Cyclic sieving and cluster multicomplexes
Abstract
Reiner, Stanton, and White RSWCSP proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu EuFu 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 FZY. The Reiner-Stanton-White and Eu-Fu results were proven using direct counting arguments. We give representation theoretic proofs of closely related results using the notion of noncrossing and semi-noncrossing tableaux due to Pylyavskyy PN as well as some geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky FZClusterII.
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