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

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 $J_{λ_1}\oplus\cdots\oplus J_{λ_r}$ where $λ_1\geq\cdots\geqλ_r>0$ and $\sum_{i=1}^rλ_i=rs$. The partition $λ(r,s,p):=(λ_1,\dots,λ_r)$ 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 $λ(r,s,p)$ 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 $λ(r,s,p)$ for all $s$ with $s\geq r$, and all primes $p$. This generalizes work of K-i. Iima and R. Iwamatsu.