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<p>In this work, we investigate a novel approach to the Combinatorial Invariance Conjecture of Kazhdan–Lusztig polynomials for the symmetric group. Using the new concept of flipclasses, we introduce some combinatorial invariants of intervals in the symmetric group whose analysis leads us to a recipe to compute the coefficients of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(q^h\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mi>h</mi> </msup> </math></EquationSource> </InlineEquation> of the Kazhdan–Lusztig <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\widetilde{R}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>R</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>-polynomials, for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(h\le 6\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>≤</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>. This recipe depends only on the isomorphism class (as a poset) of the interval indexing the polynomial and thus provides new evidence for the Combinatorial Invariance Conjecture.</p>

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Flipclasses and Combinatorial Invariance for Kazhdan–Lusztig polynomials

  • 期刊论文

  • 发表日期：

    2025年10月22日

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  • Francesco Esposito,
  • Mario Marietti

摘要

In this work, we investigate a novel approach to the Combinatorial Invariance Conjecture of Kazhdan–Lusztig polynomials for the symmetric group. Using the new concept of flipclasses, we introduce some combinatorial invariants of intervals in the symmetric group whose analysis leads us to a recipe to compute the coefficients of \(q^h\) q h of the Kazhdan–Lusztig \({\widetilde{R}}\) R ~ -polynomials, for \(h\le 6\) h ≤ 6 . This recipe depends only on the isomorphism class (as a poset) of the interval indexing the polynomial and thus provides new evidence for the Combinatorial Invariance Conjecture.