Extending the 𝐚𝐛-index
Abstract
<p> We prove for finite, graded, bounded posets, that the Poincaré-extended <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold a bold b"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">a</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">b</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {a}\mathbf {b}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -index is obtained from the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold a bold b"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">a</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">b</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {a}\mathbf {b}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -index via the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi> ω </mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -transformation as conjectured by Dorpalen-Barry, Maglione, and the second author. This <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi> ω </mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -transformation is a generalization of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis bold c"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">c</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbf {c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> - <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 bold d right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">d</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2\mathbf {d})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -index, and it also generalizes the numerical canonical decomposition of the Chow ring. It provides a conceptually new approach to extended <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold a bold b"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">a</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">b</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {a}\mathbf {b}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -indices and Chow polynomials, beyond <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -labeled posets, based on a well-known identity in the incidence algebra. </p>