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

Let J(r) denote an r x r matrix with minimal and characteristic polynomials (t - 1)(r). Suppose r = ... >=lambda(r), > 0 and Sigma(r)(i=1) lambda(i) = rs. The partition lambda(r, s, p) := (lambda 1, ... , lambda(r)) 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 lambda(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 lambda(r, s, p) for all s and p, with s >= r and p prime. (C) 2015 Elsevier Inc. All rights reserved.