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
SN - 0021-8693
VL - 450
SP - 570
EP - 587
JO - Journal of Algebra
JF - Journal of Algebra
ER -