Cyclic sieving phenomena on parabolic classes of faces of the cluster complex

TL;DR

This paper develops a uniform $q$-analogue for cyclic sieving phenomena on faces of generalized cluster complexes, extending previous results and providing type-by-type proofs based on Coxeter group classifications.

Contribution

It introduces a uniform $q$-analogue for cyclic sieving on parabolic classes of faces, generalizing prior case-specific results using Coxeter group theory.

Findings

Provides a uniform $q$-analogue formula for face enumeration

Establishes cyclic sieving phenomena for classical types

Offers a type-by-type proof based on Coxeter group classification

Abstract

The cyclic sieving phenomenon was introduced by Reiner, Stanton and White in 2004 as a generalization of Stembridge's $q=-1$ phenomenon. In a paper from 2008, Eu and Fu studied many occurrences of this phenomenon on the faces of the generalized cluster complex with the action of the Fomin-Reading rotation in the classical types $A_n$, $B_n$, $D_n$ and $I_2(k)$. There was yet no known uniform $q$-analogue of the $k$-face numbers of these complexes. In a more recent paper from 2023, Douvropoulos and Josuat-Verg\`es provided a refinement of the enumeration of the faces of the generalized cluster complex using a uniform formula. For a parabolic subgroup $W_X \subset W$ of the associated Coxeter group $W$, their formula factorises nicely under the assumption that $N_W(W_X)/W_X$ acts as a reflection group on $X$, which is very often the case. Using this condition, we provide a uniform refinement of these cyclic sieving phenomena using a $q$-analogue of their main formula with a type by type proof based on the classification of finite irreducible Coxeter groups.