Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices

Authors

Martha Yip, University of KentuckyFollow

Abstract

The set of n by n upper-triangular nilpotent matrices with entries in a finite field 𝔽_q has Jordan canonical forms indexed by partitions λ ⊢ n. We present a combinatorial formula for computing the number F_λ(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 F_λ(q).

Document Type

Article

Publication Date

3-29-2018

Notes/Citation Information

Published in The Electronic Journal of Combinatorics, v. 25, issue 1, paper #P1.68, p. 1-25.

The publisher has granted the permission for posting the article here.

Funding Information

Simons Collaboration Grant 429920.

Repository Citation

Yip, Martha, "Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices" (2018). Mathematics Faculty Publications. 30.
https://uknowledge.uky.edu/math_facpub/30