The Brady-Watt ordered root complex X(c), its subcomplexes X(sigma) and X(sigma,rho), and their positive-cone realizations

Let $(W,S)$ be an irreducible Coxeter system of finite type with $S$ finite, $n:=\mathrm{\mid }S\mathrm{\mid }\ge 1$, and let ${\alpha }_i$, $R_i$, $c$, $h$, $({\rho }_i)$, $({\mu }_i)$, $\mu$, ${\mathrm{\Phi }}_{+}$, ${\le }_T$, $M$ and $t_{\alpha }$ be as in The bipartite Coxeter element, its ordered prefix roots, and the conditional vector map mu(a) = -2(c-1)^{-1}a, The Coxeter plane, ordered-root enumeration, and invertibility of rho(c) - id and The mu-dot-root identities, the cone separation, and the canonical simple systems of the subintervals [1, sigma], so ${\mathrm{\Phi }}_{+}=\{{\rho }_1<{\rho }_2<\cdots <{\rho }_{nh\mathrm{/}2}\}$ in the global order. Write $R(\alpha )$ for the reflection with normal $\alpha$ and $S^{n-1}:=\{x\in V:B(x,x)=1\}$.

(1) The complex $X(c)$. Its vertex set is ${\mathrm{\Phi }}_{+}$. For $1\le i<j\le nh\mathrm{/}2$, join ${\rho }_i$ to ${\rho }_j$ by an edge exactly when $R({\rho }_j)R({\rho }_i){\le }_{T}c$. Let $X(c)$ contain the empty simplex and every finite nonempty set of vertices whose every two-element subset is an edge. Thus $X(c)$ is the abstract simplicial complex (An abstract simplicial complex) determined by this ordered edge relation.

(2) The subcomplexes $X(\sigma )$. For $\sigma {\le }_{T}c$, put $P_{\sigma }:=\{\alpha \in {\mathrm{\Phi }}_{+}:t_{\alpha }{\le }_T\sigma \}$ and let $X(\sigma )$ be the full subcomplex of $X(c)$ on the vertex set $P_{\sigma }$. For a positive root $\rho$ in the global order, let $X(\sigma ,\rho )$ be the full subcomplex on vertices in $P_{\sigma }\cap M(\sigma )$ that are less than or equal to $\rho$; thus $X(\sigma ,{\tau }_i)$ has vertex set $\{{\tau }_1,\dots ,{\tau }_i\}$ when $P_{\sigma }=\{{\tau }_1<\cdots <{\tau }_t\}$.

(3) Positive cones and realizations. Every vertex is a unit vector. For a finite set $F$ of vertices define $c[F]:=\{{\sum }_{v\in F}{\lambda }_{v}v:{\lambda }_v\ge 0\},c[\mathrm{\varnothing }]:=\{0\}\mathrm{.}$ For a subcomplex $Y$ put $c[Y]:={\bigcup }_{F\in Y}c[F]$ and $\mathrm{\mid }Y\mathrm{\mid }:=c[Y]\cap S^{n-1}$, its positive-cone realization in the unit sphere. For $\sigma {\le }_{T}c$ and a positive root $\rho$, write $Y(\sigma ,\rho ):=c[\{\tau \in P_{\sigma }\cap M(\sigma ):\tau \le \rho \}]\mathrm{.}$

(4) Abstentions. This definition does not assert that $c[F]$ is nondegenerate for each simplex $F$, that $\mathrm{\mid }F\mathrm{\mid }$ is a spherical simplex, that the cone realization embeds $X(c)$ or any $X(\sigma )$, or that these realizations are convex or have dimension ${\mathrm{\ell }}_T(\sigma )-1$. It also does not assert an equivalence between higher simplex membership and a single full-tuple product condition. No Choice is used.