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Let $J_r$ denote an $r\times r$ matrix over a finite field $F$ with minimaland characteristic polynomials $(t-1)^r$. Suppose $r\leq s$. It is not hard toshow 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 $\sum_{i=1}^r\lambda_i=rs$. Thepartition $\lambda(r,s,p):=(\lambda_1,\dots,\lambda_r)$ of $rs$, which dependsonly on $r,s$ and the characteristic $p$ of $F$, has many applicationsincluding to the study of algebraic groups. We prove new periodicity andduality results for $\lambda(r,s,p)$ that depend on the smallest $p$-powerexceeding $r$. This generalizes results of J. A. Green, B. Srinivasan, andothers which depend on the smallest $p$-power exceeding the (potentially large)integer $s$. We show that for fixed $r$ we can construct a finite tableallowing the computation of $\lambda(r,s,p)$ for all $s$ with $s\geq r$, andall primes $p$. This generalizes work of K-i. Iima and R. Iwamatsu.

Decomposing modular tensor products, and periodicity of `Jordan partitions'