Counting $\mathbb F_q$-points of orbital varieties in ad-nilpotent ideals of type $A_n$
Mohammad Bardestani, Keivan Mallahi-Karai, Samrith Ram, Hadi Salmasian
TL;DR
This work delivers a unified, polyhedral- and symmetric-function–driven framework for counting ${\mathbb F}_q$-points in ad-nilpotent ideals of ${\mathfrak u}_n({\mathbb F}_q)$. By developing a parabolic version of Borodin’s division algorithm, the authors reduce complex orbit-count questions to tractable counts expressed via Macdonald and Hall–Littlewood polynomials and chromatic quasisymmetric functions. They prove explicit recurrences for ${\mathrm F}_{\mu\Lambda}(q)$ and ${\mathrm F}_{\mu\mathsf h}(q)$, establish symmetry under permutation of parabolic blocks, and connect these counts to Kostka and dual Macdonald data, yielding nontrivial positivity and structural results. The paper also furnishes four applications: counts on nilpotent Hessenberg varieties, a derivation of Kirillov’s recurrence for fixed Jordan type, a closed form for squaring-zero matrices in nilradicals, and a formula for unipotent double cosets, thereby linking representation theory, algebraic geometry, and symmetric-function theory with new, direct proofs. Overall, the approach provides a powerful, computationally effective bridge between orbit geometry and Macdonald-theoretic combinatorics with broad implications for counting problems in finite linear groups.
Abstract
Let $\mathfrak b_n(\mathbb F_q)$ denote the Lie algebra of upper triangular $n \times n$ matrices over $\mathbb F_q$, and let $\mathfrak u_n(\mathbb F_q)$ be the subalgebra of strictly upper triangular matrices. For every $\mathfrak b_n(\mathbb F_q)$-stable ideal $\mathfrak a \subseteq \mathfrak u_n(\mathbb F_q)$ and partition $μ$ of $n$, we give an explicit formula for the number of elements in $\mathfrak a$ of Jordan type $μ$. Up to a power of $q$, the number of points is given by the Hall scalar product of a modified Hall-Littlewood function indexed by $μ$ and a chromatic quasisymmetric function associated to $\mathfrak a$. In the special case $\mathfrak a = \mathfrak u_Λ(\mathbb F_q)$, the nilradical of the standard parabolic subalgebra of $\mathfrak{gl}_n(\mathbb F_q)$ corresponding to a composition $Λ$ of $n$, our formula specializes to a result of Karp and Thomas: up to a polynomial in $q$, the number of elements in $\mathfrak u_Λ(\mathbb F_q)$ of Jordan type $μ$ equals the coefficient of $\mathbf x^Λ$ in the specialization of the dual Macdonald symmetric function $\mathrm Q_{μ'}(\mathbf x; q^{-1}, t)$ at $t = 0$. We give a new and shorter proof using a parabolic version of Borodin's division algorithm. We present four applications: (1) a formula for the number of points on a nilpotent Hessenberg variety; (2) a derivation of Kirillov's recurrence for counting nilpotent matrices of fixed Jordan type; (3) a formula for the number of $X \in \mathfrak u_Λ(\mathbb F_q)$ with $X^2 = 0$, yielding a new proof of the Kirillov-Melnikov-Ekhad-Zeilberger formula via two-row Macdonald polynomials; (4) a formula for the number of double cosets $\mathsf U_1 \backslash \mathsf{GL}_n(\mathbb F_q) / \mathsf U_2$, where $\mathsf U_1$ and $\mathsf U_2$ are unipotent subgroups from $\mathfrak b_n(\mathbb F_q)$-stable ideals.