`Norman involutions' and tensor products of unipotent Jordan blocks

TL;DR

This paper explores the permutation structure associated with the Jordan canonical form of tensor products of Jordan blocks in characteristic p, providing conditions for triviality, involution properties, and a wreath product decomposition of the generated group.

Contribution

It characterizes when the Norman involution permutation is trivial or nontrivial, describes its involution structure, and establishes a wreath product factorization of the associated permutation group.

Findings

Necessary and sufficient conditions for triviality of the permutation.

Nontrivial permutations are involutions involving reversals.

The permutation group factors as a wreath product based on the factorization of r.

Abstract

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 $p'$-part $a$ and $p$-part $b$: precisely $G(r, p)={\sf S}_a\wr {\sf D}_b$ where ${\sf S}_a$ is a symmetric group of degree $a$, and ${\sf D}_b$ is a dihedral group of degree $b$.