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• W1594896360 abstract "Let<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding=application/x-tex>G</mml:annotation></mml:semantics></mml:math></inline-formula>be a semisimple simply connected algebraic group defined and split over the field<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper F Subscript p><mml:semantics><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>F</mml:mi></mml:mrow></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding=application/x-tex>{mathbb {F}}_p</mml:annotation></mml:semantics></mml:math></inline-formula>with<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding=application/x-tex>p</mml:annotation></mml:semantics></mml:math></inline-formula>elements, let<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G left-parenthesis double-struck upper F Subscript q Baseline right-parenthesis><mml:semantics><mml:mrow><mml:mi>G</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>F</mml:mi></mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>G(mathbb {F}_{q})</mml:annotation></mml:semantics></mml:math></inline-formula>be the finite Chevalley group consisting of the<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper F Subscript q><mml:semantics><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>F</mml:mi></mml:mrow></mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:annotation encoding=application/x-tex>{mathbb {F}}_{q}</mml:annotation></mml:semantics></mml:math></inline-formula>-rational points of<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding=application/x-tex>G</mml:annotation></mml:semantics></mml:math></inline-formula>where<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q equals p Superscript r><mml:semantics><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mi>r</mml:mi></mml:msup></mml:mrow><mml:annotation encoding=application/x-tex>q = p^r</mml:annotation></mml:semantics></mml:math></inline-formula>, and let<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G Subscript r><mml:semantics><mml:msub><mml:mi>G</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:annotation encoding=application/x-tex>G_{r}</mml:annotation></mml:semantics></mml:math></inline-formula>be the<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r><mml:semantics><mml:mi>r</mml:mi><mml:annotation encoding=application/x-tex>r</mml:annotation></mml:semantics></mml:math></inline-formula>th Frobenius kernel. The purpose of this paper is to relate extensions between modules in<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Mod left-parenthesis upper G left-parenthesis double-struck upper F Subscript q Baseline right-parenthesis right-parenthesis><mml:semantics><mml:mrow><mml:mtext>Mod</mml:mtext><mml:mo stretchy=false>(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>F</mml:mi></mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=false>)</mml:mo><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>text {Mod}(G(mathbb {F}_{q}))</mml:annotation></mml:semantics></mml:math></inline-formula>and<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Mod left-parenthesis upper G Subscript r Baseline right-parenthesis><mml:semantics><mml:mrow><mml:mtext>Mod</mml:mtext><mml:mo stretchy=false>(</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>text {Mod}(G_{r})</mml:annotation></mml:semantics></mml:math></inline-formula>with extensions between modules in<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Mod left-parenthesis upper G right-parenthesis><mml:semantics><mml:mrow><mml:mtext>Mod</mml:mtext><mml:mo stretchy=false>(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>text {Mod}(G)</mml:annotation></mml:semantics></mml:math></inline-formula>. Among the results obtained are the following: for<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r greater-than 2><mml:semantics><mml:mrow><mml:mi>r</mml:mi><mml:mo>></mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>r >2</mml:annotation></mml:semantics></mml:math></inline-formula>and<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than-or-equal-to 3 left-parenthesis h minus 1 right-parenthesis><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>≥<!-- ≥ --></mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=false>(</mml:mo><mml:mi>h</mml:mi><mml:mo>−<!-- − --></mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>pgeq 3(h-1)</mml:annotation></mml:semantics></mml:math></inline-formula>, the<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G left-parenthesis double-struck upper F Subscript q Baseline right-parenthesis><mml:semantics><mml:mrow><mml:mi>G</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>F</mml:mi></mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>G(mathbb {F}_{q})</mml:annotation></mml:semantics></mml:math></inline-formula>-extensions between two simple<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G left-parenthesis double-struck upper F Subscript q Baseline right-parenthesis><mml:semantics><mml:mrow><mml:mi>G</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>F</mml:mi></mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>G(mathbb {F}_{q})</mml:annotation></mml:semantics></mml:math></inline-formula>-modules are isomorphic to the<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding=application/x-tex>G</mml:annotation></mml:semantics></mml:math></inline-formula>-extensions between two simple<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p Superscript r><mml:semantics><mml:msup><mml:mi>p</mml:mi><mml:mi>r</mml:mi></mml:msup><mml:annotation encoding=application/x-tex>p^r</mml:annotation></mml:semantics></mml:math></inline-formula>-restricted<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding=application/x-tex>G</mml:annotation></mml:semantics></mml:math></inline-formula>-modules with suitably “twisted highest weights. For<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than-or-equal-to 3 left-parenthesis h minus 1 right-parenthesis><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>≥<!