TY - JOUR

T1 - Decomposing modular tensor products, and periodicity of 'Jordan partitions'

AU - GLASBY, Stephen

AU - Praeger, Cheryl

AU - Xia, Binzhou

N1 - Funding Information:
We would like to thank Gary Seitz and Martin Liebeck for helpful conversations. We thank Neil Strickland for his answer to a question posed on MathOverflow 5 5 by the third author. We also thank Michael Barry for showing us his paper [2] . The first and second authors acknowledge the support of the Australian Research Council Discovery Grant DP110101153 . This work was done during the visit of the third author to School of Mathematics and Statistics, University of Western Australia, and he would like to thank the China Scholarship Council for its financial support.
Publisher Copyright:
© 2015 Elsevier Inc.

PY - 2016/3/15

Y1 - 2016/3/15

N2 - 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 J¿1¿¿¿J¿r where ¿1=¿=¿r>0 and ¿i=1r¿i=rs. The partition ¿(r, s, p):=(¿1, . . ., ¿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 ¿(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 ¿(r, s, p) for all s and p, with s=r and p prime.

AB - 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 J¿1¿¿¿J¿r where ¿1=¿=¿r>0 and ¿i=1r¿i=rs. The partition ¿(r, s, p):=(¿1, . . ., ¿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 ¿(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 ¿(r, s, p) for all s and p, with s=r and p prime.

KW - Green ring

KW - Jordan blocks

KW - Jordan canonical form

KW - Jordan partition

KW - Tensor product

UR - http://www.scopus.com/inward/record.url?scp=85000577851&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/decomposing-modular-tensor-products-periodicity-jordan-partitions

U2 - 10.1016/j.jalgebra.2015.11.025

DO - 10.1016/j.jalgebra.2015.11.025

M3 - Article

VL - 450

SP - 570

EP - 587

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -