More on Periodicity and Duality associated with Jordan partitions

15 Jul 2019  ·  Barry Michael J. J. ·

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. read more

PDF Abstract
No code implementations yet. Submit your code now

Categories


Representation Theory