Determining the Jordan canonical form of the tensor product of Jordan blockshas many applications including to the representation theory of algebraicgroups, and to tilting modules. Although there are several algorithms forcomputing this decomposition in literature, it is difficult to predict theoutput of these algorithms. We call a decomposition of the form $J_r\otimesJ_s=J_{\lambda_1}\oplus\cdots\oplus J_{\lambda_b}$ a `Jordan partition'. Weprove several deep results concerning the $p$-parts of the $\lambda_i$ where$p$ is the characteristic of the underlying field. Our main results include theproof of two conjectures made by McFall in 1980, and the proof that ${\rmlcm}(r,s)$ and $\gcd(\lambda_1,\dots,\lambda_b)$ have equal $p$-parts. Finally,we establish some explicit formulas for Jordan partitions when $p=2$.
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