More on Periodicity and Duality associated with Jordan partitions

TL;DR

This paper investigates the structure and periodicity of Jordan partitions associated with tensor products of Jordan blocks over fields of characteristic p, revealing new partial periodic and reflective behaviors.

Contribution

It determines the least period length of the Jordan partition's composition and uncovers new partial subperiodic and subreflective properties.

Findings

Least period length of the Jordan partition's composition is established.

New partial subperiodic behavior is identified.

New partial subreflective behavior is demonstrated.

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 $J_{\lambda_1} \oplus J_{\lambda_2} \oplus \dots \oplus J_{\lambda_{r}}$ where $\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_{r}>0$. This decomposition determines a partition $\lambda(r,s,p)=(\lambda_1,\lambda_2,\dots, \lambda_{r})$ of $r s$, known as the \textbf{Jordan partition}, but the values of the parts depend on $r$, $s$, and $p$. Write \[(\lambda_1,\lambda_2,\dots, \lambda_{r})=(\overbrace{\mu_1,\dots,\mu_1}^{m_1},\overbrace{\mu_2,\dots,\mu_2}^{m_2},\dots, \overbrace{\mu_k,\dots,\mu_k}^{m_k}) =(m_1 \cdot \mu_1, \dots,m_k \cdot \mu_k),\] where $\mu_1>\mu_2>\dots>\mu_k>0$, 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 $r \leq p^\beta$, $c(r,s,p)$ 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.