1999/06/22 by Patrick Polo · 47 citations
Mathematics · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algorithm #Annotation #Type (biology) #Artificial intelligence #Computer science #Mathematics #Biology
paper · doi:10.1090/s1088-4165-99-00074-6
published in Representation Theory of the American Mathematical Society 3(4), 90-104 (Serbian Mathematical Society)
openalex publication_date 1999/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21
To each polynomial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with integral nonnegative coefficients and constant term equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we associate a certain pair of elements <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis y comma w right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</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">(y,w)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the 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:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">Sn</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="n equals 1 plus d plus upper P left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n = 1 + d + P(1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, such that 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 y comma w"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Py,w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> equals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This pair satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l left-parenthesis w right-parenthesis minus script l left-parenthesis y right-parenthesis equals 2 d plus upper P left-parenthesis 1 right-parenthesis minus 1"> <mml:semantics> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">ℓ (w) - ℓ (y) = 2d + P(1) - 1</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="script l left-parenthesis w right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">ℓ (w)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the number of inversions of <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>.