Seminar: Hadi Salmasian - Counting (F_q)-points of orbital varieties (May 8, 2026)

Speaker: Hadi Salmasian (University of Ottawa)
Title: Counting (F_q)-points of orbital varieties
Time: May 8, 2026 at 13:30
Place: Boğaziçi University Mathematics Department, TB130

Abstract:
Let (g = gl(n,F_q)) denote the Lie algebra of (n \times n) matrices with entries over the finite field (F_q). Let (b) be the Borel subalgebra of upper triangular matrices in (g), and let (u) be the nilradical of (b), i.e. the subalgebra of strictly upper triangular matrices in (g). Finally, let (a) be any (b)-stable ideal of (u). By an orbital variety, we mean the intersection of a nilpotent (GL(n,F_q))-conjugacy class of (g) with (a).

In this talk, two formulas are obtained for the number of (F_q)-points of these orbital varieties: one formula in terms of the scalar product of modified Hall–Littlewood and chromatic quasisymmetric polynomials, and another formula as a summation over a certain class of standard Young tableaux.

In the special case where (a) is the nilradical of a parabolic subalgebra, the result specializes to a recent theorem of Karp and Thomas that provides a formula in terms of coefficients of Macdonald polynomials. Some applications are also discussed, including a generalization, with a new proof, of the Kirillov–Melnikov–Ekhad–Zeilberger formula for the number of matrices (X \in u) such that (X^2 = 0) and (rank(X)=r).

The talk will be accessible to a broad audience, including graduate students working in algebra.