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Infiniteness of double coset collections in algebraic groups

Let G be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups X and P, is the double coset collection X⧹ G/ P finite or infinite? We limit ourselves to the case where X is maximal rank and redu... Full description

Journal Title: Journal of Algebra 2004, Vol.273(2), pp.718-733
Main Author: Duckworth, W.Ethan
Format: Electronic Article Electronic Article
Language: English
Subjects:
Quelle: ScienceDirect Journals (Elsevier)
ID: ISSN: 0021-8693 ; E-ISSN: 1090-266X ; DOI: 10.1016/j.jalgebra.2003.08.011
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recordid: elsevier_sdoi_10_1016_j_jalgebra_2003_08_011
title: Infiniteness of double coset collections in algebraic groups
format: Article
creator:
  • Duckworth, W.Ethan
subjects:
  • Algebraic Groups
  • Finite Groups of Lie Type
  • Double Cosets
  • Spherical Subgroups
  • Finite Orbit Modules
  • Algebraic Groups
  • Finite Groups of Lie Type
  • Double Cosets
  • Spherical Subgroups
  • Finite Orbit Modules
  • Mathematics
ispartof: Journal of Algebra, 2004, Vol.273(2), pp.718-733
description: Let G be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups X and P, is the double coset collection X⧹ G/ P finite or infinite? We limit ourselves to the case where X is maximal rank and reductive and P parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those X which are spherical subgroups. Finally, excluding a case in F 4, we show that if X⧹ G/ P is finite then X is spherical or the Levi factor of P is spherical. This places great restrictions on X and P for X⧹ G/ P to be finite. The primary method is to descend to calculations at the finite group level and then to use elementary character theory.
language: eng
source: ScienceDirect Journals (Elsevier)
identifier: ISSN: 0021-8693 ; E-ISSN: 1090-266X ; DOI: 10.1016/j.jalgebra.2003.08.011
fulltext: fulltext
issn:
  • 0021-8693
  • 00218693
  • 1090-266X
  • 1090266X
url: Link


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descriptionLet G be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups X and P, is the double coset collection X⧹ G/ P finite or infinite? We limit ourselves to the case where X is maximal rank and reductive and P parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those X which are spherical subgroups. Finally, excluding a case in F 4, we show that if X⧹ G/ P is finite then X is spherical or the Levi factor of P is spherical. This places great restrictions on X and P for X⧹ G/ P to be finite. The primary method is to descend to calculations at the finite group level and then to use elementary character theory.
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Let G be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups X and P, is the double coset collection X⧹ G/ P finite or infinite? We limit ourselves to the case where X is maximal rank and reductive and P parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those X which are spherical subgroups. Finally, excluding a case in F 4, we show that if X⧹ G/ P is finite then X is spherical or the Levi factor of P is spherical. This places great restrictions on X and P for X⧹ G/ P to be finite. The primary method is to descend to calculations at the finite group level and then to use elementary character theory.

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Let G be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups X and P, is the double coset collection X⧹ G/ P finite or infinite? We limit ourselves to the case where X is maximal rank and reductive and P parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those X which are spherical subgroups. Finally, excluding a case in F 4, we show that if X⧹ G/ P is finite then X is spherical or the Levi factor of P is spherical. This places great restrictions on X and P for X⧹ G/ P to be finite. The primary method is to descend to calculations at the finite group level and then to use elementary character theory.

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