Assume we glue an $n$-dimensional simplicial complex $K$ from copies of an $n$-simplex $\Delta$ with fixed spherical metric. We may think that $\Delta$ has colored vertices and we glue so that the colors match.
If $\Delta$ is right angled then $K$ forms a $\mathrm{CAT}(1)$ space if and only if $K$ is flag; (i.e., $K$ and all its links have no triangles).
Is there a similar condition (if and/or only if) in the case if $\Delta$ is a spherical Coxeter simplex? (say for the quotient $\Delta=\mathbb S^{n}/S_{n+2}$)
Comments.
It is easy to make a similar condition if $\Delta$ is 1-dimensional. After that you can get an iff condition for the joints of finite number of 1-simplexes. Note that right-angles simples is joint of finite number of 0-dimensional simplexes, so this is a bit more general and the proof is almost identical.
At the moment I do not see a condition for the spherical triangle with angles $\tfrac\pi2$, $\tfrac\pi3$ and $\tfrac\pi3$, which is $\mathbb S^{2}/S_4$. This should be something like this: "no quadrilateral for the vertices with angle $\tfrac\pi3$ and yet something". The "no quadrilateral" condition guarantees* that $K$ is locally $\mathrm{CAT}(1)$ and something has to forbid a finite number of finite subcomplexes. (It should be possible to list such subcomplexes, and likely they are known, but I am interested in $n$-dimensional case.)
(*) In this case the "no triangle" condition is automatic since colors match along the gluing,
