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

Stephen GLASBY, Cheryl Praeger, Binzhou Xia

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)570-587
Number of pages18
JournalJournal of Algebra
Volume450
DOIs
Publication statusPublished - 15 Mar 2016
Externally publishedYes

Fingerprint

Periodicity
Tensor Product
Partition
Jordan Canonical Form
Minimal polynomial
Characteristic polynomial
Algebraic Groups
Table
Duality
Denote
Imply
Generalise
Integer

Cite this

GLASBY, Stephen ; Praeger, Cheryl ; Xia, Binzhou. / Decomposing modular tensor products, and periodicity of 'Jordan partitions'. In: Journal of Algebra. 2016 ; Vol. 450. pp. 570-587.
@article{e99db959b032402b81d86622f1105eb9,
title = "Decomposing modular tensor products, and periodicity of 'Jordan partitions'",
abstract = "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.",
keywords = "Green ring, Jordan blocks, Jordan canonical form, Jordan partition, Tensor product",
author = "Stephen GLASBY and Cheryl Praeger and Binzhou Xia",
year = "2016",
month = "3",
day = "15",
doi = "10.1016/j.jalgebra.2015.11.025",
language = "English",
volume = "450",
pages = "570--587",
journal = "Journal of Algebra",
issn = "0021-8693",
publisher = "Academic Press Inc.",

}

Decomposing modular tensor products, and periodicity of 'Jordan partitions'. / GLASBY, Stephen; Praeger, Cheryl; Xia, Binzhou.

In: Journal of Algebra, Vol. 450, 15.03.2016, p. 570-587.

Research output: Contribution to journalArticle

TY - JOUR

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

AU - GLASBY, Stephen

AU - Praeger, Cheryl

AU - Xia, Binzhou

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 -