Abstract
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 $F_\lambda(q)$.
Highlights
In the beautiful paper Variations on the Triangular Theme [7], Kirillov studied various structures on the set of triangular matrices
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)
Summary
In the beautiful paper Variations on the Triangular Theme [7], Kirillov studied various structures on the set of triangular matrices. Suppose X is a matrix of the form v 0 of Jordan type λ such that λ is obtained by adding a box to μ in the ith row and jth column. In order to satisfy the first condition, the entries in the vector vj−1 corresponding to the boxes in the (j − 1)th column and rows i must not simultaneously be zero (refer to Equation (4). The necessary and sufficient condition that X and Jμ must satisfy is that rank(Xk) = rank(Jμk) for all k 1, so the entries in the vector v corresponding to the boxes in the first column of the diagram for v1 must all be zero, while the remaining n−1−μ1 entries are free to be any element in Fq, so there are qn−1−μ1 matrices X whose leading principal submatrix is Jμ in this case.
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