Geodesic
Rook placements and Jordan forms of upper-triangular nilpotent matrices
Martha Yip1
1University of Kentucky
The electronic journal of combinatorics, Tome 25 (2018) no. 1
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The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $\mathbb{F}_q$ has Jordan canonical forms indexed by partitions $\lambda \vdash n$. We present a combinatorial formula for computing the number $F_\lambda(q)$ of matrices of Jordan type $\lambda$ 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 $F_\lambda(q)$.
DOI : 10.37236/6888
Classification : 05A19
Mots-clés : nilpotent matrices, finite fields, Jordan form, rook placements, Young tableaux, set partitions

Martha Yip  1

1 University of Kentucky 
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     author = {Martha Yip},
     title = {Rook placements and {Jordan} forms of upper-triangular nilpotent matrices},
     journal = {The electronic journal of combinatorics},
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Martha Yip. Rook placements and Jordan forms of upper-triangular nilpotent matrices. The electronic journal of combinatorics, Tome 25 (2018) no. 1. doi: 10.37236/6888

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