q-Rook placements and Jordan forms of upper-triangular nilpotent matrices

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 $λ \vdash n$. We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number$ F_λ (q)$ of matrices of fixed Jordan type as a weighted sum over rook placements.

Keywords

• q-rook placements
• jordan canonical form
• nilpotent matrices
• set partitions
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]