Finite large spherical complexes, their almost-negative matrices, the metric flag condition, and links

Fix the following notions; no theorem about them is asserted here beyond well-definedness.

(1) Finite spherical complexes. Let $K$ be a finite abstract simplicial complex (An abstract simplicial complex) whose simplices $\sigma$ carry positive-definite Gram matrices $C_{\sigma }$ of diagonal $1$ on their vertices, compatible on common faces, and let $X=\mathrm{\mid }K{\mathrm{\mid }}_C$ be the associated finite spherical complex with its chain metric $d$ and its truncated angular metric $d_{\pi }=\min \{\pi ,d\}$ (Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas(iii), Spherical Gram simplices and angular links of Euclidean faces, The angular path metric, the Euclidean cone and spherical joins(2)). $X$ is large when for every simplex $\sigma$ and all distinct $s,t\in \sigma$ the edge length $\arccos c_{st}$ is at least $\pi \mathrm{/}2$, equivalently every off-diagonal entry of every $C_{\sigma }$ is at most $0$. Here “large” names this edge-length condition; it does not assert the distinct unique-geodesic-below-$\pi$ property called “large” for piecewise-spherical spaces in Charney–Davis §2.1.1.

(2) The associated almost-negative matrix. Let $X$ be large. Since $K$ is simplicial, a pair of vertices spans at most one edge. For an edge $\{s,t\}$, let $c_{st}=c_{ts}$ be the corresponding entry of its prescribed Gram matrix, and let ${\mathrm{\ell }}_{st}:=\arccos (c_{st})\in [\pi \mathrm{/}2,\pi )$ be its one-cell spherical length. Define the symmetric matrix $C=C(X)$ on the vertex set by $c_{ss}=1$, by the prescribed edge entry $c_{st}$ when $\{s,t\}$ is an edge of $K$, and by $c_{st}=c_{ts}:=-1$ when $\{s,t\}$ is not an edge. Its off-diagonal entries are non-positive: this is the almost-negative matrix associated with $X$. The edge entry is taken from the prescribed local Gram data because a gluing's global chain metric need not restrict to a cell metric in general (Abstract isometric polyhedral gluings and the chain metric).

(3) The metric flag condition. Let $X$ be large. A set $T$ of vertices of $K$ is pairwise adjacent when every two distinct members of $T$ span an edge; in that case $C_T:=(c_{st}{)}_{s,t\in T}$ is the cosine matrix of $T$. We regard the empty matrix as positive definite, so the empty set passes this test. Say that $X$ is metric flag when for every pairwise adjacent $T$: $T$ is the vertex set of a simplex of $K$ if and only if $C_T$ is positive definite (Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form). The forward implication is automatic for complexes of spherical simplices, since a principal submatrix of a positive-definite Gram matrix is positive definite; the content of the condition is the converse. A large metric flag complex is a large finite spherical complex satisfying (3).

(4) Links. For a face $F$ of $X$ (that is, a simplex of $K$) the link ${\mathrm{Lk}}_X(F)$ is the finite spherical complex whose simplices are the links ${\mathrm{Lk}}_{\sigma }(F)$ of the simplices $\sigma \supseteq F$ (Subcomplexes, closures, stars, and links in a simplicial complex), carrying the Gram matrices obtained by the iterated Schur complement of Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas(iv): its vertices are the vertices $t\mathrm{\notin }F$ for which $F\cup \{t\}$ is a simplex of $K$, and its cells are the sets $T$ disjoint from $F$ with $F\cup T$ a simplex of $K$. Adjacency to every vertex of $F$ alone does not suffice: the resulting clique may have a non-positive-definite cosine matrix. No assertion is made here about complexes with edges shorter than $\pi \mathrm{/}2$; the metric flag test is used only in the large case.