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

Paul Sobaje

doi:10.1007/978-3-319-94033-5_15

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

Paul Sobaje

Mathematical Sciences

University of Georgia

Research output: Contribution to book or proceeding › Conference article › peer-review

Abstract

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.

Original language	English
Title of host publication	Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016
Editors	Srikanth B. Iyengar, Julia Pevtsova, Jon F. Carlson
Publisher	Springer New York LLC
Pages	377-392
Number of pages	16
ISBN (Print)	9783319940328
DOIs	https://doi.org/10.1007/978-3-319-94033-5_15
State	Published - 2018
Event	PIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016 - Vancouver, Canada Duration: Jul 27 2016 → Aug 5 2016

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	242
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	PIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016
Country/Territory	Canada
City	Vancouver
Period	07/27/16 → 08/5/16

Keywords

17B10 (primary )
20G05 (secondary)

Access to Document

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

Cite this

Sobaje, P. (2018). Varieties Related to the Problem of Lifting $$G:r$$ -Modules to G. In S. B. Iyengar, J. Pevtsova, & J. F. Carlson (Eds.), Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016 (pp. 377-392). (Springer Proceedings in Mathematics and Statistics; Vol. 242). Springer New York LLC. https://doi.org/10.1007/978-3-319-94033-5_15

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abstract = "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{\textquoteright}s tilting module conjecture is equivalent to the linearizability of G-actions on certain affine spaces.",

keywords = "17B10 (primary ), 20G05 (secondary)",

author = "Paul Sobaje",

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year = "2018",

doi = "10.1007/978-3-319-94033-5_15",

language = "English",

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}

Sobaje, P 2018, Varieties Related to the Problem of Lifting $$G:r$$ -Modules to G. in SB Iyengar, J Pevtsova & JF Carlson (eds), Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016. Springer Proceedings in Mathematics and Statistics, vol. 242, Springer New York LLC, pp. 377-392, PIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016, Vancouver, Canada, 07/27/16. https://doi.org/10.1007/978-3-319-94033-5_15

Varieties Related to the Problem of Lifting $$G:r$$ -Modules to G. / Sobaje, Paul.
Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016. ed. / Srikanth B. Iyengar; Julia Pevtsova; Jon F. Carlson. Springer New York LLC, 2018. p. 377-392 (Springer Proceedings in Mathematics and Statistics; Vol. 242).

Research output: Contribution to book or proceeding › Conference article › peer-review

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AU - Sobaje, Paul

PY - 2018

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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.

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KW - 20G05 (secondary)

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Sobaje P. Varieties Related to the Problem of Lifting $$G:r$$ -Modules to G. In Iyengar SB, Pevtsova J, Carlson JF, editors, Geometric and Topological Aspects of the Representation Theory of Finite Groups - PIMS Summer School and Workshop, 2016. Springer New York LLC. 2018. p. 377-392. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-319-94033-5_15

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

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