Decomposing modular tensor products, and periodicity of ‘Jordan partitions’

Let JrJr denote an r×rr×r matrix with minimal and characteristic polynomials (t−1)r(t−1)r. Suppose r⩽sr⩽s. It is not hard to show that the Jordan canonical form of Jr⊗JsJr⊗Js is similar to Jλ1⊕⋯⊕JλrJλ1⊕⋯⊕Jλr where λ1⩾⋯⩾λr>0λ1⩾⋯⩾λr>0 and ∑i=1rλi=rs. The partition λ(r,s,p):=(λ1,…,λr)λ(r,s,p):=(λ1,…,λr) of rs  , which depends only on r,sr,s and the characteristic p:=char(F)p:=char(F), has many applications including the study of algebraic groups. We prove new periodicity and duality results for λ(r,s,p)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)λ(r,s,p) for all s and p  , with s⩾rs⩾r and p prime.