The set of n by n upper-triangular nilpotent matrices with entries in a finite field 𝔽q has Jordan canonical forms indexed by partitions λ ⊢ n. We present a combinatorial formula for computing the number Fλ(q) of matrices of Jordan type λ as a weighted sum over standard Young tableaux. We construct a bijection between paths in a modified version of Young's lattice and non-attacking rook placements, which leads to a refinement of the formula for Fλ(q).
Simons Collaboration Grant 429920.
Yip, Martha, "Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices" (2018). Mathematics Faculty Publications. 30.