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 $r\le s$, the Jordan canonical form of the $rs\times rs$ matrix $J_r\otimes J_s$ has the form $J_{{\lambda }_1}\oplus J_{{\lambda }_2}\oplus \cdots \oplus J_{{\lambda }_r}$ where ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_r>0$. This decomposition determines a partition $\lambda (r,s,p)=({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_r)$ of $rs$, 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 ${\mu }_1>{\mu }_2>\cdots >{\mu }_k>0$, 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 $r\le p^{\beta }$, $c(r,s,p)$ is periodic in the second variable $s$ with period length $p^{\beta }$ and exhibits a reflection property within that period. We determine the least period length and we exhibit new partial subperiodic and partial subreflective behavior.

6 pages