More on Periodicity and Duality associated with Jordan partitions

Abstract

Let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field $F$ of characteristic $p$. For positive integers $r$ and $s$ with $r \leq s$, the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has the form $J_{λ_1} \oplus J_{λ_2} \oplus \dots \oplus J_{λ_{r}}$ where $λ_1 \geq λ_2 \geq \dots \geq λ_{r}>0$. This decomposition determines a partition $λ(r,s,p)=(λ_1,λ_2,\dots, λ_{r})$ of $r s$, known as the \textbf{Jordan partition}, but the values of the parts depend on $r$, $s$, and $p$. Write \[(λ_1,λ_2,\dots, λ_{r})=(\overbrace{μ_1,\dots,μ_1}^{m_1},\overbrace{μ_2,\dots,μ_2}^{m_2},\dots, \overbrace{μ_k,\dots,μ_k}^{m_k}) =(m_1 \cdot μ_1, \dots,m_k \cdot μ_k),\] where $μ_1>μ_2>\dots>μ_k>0$, and denote the composition $(m_1,\dots,m_k)$ of $r$ by $c(r,s,p)$. A recent result of Glasby, Praeger, and Xia in \cite{GPX} implies that if $r \leq p^β$, $c(r,s,p)$ is periodic in the second variable $s$ with period length $p^β$ and exhibits a reflection property within that period. We determine the least period length and we exhibit new partial subperiodic and partial subreflective behavior.