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• W2123203003 abstract "For permutations <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</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=w> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding=application/x-tex>w</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu left-parenthesis x comma w right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mu (x,w)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the coefficient of highest possible degree in the Kazhdan-Lusztig polynomial <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P Subscript x comma w> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>P_{x,w}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is well-known that the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu left-parenthesis x comma w right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mu (x,w)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> arise as the edge labels of certain graphs encoding the 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:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The 0-1 Conjecture states that the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu left-parenthesis x comma w right-parenthesis element-of StartSet 0 comma 1 EndSet> <mml:semantics> <mml:mrow> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mu (x,w) in {0,1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We present two counterexamples to this conjecture, the first in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S 16> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>16</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>S_{16}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for which <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</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=w> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding=application/x-tex>w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are in the same left cell, and the second in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S 10> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>10</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>S_{10}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The proof of the counterexample in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S 16> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>16</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>S_{16}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> relies on computer calculations." @default.
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• W2123203003 date "2003-05-07" @default.
• W2123203003 modified "2023-09-26" @default.
• W2123203003 title "Counterexamples to the 0-1 Conjecture" @default.
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