Spherical subsets, the nerve, the poset of spherical cosets, and the Davis realization

Let $(S,m)$ be a Coxeter matrix with $S$ finite, let $W$ be the presented group with length function $\mathrm{\ell }$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), and for $T\subseteq S$ let $W_T=⟨s:s\in T⟩\le W$ (Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups (1), The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups); recall $W_T=\{w\in W:S(w)\subseteq T\}$ for the support $S(w)$ of a reduced expression (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (1)).

(1) Spherical subsets and the nerve. A subset $T\subseteq S$ is spherical when $W_T$ is finite. Let $\mathbb{S}$ denote the set of spherical subsets, partially ordered by inclusion; it has least element $\mathrm{\varnothing }$ because $W_{\mathrm{\varnothing }}=\{1\}$. If $T\in \mathbb{S}$ and $T^{\mathrm{\prime }}\subseteq T$, then $W_{T^{\mathrm{\prime }}}\le W_T$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups), so $T^{\mathrm{\prime }}$ is spherical by In a finite group, the subgroup, every coset and the set of cosets are finite: $\mathbb{S}$ is downward closed. For each $s\in S$, the relation $s^2=1$ makes $W_{\{s\}}$ finite, so every singleton is spherical (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups). The nerve $L$ is the abstract simplicial complex (An abstract simplicial complex) on vertex set $S$ whose nonempty simplices are the nonempty spherical subsets; it also contains the empty simplex by the library's complex convention. Since $S$ is finite, $L$ is finite.

(2) The poset of spherical cosets. For $T\in \mathbb{S}$ and $w\in W$ let $w{W}_T$ be the left coset (Left and right cosets $gH$ and $Hg$ of a subgroup), and put $WS:=\{w{W}_T:w\in W,T\in \mathbb{S}\},$ partially ordered by inclusion of subsets of $W$. For $T=\mathrm{\varnothing }$ this gives $w{W}_{\mathrm{\varnothing }}=\{w\}$, so $W$ sits inside $WS$ as the set of minimal elements. A member of $WS$ is the resulting subset of $W$, not a choice of representative pair $(w,T)$; the equality, inclusion, and intersection criteria for these cosets are proved in Equality, inclusion and intersection of spherical cosets, and the quotient poset ↗.

(3) The Davis realization. $\mathrm{\Sigma }:=\mathrm{\mid }WS\mathrm{\mid }$ is the geometric realization of the order complex of the poset $WS$ (Face poset and order complex, The geometric realization of an abstract simplicial complex): its vertices are the cosets $w{W}_T$, and its simplices are the finite chains in $WS$. The chamber is $K:=\mathrm{\mid }\mathbb{S}\mathrm{\mid }$, the order complex of the poset of spherical subsets, and $j\text{\hspace{0.17em}⁣}:K\to \mathrm{\Sigma }$ is the simplicial map induced by $T\mapsto W_T$; this is simplicial because $T\subseteq T^{\mathrm{\prime }}$ implies $W_T\subseteq W_{T^{\mathrm{\prime }}}$. As an abstract complex, $K$ is the cone with apex $\mathrm{\varnothing }$ over the barycentric subdivision of $L$, and it is finite, hence compact and Hausdorff (A finite simplicial complex has a compact Hausdorff realization).

(4) The $W$-action. Left multiplication $(v,w{W}_T)\mapsto (vw)W_T$ is a well-defined left action of $W$ on the set $WS$ by order-preserving bijections (Group and abelian group, Left and right cosets $gH$ and $Hg$ of a subgroup); it induces a simplicial action of $W$ on $\mathrm{\Sigma }$. The chambers of $\mathrm{\Sigma }$ are the images $wj(K)=wK$, $w\in W$, and the map $w\mapsto wK$ is injective: the vertex $W_{\mathrm{\varnothing }}=\{1\}$ of $K$ is carried to $w{W}_{\mathrm{\varnothing }}=\{w\}$, and $\{v\}=\{w\}{W}_{\mathrm{\varnothing }}$ forces $v=w$ (Left and right cosets $gH$ and $Hg$ of a subgroup).

• (5) Abstentions. Nothing beyond these constructions is asserted here: not that the chambers meet one another in faces, not that the spherical cosets $w{W}_T$ carry the structure of the Coxeter cells $C_T$, not that the action on $\mathrm{\Sigma }$ is proper with compact quotient, and not that $\mathrm{\Sigma }$ is simply connected. Those assertions are the content of The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K) ↗, a recorded justifier of this definition; simple connectivity is proved later on this page.
• Choice. No Choice is used in (1)-(4): all constructions are set-theoretic over the finite set $S$ and the fixed group $W$.