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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 $b$., Comment: 27 pages, 5 tables, 1 figure Minor typos corrected

Determining the Jordan canonical form of the tensor product of Jordan blocks has many applications including to the representation theory of algebraic groups, and to tilting modules. Although there are several algorithms for computing this decomposition in literature, it is difficult to predict the output of these algorithms. We call a decomposition of the form $J_r\otimes J_s=J_{\lambda_1}\oplus\cdots\oplus J_{\lambda_b}$ a `Jordan partition'. We prove several deep results concerning the $p$-parts of the $\lambda_i$ where $p$ is the characteristic of the underlying field. Our main results include the proof of two conjectures made by McFall in 1980, and the proof that ${\rm lcm}(r,s)$ and $\gcd(\lambda_1,\dots,\lambda_b)$ have equal $p$-parts. Finally, we establish some explicit formulas for Jordan partitions when $p=2$., Comment: Old Theorems 14 and 16 in v1 have been shortened and combined using comments by M.J.J. Barry. A reference of Gow and Laffey added, and an additional ARC grant acknowledgement

Let $J_r$ denote an $r\times r$ matrix over a finite field $F$ with minimal and characteristic polynomials $(t-1)^r$. Suppose $r\leq s$. It is not hard to show that the Jordan canonical form of $J_r\otimes J_s$ is similar to $J_{\lambda_1}\oplus\cdots\oplus J_{\lambda_r}$ where $\lambda_1\geq\cdots\geq\lambda_r>0$ and $\sum_{i=1}^r\lambda_i=rs$. The partition $\lambda(r,s,p):=(\lambda_1,\dots,\lambda_r)$ of $rs$, which depends only on $r,s$ and the characteristic $p$ of $F$, has many applications including to the study of algebraic groups. We prove new periodicity and duality results for $\lambda(r,s,p)$ that depend on the smallest $p$-power exceeding $r$. This generalizes results of J. A. Green, B. Srinivasan, and others which depend on the smallest $p$-power exceeding the (potentially large) integer $s$. We show that for fixed $r$ we can construct a finite table allowing the computation of $\lambda(r,s,p)$ for all $s$ with $s\geq r$, and all primes $p$. This generalizes work of K-i. Iima and R. Iwamatsu., Comment: 17 pages To appear J. Algebra