On Medium-Rank Lie Primitive and Maximal Subgroups of Exceptional Groups of Lie Type

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2021-02-22 DOI:10.1090/memo/1434

David A. Craven

{"title":"On Medium-Rank Lie Primitive and Maximal Subgroups of Exceptional Groups of Lie Type","authors":"David A. Craven","doi":"10.1090/memo/1434","DOIUrl":null,"url":null,"abstract":"<p>We study embeddings of groups of Lie type <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\n <mml:semantics>\n <mml:mi>H</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> into exceptional algebraic groups <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper G\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"bold\">G</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbf {G}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of the same characteristic. We exclude the case where <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\n <mml:semantics>\n <mml:mi>H</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is of type <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal upper L Subscript 2\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">P</mml:mi>\n <mml:mi mathvariant=\"normal\">S</mml:mi>\n <mml:mi mathvariant=\"normal\">L</mml:mi>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {PSL}_2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. A subgroup of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper G\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"bold\">G</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbf {G}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is <italic>Lie primitive</italic> if it is not contained in any proper, positive-dimensional subgroup of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper G\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"bold\">G</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbf {G}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>\n\n<p>With a few possible exceptions, we prove that there are no Lie primitive subgroups <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\n <mml:semantics>\n <mml:mi>H</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper G\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"bold\">G</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbf {G}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, with the conditions on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\n <mml:semantics>\n <mml:mi>H</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper G\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"bold\">G</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbf {G}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> given above. The exceptions are for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\n <mml:semantics>\n <mml:mi>H</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> one of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal upper L Subscript 3 Baseline left-parenthesis 3 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">P</mml:mi>\n <mml:mi mathvariant=\"normal\">S</mml:mi>\n <mml:mi mathvariant=\"normal\">L</mml:mi>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>3</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {PSL}_3(3)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal upper U Subscript 3 Baseline left-parenthesis 3 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">P</mml:mi>\n <mml:mi mathvariant=\"normal\">S</mml:mi>\n <mml:mi mathvariant=\"normal\">U</mml:mi>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>3</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {PSU}_3(3)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal upper L Subscript 3 Baseline left-parenthesis 4 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">P</mml:mi>\n <mml:mi mathvariant=\"normal\">S</mml:mi>\n <mml:mi mathvariant=\"normal\">L</mml:mi>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>4</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {PSL}_3(4)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal upper U Subscript 3 Baseline left-parenthesis 4 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">P</mml:mi>\n <mml:mi mathvariant=\"normal\">S</mml:mi>\n <mml:mi mathvariant=\"normal\">U</mml:mi>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>4</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {PSU}_3(4)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal upper U Subscript 3 Baseline left-parenthesis 8 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">P</mml:mi>\n <mml:mi mathvariant=\"normal\">S</mml:mi>\n <mml:mi mathvariant=\"normal\">U</mml:mi>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>8</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {PSU}_3(8)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal upper U Subscript 4 Baseline left-parenthesis 2 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">P</mml:mi>\n <mml:mi mathvariant=\"normal\">S</mml:mi>\n <mml:mi mathvariant=\"normal\">U</mml:mi>\n </mml:mrow>\n <mml:mn>4</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {PSU}_4(2)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal upper S normal p Subscript 4 Baseline left-parenthesis 2 right-parenthesis prime\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">P</mml:mi>\n <mml:mi mathvariant=\"normal\">S</mml:mi>\n <mml:mi mathvariant=\"normal\">p</mml:mi>\n </mml:mrow>\n <mml:mn>4</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>′</mml:mo>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio 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引用次数: 5

Abstract

We study embeddings of groups of Lie type H H in characteristic p p into exceptional algebraic groups G \mathbf {G} of the same characteristic. We exclude the case where H H is of type P S L 2 \mathrm {PSL}_2 . A subgroup of G \mathbf {G} is Lie primitive if it is not contained in any proper, positive-dimensional subgroup of G \mathbf {G} .

With a few possible exceptions, we prove that there are no Lie primitive subgroups H H in G \mathbf {G} , with the conditions on H H and G \mathbf {G} given above. The exceptions are for H H one of P S L 3 ( 3 ) \mathrm {PSL}_3(3) , P S U 3 ( 3 ) \mathrm {PSU}_3(3) , P S L 3 ( 4 ) \mathrm {PSL}_3(4) , P S U 3 ( 4 ) \mathrm {PSU}_3(4) , P S U 3 ( 8 ) \mathrm {PSU}_3(8) , P S U 4 ( 2 ) \mathrm {PSU}_4(2) , P S p 4 ( 2 ) ′ \mathrm {

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李型例外群的中秩李基元和极大子群

我们研究了特征p p中H H李型群嵌入到相同特征的例外代数群G\mathbf｛G｝中的问题。我们排除了H H是P S L 2 \ mathrm类型的情况{PSL}_2。如果G\mathbf｛G｝的子群不包含在G\mathbf｛G｛的任何正维子群中，则它是李原子群。除了几个可能的例外，我们证明了在G\math bf｛｝中不存在李原子群H H，并且给出了关于H H和G\math BF｛G｝的条件。例外情况是P S L 3（3）\mathrm的H H之一{PSL}_3（3），P S U 3（3）\数学{PSU}_3（3），P S L 3（4）\数学{PSL}_3（4），P S U 3（4）\数学{PSU}_3（4），P S U 3（8）\数学{PSU}_3（8），P S U 4（2）\数学{PSU}_4（2），P S P 4（2）′\数学{

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来源期刊

ACS Applied Bio Materials Chemistry-Chemistry (all)

CiteScore

9.40

自引率

2.10%

发文量

464