EQUIVALENCES BETWEEN FUSION SYSTEMS OF FINITE GROUPS OF LIE TYPE

作者：Broto Carles^*; Moller Jesper M; Oliver Bob

来源：Journal of the American Mathematical Society, 2012, 25(1): 1-20.

DOI：10.1090/S0894-0347-2011-00713-3

• groups of Lie type
• fusion systems
• classifying spaces
• p-completion

We prove, for certain pairs<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G comma upper G prime"><mml:semantics><mml:mrow><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>G</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:annotation encoding="application/x-tex">G,G’</mml:annotation></mml:semantics></mml:math>of finite groups of Lie type, that the<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>-fusion systems<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F Subscript p Baseline left-parenthesis upper G right-parenthesis"><mml:semantics><mml:mrow><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi></mml:mrow><mml:mi>p</mml:mi></mml:msub><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">\mathcal {F}_p(G)</mml:annotation></mml:semantics></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F Subscript p Baseline left-parenthesis upper G prime right-parenthesis"><mml:semantics><mml:mrow><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>G</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {F}_p(G’)</mml:annotation></mml:semantics></mml:math>are equivalent. In other words, there is an isomorphism between a Sylow<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>-subgroup of<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>and one of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G prime"><mml:semantics><mml:msup><mml:mi>G</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">G’</mml:annotation></mml:semantics></mml:math>which preserves<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>-fusion. This occurs, for example, when<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals double-struck upper G left-parenthesis q right-parenthesis"><mml:semantics><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">G</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">G=\mathbb {G}(q)</mml:annotation></mml:semantics></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G prime equals double-struck upper G left-parenthesis q prime right-parenthesis"><mml:semantics><mml:mrow><mml:msup><mml:mi>G</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">G</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>q</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">G’=\mathbb {G}(q’)</mml:annotation></mml:semantics></mml:math>for a simple Lie “type”<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">G</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation></mml:semantics></mml:math>, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"><mml:semantics><mml:mi>q</mml:mi><mml:annotation encoding="application/x-tex">q</mml:annotation></mml:semantics></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q prime"><mml:semantics><mml:msup><mml:mi>q</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">q’</mml:annotation></mml:semantics></mml:math>are prime powers, both prime to<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>, which generate the same closed subgroup of<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>-adic units. Our proof uses homotopy-theoretic properties of the<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>-completed classifying spaces of<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>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G prime"><mml:semantics><mml:msup><mml:mi>G</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:annotation encoding="application/x-tex">G’</mml:annotation></mml:semantics></mml:math>, and we know of no purely algebraic proof of this result.

• 出版日期2012-1