Reductive pairs arising from representations

TY - JOUR

T1 - Reductive pairs arising from representations

AU - Goodbourn, Oliver

PY - 2016/6/1

Y1 - 2016/6/1

N2 - Let G be a reductive algebraic group and V a G-module. We consider the question of when (GL(V), ρ(G)) is a reductive pair of algebraic groups, where ρ is the representation afforded by V. We first make some observations about general G and V, then specialise to the group SL2(K) with K algebraically closed of positive characteristic p. For this group we provide complete answers for the classes of simple and Weyl modules, the behaviour being determined by the base p expansion of the highest weight of the module. We conclude by illustrating some of the results from the first section with examples for the group SL3(K).

AB - Let G be a reductive algebraic group and V a G-module. We consider the question of when (GL(V), ρ(G)) is a reductive pair of algebraic groups, where ρ is the representation afforded by V. We first make some observations about general G and V, then specialise to the group SL2(K) with K algebraically closed of positive characteristic p. For this group we provide complete answers for the classes of simple and Weyl modules, the behaviour being determined by the base p expansion of the highest weight of the module. We conclude by illustrating some of the results from the first section with examples for the group SL3(K).

KW - Reductive pairs of algebraic groups

KW - Representations of algebraic groups

UR - http://www.scopus.com/inward/record.url?scp=84959449410&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2016.02.006

DO - 10.1016/j.jalgebra.2016.02.006

M3 - Article

AN - SCOPUS:84959449410

SN - 0021-8693

VL - 455

SP - 93

EP - 107

JO - Journal of Algebra

JF - Journal of Algebra

ER -