Cyclic sieving and cluster multicomplexes

TL;DR

This paper provides representation theoretic proofs for enumeration results related to polygon dissections and cluster complexes, extending previous combinatorial results to a broader algebraic framework.

Contribution

It introduces representation theoretic methods and geometric realizations to prove enumeration results in cluster algebra combinatorics, generalizing prior counting arguments.

Findings

Representation theoretic proofs of enumeration results

Extension of results to finite type cluster algebras

Connections with noncrossing tableaux and geometric realizations

Abstract

Reiner, Stanton, and White \cite{RSWCSP} proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu \cite{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 \cite{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 \cite{PN} as well as some geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky \cite{FZClusterII}.