  Coxeter matroids generalize matroids just as flag varieties of Lie groups
generalize Grassmannians. Valuations of Coxeter matroids are functions that
behave well with respect to subdivisions of a Coxeter matroid into smaller
ones. We compute the universal valuative invariant of Coxeter matroids. A key
ingredient is the family of Coxeter Schubert matroids, which correspond to the
Bruhat cells of flag varieties. In the process, we compute the universal
valuation of generalized Coxeter permutohedra, a larger family of polyhedra
that model Coxeter analogues of combinatorial objects such as matroids,
clusters, and posets.
