TY - JOUR

T1 - A combinatorial proof of the cyclic sieving phenomenon for faces of Coxeterhedra

AU - Eu, Sen Peng

AU - Fu, Tung Shan

AU - Pan, Yeh Jong

N1 - Funding Information:
Research partially supported by NSC grants 98-2115-M-390-002 (S.-P. Eu), 99-2115-M-251-001 (T.-S. Fu), and 99-2115-M-127-001 (Y.-J. Pan).

PY - 2013/5

Y1 - 2013/5

N2 - 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 } wεW,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. (J. Comb. Theory, Ser. A 108:17-50, 2004). The original proof is proved by an algebraic method that involves representation theory and Springer's theorem on regular elements.

AB - 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 } wεW,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. (J. Comb. Theory, Ser. A 108:17-50, 2004). The original proof is proved by an algebraic method that involves representation theory and Springer's theorem on regular elements.

KW - Boxed ordered partition

KW - Coxeterhedron

KW - Cyclic sieving phenomenon

KW - Permutohedron

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U2 - 10.1007/s10878-012-9495-6

DO - 10.1007/s10878-012-9495-6

M3 - Article

AN - SCOPUS:84877816392

VL - 25

SP - 617

EP - 638

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

IS - 4

ER -