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

Lucas Pouillart

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-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.

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

Abstract

The cyclic sieving phenomenon was introduced by Reiner, Stanton and White in 2004 as a generalization of Stembridge's 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 , , and . There was yet no known uniform -analogue of the -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 of the associated Coxeter group , their formula factorises nicely under the assumption that acts as a reflection group on , which is very often the case. Using this condition, we provide a uniform refinement of these cyclic sieving phenomena using a -analogue of their main formula with a type by type proof based on the classification of finite irreducible Coxeter groups.

Paper Structure

This paper contains 24 sections, 42 theorems, 45 equations, 9 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The cluster complex
  3. Normalizers of parabolic subgroups
  4. Howlett's decomposition
  5. Douglass-Pfeiffer-Röhrle decomposition
  6. Parabolic quotients in type $A_{n-1}$
  7. Parabolic quotients in type $B_n$
  8. Parabolic quotients in type $D_n$
  9. Parabolic quotients in dihedral type
  10. $q$-analogues of counting formulas
  11. Proof of the main theorem in type $A_{n-1}$
  12. Proof of the main theorem in type $B_n$
  13. Combinatorics of the type $B_n$ cluster complexes
  14. Sieving polynomials in type $B_n$
  15. Counting stable faces of $B_n$
  16. ...and 9 more sections

Key Result

Theorem 1.1

RSW04 Let $X$ be the set of triangulations of the $(n+2)$-gon using noncrossing diagonals. Let Let the group $C_{n+2}$ act on $X$ as the polygon rotation. Then $(X, C_{n+2}, \mathop{\mathrm{Cat}}\nolimits_n(q))$ exhibits the CSP.

Figures (9)

  • Figure 1: The orbit under the rotation of a face of $\Gamma(A_{5})$.
  • Figure 2: The intersection lattice of type $I_2(k)$
  • Figure 3: A face of parabolic type $A_1^2$ (or $[1,1,2,2]$) of $\Gamma(A_5)$.
  • Figure 4: Faces of type $[1,2]$ on the left, and $[1,1]$ on the middle and the right, of $\Gamma(B_3)$.
  • Figure 5: A vertex and a facet of $\Gamma^{(3)}(I_2(5))$.
  • ...and 4 more figures

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark
  • ...and 71 more