A graph-covering argument proves that the product of two recoil classes in a symmetric group is a sum of recoil classes, with a refinement of the structure coefficients, and the geometric core extends to arbitrary Coxeter groups.
Solomon, A Mackey formula in the group ring of a Coxeter group, J
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Combinatorics of descent algebras and graph coverings
A graph-covering argument proves that the product of two recoil classes in a symmetric group is a sum of recoil classes, with a refinement of the structure coefficients, and the geometric core extends to arbitrary Coxeter groups.