TY - GEN

T1 - Varieties Related to the Problem of Lifting $$G:r$$ -Modules to G

AU - Sobaje, Paul

N1 - Publisher Copyright:
© 2018, Springer International Publishing AG.

PY - 2018

Y1 - 2018

N2 - Let G be a simple simply connected algebraic group over an algebraically closed field k of characteristic p, with rth Frobenius kernel $$G:r$$. Let M be a $$G:r$$ -module and V a rational G-module. We put a variety structure on the set of all $$G:r$$ -summands of V that are isomorphic to M, and study basic properties of these varieties. This is primarily to set the stage for later work that will bring techniques from geometric invariant theory to bear on the problem of lifting $$G:r$$ -modules to G. However, we do give a few applications of the work in this paper to the representation theory of G, in particular noting that the truth of Donkin’s tilting module conjecture is equivalent to the linearizability of G-actions on certain affine spaces.

AB - Let G be a simple simply connected algebraic group over an algebraically closed field k of characteristic p, with rth Frobenius kernel $$G:r$$. Let M be a $$G:r$$ -module and V a rational G-module. We put a variety structure on the set of all $$G:r$$ -summands of V that are isomorphic to M, and study basic properties of these varieties. This is primarily to set the stage for later work that will bring techniques from geometric invariant theory to bear on the problem of lifting $$G:r$$ -modules to G. However, we do give a few applications of the work in this paper to the representation theory of G, in particular noting that the truth of Donkin’s tilting module conjecture is equivalent to the linearizability of G-actions on certain affine spaces.

KW - 17B10 (primary )

KW - 20G05 (secondary)

UR - http://www.scopus.com/inward/record.url?scp=85055093723&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-94033-5_15

DO - 10.1007/978-3-319-94033-5_15

M3 - Conference article

AN - SCOPUS:85055093723

SN - 9783319940328

T3 - Springer Proceedings in Mathematics and Statistics

SP - 377

EP - 392

BT - Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016

A2 - Iyengar, Srikanth B.

A2 - Pevtsova, Julia

A2 - Carlson, Jon F.

PB - Springer New York LLC

T2 - PIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016

Y2 - 27 July 2016 through 5 August 2016

ER -