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
UR - http://www.scopus.com/inward/record.url?scp=84877816392&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84877816392&partnerID=8YFLogxK
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 -