-- ≥ --></mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=false>(</mml:mo><mml:mi>h</mml:mi><mml:mo>−<!-- − --></mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>p geq 3(h-1)</mml:annotation></mml:semantics></mml:math></inline-formula>, we provide a complete characterization of<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 1 Baseline left-parenthesis upper G left-parenthesis double-struck upper F Subscript q Baseline right-parenthesis comma upper H Superscript 0 Baseline left-parenthesis lamda right-parenthesis right-parenthesis><mml:semantics><mml:mrow><mml:msup><mml:mtext>H</mml:mtext><mml:mrow class=MJX-TeXAtom-ORD><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=false>(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>F</mml:mi></mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=false>)</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=false>(</mml:mo><mml:mi>λ<!-- λ --></mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>text {H}^{1}(G(mathbb {F}_{q}),H^{0}(lambda ))</mml:annotation></mml:semantics></mml:math></inline-formula>where<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 0 Baseline left-parenthesis lamda right-parenthesis equals ind Subscript upper B Superscript upper G Baseline lamda><mml:semantics><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=false>(</mml:mo><mml:mi>λ<!-- λ --></mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mtext>ind</mml:mtext><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>B</mml:mi></mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>G</mml:mi></mml:mrow></mml:msubsup><mml:mtext> </mml:mtext><mml:mi>λ<!-- λ --></mml:mi></mml:mrow><mml:annotation encoding=application/x-tex>H^{0}(lambda )=text {ind}_{B}^{G} lambda</mml:annotation></mml:semantics></mml:math></inline-formula>and<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda><mml:semantics><mml:mi>λ<!-- λ --></mml:mi><mml:annotation encoding=application/x-tex>lambda</mml:annotation></mml:semantics></mml:math></inline-formula>is<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p Superscript r><mml:semantics><mml:msup><mml:mi>p</mml:mi><mml:mi>r</mml:mi></mml:msup><mml:annotation encoding=application/x-tex>p^r</mml:annotation></mml:semantics></mml:math></inline-formula>-restricted. Furthermore, for<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than-or-equal-to 3 left-parenthesis h minus 1 right-parenthesis><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>≥<!-- ≥ --></mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=false>(</mml:mo><mml:mi>h</mml:mi><mml:mo>−<!-- − --></mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>p geq 3(h-1)</mml:annotation></mml:semantics></mml:math></inline-formula>, necessary and sufficient bounds on the size of the highest weight of a<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding=application/x-tex>G</mml:annotation></mml:semantics></mml:math></inline-formula>-module<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V><mml:semantics><mml:mi>V</mml:mi><mml:annotation encoding=application/x-tex>V</mml:annotation></mml:semantics></mml:math></inline-formula>are given to insure that the restriction map<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper H Superscript 1 Baseline left-parenthesis upper G comma upper V right-parenthesis right-arrow normal upper H Superscript 1 Baseline left-parenthesis upper G left-parenthesis double-struck upper F Subscript q Baseline right-parenthesis comma upper V right-parenthesis><mml:semantics><mml:mrow><mml:msup><mml:mi mathvariant=normal>H</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>⁡<!-- ⁡ --></mml:mo><mml:mo stretchy=false>(</mml:mo><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo stretchy=false>→<!-- → --></mml:mo><mml:msup><mml:mi mathvariant=normal>H</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>⁡<!-- ⁡ --></mml:mo><mml:mo stretchy=false>(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:msub><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>F</mml:mi></mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=false>)</mml:mo><mml:mo>,</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>operatorname {H}^{1}(G,V)rightarrow operatorname {H}^{1}(G(mathbb {F}_{q}),V)</mml:annotation></mml:semantics></mml:math></inline-formula>is an isomorphism. Finally, it is shown that the extensions between two simple<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p Superscript r><mml:semantics><mml:msup><mml:mi>p</mml:mi><mml:mi>r</mml:mi></mml:msup><mml:annotation encoding=application/x-tex>p^r</mml:annotation></mml:semantics></mml:math></inline-formula>-restricted<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding=application/x-tex>G</mml:annotation></mml:semantics></mml:math></inline-formula>-modules coincide in all three categories provided the highest weights are “close together." @default.
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• W1594896360 title "Extensions for finite Chevalley groups II" @default.
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