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We study the interplay between the discrete geometry of Bruhat posetintervals and subword complexes of finite Coxeter systems. We establishconnections between the cones generated by cover labels for Bruhat intervalsand of root configurations for subword complexes, culminating in the notion ofbrick polyhedra for general subword complexes.

Bruhat intervals, subword complexes and brick polyhedra for finite Coxeter groups