all zig-zags is given in the case where \(\Delta = \Sigma(W,S)\) is a Coxeter complex. In particular the length of any zig-zag is equal to \(h | S |\), where \(h\) is the Coxeter number of \((W,S)\). Two faces in a thin chamber complex are \(z\)-connected if they are included in two flags belonging to a same zig-zag. This property is discussed in relation with the notion of normal geodesic.