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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.28242 |
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Table of Contents:
- 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-Vergès 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.