From the introduction: We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing partitions for well-generated complex reflection groups.
Our goal is Theorem 1.1 below, whose terminology is explained briefly here, and more fully in the next section.
Let \(V=\mathbb C^n\) and let \(W\subset\text{GL}(V)\) be a finite irreducible complex reflection group, that is, \(W\) is generated by its set \(R\) of reflections. We will assume further that \(W\) is ‘well-generated’ in the sense that it can be generated by \(n\) of its reflections. Define the Coxeter number \(h:=d_n\) and the \(W\)-\(q\)-Catalan number \[ \text{Cat}(W,q):=\prod^n_{i=1}\frac{[h+d_i]_q}{[d_i]_q}, \] where \([n]_q:=1+q+q^2+\cdots+q^{n-1}=\tfrac{q^n-1}{q-1}\). Let \(c\) be a regular element of \(W\) in the sense of T. A. Springer [Invent. Math. 25, 159-198 (1974; Zbl 0287.20043)], of order \(h\); such a regular element of order \(h\) will exist because \(W\) is well-generated; see Subsection 2.3. In other words, \(c\) has an eigenvector \(v\in V\) fixed by none of the reflections, and whose \(c\)-eigenvalue \(\zeta_h\) is a primitive \(h\)-th root of unity.
Let \(\text{NC}(W):=\{w\in W:\ell_R(w)+\ell_R(w^{-1}c)=n\}\), where \(\ell_R\) is the absolute length function \(\ell_R(w):=\min\{\ell:w=r_1r_2\cdots r_\ell\) for some \(r_i\in R\}\). The initials “NC” in \(\text{NC}(W)\) are motivated by the special case where \(W\) is the Weyl group of type \(A_{n-1}\), and the set \(\text{NC}(W)\) bijects with the noncrossing partitions of \(\{1,2,\dots,n\}\); see Subsection 2.1 below.
The fact that \(R\) is stable under \(W\)-conjugacy implies that the cyclic group \(C:=\langle c\rangle\) acts on the set \(\text{NC}(W)\) by conjugation: \(w\in\text{NC}(W)\) implies \(cwc^{-1}\in\text{NC}(W)\).
Theorem 1.1. In the above setting, the triple \((X,X(q),C)\) given by \[ X=\text{NC}(W),\quad X(q)=\text{Cat}(W,q),\quad C=\langle c\rangle\cong\mathbb Z/h\mathbb Z, \] exhibits the cyclic sieving phenomenon defined by V. Reiner, D. Stanton, and D. White [J. Comb. Theory, Ser. A 108, No. 1, 17-50 (2004; Zbl 1052.05068)]: for any element \(c^i\) in \(C\), the subset \(X^{c^i}\) of elements in \(X\) fixed by \(c^i\) has cardinality \(|X^{c^i}|=[X(q)]_{q=\zeta^i_h}\).
After explaining a bit more of the background and definitions in Section 2, Theorem 1.1 is proven in Section 3. The (partly conjectural) interpretation of \(\text{Cat}(W,q)\) is discussed in Section 4, leading to speculation on more conceptual proofs of Theorem 1.1 in Section 5. Section 6 remarks on some conjectural variations.