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Let Jr denote an r×r matrix with minimal and characteristic polynomials (t-1)r. Suppose r≤s. It is not hard to show that the Jordan canonical form of Jr⊗Js is similar to Jλ1⊕⋯⊕Jλr where λ1≥⋯≥λr>0 and ∑i=1rλi=rs. The partition λ(r, s, p):=(λ1, . . ., λr) of rs, which depends only on r, s and the characteristic p:=char(F), has many applications including 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. It also implies that for fixed r we can construct a finite table allowing the computation of λ(r, s, p) for all s and p, with s≥r and p prime.
Glasby, S. P., Praeger, C. E., & Xia, B. (2016). Decomposing modular tensor products, and periodicity of “Jordan partitions.” Journal of Algebra, 450, 570–587. https://doi.org/10.1016/j.jalgebra.2015.11.025