`Norman involutions' and tensor products of unipotent Jordan blocks

A good knowledge of the Jordan canonical form (JCF) for a tensor product of Jordan blocks' is key to understanding the actions of $p$-groups of matrices in characteristic $p$. The JCF corresponds to a certain partition which depends on the characteristic $p$, and the study of these partitions dates back to Aitken's work in 1934. Equivalently each JCF corresponds to a certain permutation $\pi$ introduced by Norman in 1995. These permutations $\pi = \pi(r,s,p)$ depend on the dimensions $r$, $s$ of the Jordan blocks, and on $p$. We give necessary and sufficient conditions for $\pi(r,s,p)$ to be trivial, building on work of M.J. Barry. We show that when $\pi(r,s,p)$ is nontrivial, it is an involution involving reversals. Finally, we prove that the group $G(r,p)$ generated by $\pi(r,s,p)$ for all $s$, factors’ as a wreath product corresponding to the factorisation $r=ab$ as a product of its $p^{\mathrm{\prime }}$-part $a$ and $p$-part $b$: precisely $G(r,p)={\mathsf{S}}_a\wr {\mathsf{D}}_b$ where ${\mathsf{S}}_a$ is a symmetric group of degree $a$, and ${\mathsf{D}}_b$ is a dihedral group of degree $b$.