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• W1946114473 abstract "We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a <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>-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p minus> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>p-</mml:annotation> </mml:semantics> </mml:math> </inline-formula>blocks remained unclassified were the alternating groups <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A Subscript n> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>A_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here we show that these all have a <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>-block with defect 0 for every prime <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than-or-equal-to 5> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>pgeq 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This follows from proving the same result for every symmetric group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript n> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>S_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which in turn follows as a consequence of the <italic><inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding=application/x-tex>t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-core partition conjecture</italic>, that every non-negative integer possesses at least one <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding=application/x-tex>t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-core partition, for any <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t greater-than-or-equal-to 4> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>tgeq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t greater-than-or-equal-to 17> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>17</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>tgeq 17</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we reduce this problem to Lagrange’s Theorem that every non-negative integer can be written as the sum of four squares. The only case with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t greater-than 17> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>17</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>t>17</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that was not covered in previous work, was the case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t equals 13> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mn>13</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>t=13</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This we prove with a very different argument, by interpreting the generating function for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding=application/x-tex>t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne’s Theorem (née the <italic>Weil Conjectures</italic>). We also consider congruences for the number of <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>-blocks of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript n> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>S_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, proving a conjecture of Garvan, that establishes certain multiplicative congruences when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=5 less-than-or-equal-to p less-than-or-equal-to 23> <mml:semantics> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>23</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>5leq p leq 23</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime <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> and positive integer <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=application/x-tex>m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the number of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p minus> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>p-</mml:annotation> </mml:semantics> </mml:math> </inline-formula>blocks with defect 0 in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript n> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a multiple of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=application/x-tex>m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for almost all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also establish that any given prime <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> divides the number of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p minus> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>p-</mml:annotation> </mml:semantics> </mml:math> </inline-formula>modularly irreducible representations of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript n> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>S_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for almost all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W1946114473 created "2016-06-24" @default.
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• W1946114473 date "1996-01-01" @default.
• W1946114473 modified "2023-09-28" @default.
• W1946114473 title "Defect zero blocks for finite simple groups" @default.
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