Posets of decompositions in spherical buildings

We propose definitions of the common bases complex, the poset of decompositions, and the poset of partial decompositions for arbitrary spherical buildings. We show that the poset of decompositions is Cohen-Macaulay, and that the poset of partial decompositions is spherical and homotopy equivalent to the common bases complex. To prove these results, we rely on the concepts of opposition, Levi spheres, and convexity in buildings. In particular, our results extend the already known constructions for the linear case (vector spaces) to arbitrary buildings. As a byproduct, we see that the poset of ordered partial decompositions carries the square of the Steinberg representation.