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A Coxeter group is a group defined by a presentation by involutions $r_i$ with relators $(r_ir_j)^{m_{ij}}=1$ for certain family $(m_{ij})$ of integers greater than 1.

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coset poset of reflection subgroup

I assume that $W$ is finit (not just $S$) and I take parabolic subgroups rather than reflection subgroups. Then the coset poset is indeed Cohen-Macaulay. In the recent work Cluster Parking Functions, …
Matthieu Josuat-Vergès's user avatar