Enumerating matrices with prescribed entries in an adjoint orbit

We study intersections of conjugacy classes of square matrices over a finite field with affine coordinate subspaces, or equivalently matrices in a fixed adjoint orbit with prescribed entries.

Our main result treats the case of prescribed columns: for a partially defined linear map we give a Hall scalar product formula for the number of extensions to an endomorphism with prescribed similarity invariants.

This formula is expressed in terms of skew modified Hall--Littlewood functions and $q$-Whittaker functions.

As applications, we count monic matrix polynomials over $\mathbb{F}_q$ with prescribed Smith normal form and with prescribed determinant, and recover the Gerstenhaber--Reiner formula for the number of square matrices with a fixed characteristic polynomial.

We also note that known point-count formulas for Hessenberg varieties imply related formulas for Hessenberg supports involving chromatic quasisymmetric functions, motivating polynomiality questions for more general supports and prescribed affine slices.