Posets of decompositions in spherical buildings

TL;DR

This paper extends the theory of posets related to decompositions from vector spaces to arbitrary spherical buildings, establishing their topological properties and representation-theoretic implications.

Contribution

It introduces new definitions for common bases complexes and decompositions in spherical buildings, proving their Cohen-Macaulay and spherical properties, generalizing known linear case results.

Findings

Poset of decompositions is Cohen-Macaulay

Poset of partial decompositions is spherical

Poset of partial decompositions relates to Steinberg representation

Abstract

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.