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Posets of decompositions in spherical buildings

Kevin Ivan Piterman, John Shareshian, Volkmar Welker

TL;DR

The paper extends the theory of common bases and partial decompositions from linear spaces to spherical buildings by introducing CB(Δ), PD(Δ), and OPD(Δ) and proving strong topological properties. It establishes that PD(Δ) is Cohen–Macaulay and spherical, with a homotopy equivalence to CB(Δ), and that OPD(Δ) is homotopy equivalent to the join Δ*Δ, using opposition, Levi spheres, and convexity. The authors develop intrinsic, type-free constructions and demonstrate equivariant homotopy equivalences that connect the building-based posets to their linear counterparts, including applications to algebraic groups via Δ(G,k). They also derive decomposition and interval analyses, leading to structural insights such as a Levi-sphere description of decompositions and a Steinberg-square decomposition in group settings. Overall, the work unifies and extends poset/complex constructions across types and reveals new connections to representation theory and group cohomology.

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.

Posets of decompositions in spherical buildings

TL;DR

The paper extends the theory of common bases and partial decompositions from linear spaces to spherical buildings by introducing CB(Δ), PD(Δ), and OPD(Δ) and proving strong topological properties. It establishes that PD(Δ) is Cohen–Macaulay and spherical, with a homotopy equivalence to CB(Δ), and that OPD(Δ) is homotopy equivalent to the join Δ*Δ, using opposition, Levi spheres, and convexity. The authors develop intrinsic, type-free constructions and demonstrate equivariant homotopy equivalences that connect the building-based posets to their linear counterparts, including applications to algebraic groups via Δ(G,k). They also derive decomposition and interval analyses, leading to structural insights such as a Levi-sphere description of decompositions and a Steinberg-square decomposition in group settings. Overall, the work unifies and extends poset/complex constructions across types and reveals new connections to representation theory and group cohomology.

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.

Paper Structure

This paper contains 15 sections, 34 theorems, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial complexes
  4. Chamber complexes and convexity
  5. Posets
  6. Buildings
  7. Opposition
  8. Convexity
  9. The CAT(1) metric
  10. Automorphism group
  11. Posets and simplicial complexes in the case of vector spaces
  12. The common basis complex for buildings
  13. Levi spheres and decompositions
  14. An ordered partial decomposition poset for buildings
  15. OPD and PD posets for algebraic groups

Key Result

Theorem 1.1

Let $\Delta$ be a spherical building. There are order-preserving maps such that for every group $H$ acting on $\Delta$ by simplicial automorphisms, $\Gamma$ and $\phi$ are $H$-equivariant and induce homotopy equivalences between the fixed point subposets: In particular, $\Gamma$ and $\phi$ induce homotopy equivalences and equivariant isomorphisms in (co)homology.

Figures (3)

  • Figure 1: The $A_2$ Coxeter complex.
  • Figure 2: Poset $Y(\Sigma)_{\subsetneq \Sigma}$ of proper and non-empty convex subcomplexes of $\Sigma$.
  • Figure 3: Lower interval $Y(\Sigma)_{\subsetneq K}$.

Theorems & Definitions (75)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 2.1: Quillen's fiber theorem Qui78
  • Theorem 2.2
  • Theorem 2.3: Equivariant Whitehead
  • proof
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • ...and 65 more