For a Coxeter system (W,S), the subgroup W
J
generated by a subset J⊆S is called a parabolic subgroup of W. The Coxeterhedron PW associated to (W,S) is the finite poset of all cosets {wW
J
}
w∈W,J⊆S
of all parabolic subgroups of W, ordered by inclusion. This poset can be realized by the face lattice of a simple polytope, constructed as the convex hull of the orbit of a generic point in ℝ
n
under an action of the reflection group W. In this paper, for the groups W=A
n−1, B
n
and D
n
in a case‐by‐case manner, we present an elementary proof of the cyclic sieving phenomenon for faces of various dimensions of PW under the action of a cyclic group generated by a Coxeter element. This result provides a geometric, enumerative and combinatorial approach to re‐prove a theorem in Reiner et al. (2004). The original proof is proved by an algebraic method that involves representation theory and Springer’s theorem on regular elements.
