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

JOURNAL OF ALGEBRA

卷:450

Article

Glasby, S. P.¹  Praeger, Cheryl E.¹  Xia, Binzhou² 

[1] Univ Western Australia, Ctr Math Symmetry & Computat, 35 Stirling Highway, Crawley 6009, Australia

[2] Peking Univ, Beijing Int Ctr Math Res, Beijing 100871, Peoples R China

关键词: Tensor product;    Jordan blocks;    Jordan canonical form;    Jordan partition;    Green ring;   

DOI  :  10.1016/j.jalgebra.2015.11.025

来源: Elsevier

【 摘 要 】

Let J(r) denote an r x 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 J(r) circle times J(s), is similar to J(lambda 1) circle plus ... circle plus J(lambda r), where lambda(1) >= ... >=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.

【 授权许可】

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