Varieties Related to the Problem of Lifting Gr -Modules to G

Paul Sobaje

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

Varieties Related to the Problem of Lifting Gr -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 - Jan 1 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

Scopus Subject Areas

General Mathematics

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 Gr -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

@inproceedings{8b0930c46b414577aed5aa536213ef25,

title = "Varieties Related to the Problem of Lifting Gr -Modules to G",

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",

note = "Publisher Copyright: {\textcopyright} 2018, Springer International Publishing AG.; PIMS Summer School and Workshop on Geometric Methods in the Representation Theory of Finite Groups, 2016 ; Conference date: 27-07-2016 Through 05-08-2016",

year = "2018",

month = jan,

day = "1",

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

language = "English",

isbn = "9783319940328",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer New York LLC",

pages = "377--392",

editor = "Iyengar, \{Srikanth B.\} and Julia Pevtsova and Carlson, \{Jon F.\}",

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

}

Sobaje, P 2018, Varieties Related to the Problem of Lifting Gr -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 Gr -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

TY - GEN

T1 - Varieties Related to the Problem of Lifting Gr -Modules to G

AU - Sobaje, Paul

PY - 2018/1/1

Y1 - 2018/1/1

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 - https://www.scopus.com/pages/publications/85055093723

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 -

Varieties Related to the Problem of Lifting Gr -Modules to G

Abstract

Publication series

Conference

Scopus Subject Areas

Keywords

Access to Document

Other files and links

Fingerprint

Cite this