In Victor Reiner's Quotients of Coxeter Complexes and $P$-Partitions, we have the below definition for the quotient complex of a Coxeter complex by a finite subgroup of the Coxeter group. I think there is something wrong with the definition, and would like your help sorting it out.
Given a finite Coxeter system $(W,S)$, with $S$ a set of $n$ Euclidean reflections in some $n$-dimensional real inner product space that generate a finite group $W$, Reiner begins with the following pair of definitions:
$\Sigma(W,S)$ is the triangulation of the unit sphere in $\mathbb{R}^n$ obtained by intersecting it with all the hyperplanes fixed (pointwise) by any reflection in $W$.
Alternatively, $\Sigma(W,S)$ is the poset whose elements are the cosets $wW_J$ of subgroups $W_J$ of $W$ generated by subsets $J$ of $S$, ordered by reverse inclusion.
(This is on p. 9 of AMS Memoirs edition of Reiner's work, but p. 11 of the link above.)
The definitions are equivalent, in that the same abstract simplicial complex has the former definition as its geometric realization and the latter definition as its face poset. Reiner cites Kenneth Brown's book on buildings for this.
So far, so good. I both find the argument in Brown convincing and find that the result matches my hand calculations. Reiner goes on to give a parallel pair of definitions for the quotient of $\Sigma(W,S)$ by the action of a subgroup $G\subset W$.
$\Sigma(W,S)/G$ is the topological quotient of the simplicial complex $\Sigma(W,S)$ by the action of $G$.
$\Sigma(W,S)/G$ is the poset of double cosets $GwW_J$, ordered by reverse inclusion.
(p. 11 in the AMS edition and p. 13 in the link.)
Here, I have a problem. I don't see that the geometric version of the quotient complex (which is still a $\Delta$-complex since $G$ acts by simplicial homeomorphisms, but is no longer a simplicial complex) will have face poset given by the reverse inclusion relation. I agree that the double cosets $GwW_J$ will be in bijection with the simplices of the geometric version, provided we regard cosets with distinct $J$'s as distinct even if they are the same set, but I think that the incidence relations are subtler. My question is, am I right about this, and if so, what's the correct way to define the incidence relation in this combinatorial/group-theoretic definition of the quotient complex?
To make the problem concrete:
Let $W$ be $S_4$, seen as the isometries of a regular tetrahedron in $\mathbb{R}^3$ centered on the origin. Let $S=\{(12),(23),(34)\}$ with appropriate labels on the vertices of the tetrahedron. Then $\Sigma(W,S)$ is the barycentric subdivision of the boundary of the tetrahedron. Let $G = \langle (13),(1234)\rangle = D_4\subset S_4$.
The double cosets $D_4(id.)\langle(12),(23)\rangle$ and $D_4(id.)\langle (23),(34)\rangle$ both contain the entire group $S_4$; so if we order by reverse inclusion, they both sit below everything else, including each other. This doesn't make sense. In the $\Delta$-complex this would correspond to two vertices being incident to both each other (??) and also every edge.
So, to reiterate the question: am I flat wrong? And if not, then what is the partial order on the double cosets $GwW_J$ (regarding $GwW_J \neq Gw'W_{J'}$ unless we have set equality and $J=J'$) that recovers the incidence relation of the faces in the geometric definition of $\Sigma(W,S)/G$?
Addendum 5/12/16: Reiner answers the question below with a reference to a 2004 paper where he gave the right definition. Jim Humphreys has given a link in a comment, but just so the complete question/answer pair can be on this page, here is the answer:
- $\Sigma(W,S)/G$ is the poset of ordered pairs $(GwW_J,J)$, ordered by reverse inclusion of both factors.
I.e. $(GwW_J,J) \leq (Gw'W_{J'},J')$ if $J\supset J'$ and $GwW_J\supset Gw'W_{J'}$.
