The Cyclic Sieving Phenomenon for Faces of Generalized Cluster Complexes
نویسندگان
چکیده
The notion of cyclic sieving phenomenon is introduced by Reiner, Stanton, and White as a generalization of Stembridge’s q = −1 phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a generalization of the cluster complexes found by Fomin and Zelevinsky. In this paper, the faces of various dimensions of the generalized cluster complexes in type An, Bn, Dn, and I2(a) are shown to exhibit the cyclic sieving phenomenon under a cyclic group action. For the cluster complexes of exceptional type E6, E7, E8, F4, H3, and H4, a verification for such a phenomenon on their maximal faces is given.
منابع مشابه
The Cyclic Sieving Phenomenon for Faces of Cyclic Polytopes
A cyclic polytope of dimension d with n vertices is a convex polytope combinatorially equivalent to the convex hull of n distinct points on a moment curve in R. In this paper, we prove the cyclic sieving phenomenon, introduced by Reiner-StantonWhite, for faces of an even-dimensional cyclic polytope, under a group action that cyclically translates the vertices. For odd-dimensional cyclic polytop...
متن کاملCyclic Sieving and Cluster Multicomplexes
Reiner, Stanton, and White [10] proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu [2] generalized these results to Cartan-Killing types other than A by means of actions of deformed Coxeter elements on cluster complexes of Fomin and Zelevinsky [6]. The ReinerStanton-White and Eu-Fu results were proven using direct counting arguments. We give rep...
متن کاملDihedral Sieving Phenomena
Abstract. Cyclic sieving is a well-known phenomenon where certain interesting polynomials, especially qanalogues, have useful interpretations related to actions and representations of the cyclic group. We define sieving for an arbitrary group G and study it for the dihedral group I2pnq of order 2n. This requires understanding the generators of the representation ring of the dihedral group. For ...
متن کاملSpringer’s Regular Elements over Arbitrary Fields
Springer’s theory of regular elements in complex reflection groups is generalized to arbitrary fields. Consequences for the cyclic sieving phenomenon in combinatorics are discussed.
متن کاملThe Cyclic Sieving Phenomenon for Non-Crossing Forests
In this paper we prove that the set of non-crossing forests together with a cyclic group acting on it by rotation and a natural q-analogue of the formula for their number exhibits the cyclic sieving phenomenon, as conjectured by Alan Guo.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007