Cyclic sieving for q-Kreweras numbers in the exceptional types

Let $W$ be a finite Weyl group with Coxeter number $h$ and simple Lie algebra $\mathfrak g$. For each $s\ge1$ the cyclic group $\mathbb{Z}/sh\mathbb{Z}$ acts on the multichains $w_1\le\dots\le w_s$ of noncrossing partitions of $W$. Reiner and Sommers defined, for each nilpotent orbit $\mathcal{O}$ of $\mathfrak g$ and each positive integer $m$, a $q$-Kreweras number, a polynomial in $q$, and tabulated it in the exceptional types. Each $W$-orbit of flats $X$ determines a nilpotent orbit $\mathcal{O}_X$, principal in the Levi subalgebra of $X$. Their Conjecture 1.4 says that the $q$-Kreweras number of $\mathcal{O}_X$ with $m=sh+1$, at a primitive $d$-th root of unity, counts the multichains fixed by an element of order $d$ whose bottom element has fixed space in the orbit of $X$. Reiner and Sommers proved it in types $A$, $B$, $C$ and $D$. We prove it in types $G_2$, $F_4$, $E_6$, $E_7$ and $E_8$, so it holds for every irreducible Weyl group. At $q=1$ the statement is a count proved by Douvropoulos and Josuat-Vergès; what is new is every $d>1$. Krattenthaler and Müller proved the unrefined cyclic sieving, Armstrong's conjecture, in the exceptional types by reducing it to finitely many cases. Using their parametrization of $s$ and of the acting element by three integers, and their description of the fixed points, we show that with two of the integers fixed the fixed-point count is a polynomial of degree at most the rank in the third. A finite-difference argument shows the same for the tabulated $q$-Kreweras number at the root of unity, once it holds for rank plus one consecutive values of the third integer. Only finitely many choices of the fixed pair need a direct check. The resulting finitely many identities are verified by two independent exact programs, and the reduction is checked in Lean. This work has not yet been reviewed by experts. The proofs, the Lean formalization, the computer checks and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction.

Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.