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<p>Based on the vertex operator realization of the Schur functions, a determinant-type plethystic Murnaghan–Nakayama rule is obtained and utilized to derive a general formula of the expansion coefficients of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(s_{\nu }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>s</mi> <mi>ν</mi> </msub> </math></EquationSource> </InlineEquation> in the plethysm product <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((p_{n}\circ h_{k})s_{\mu }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>∘</mo> <msub> <mi>h</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mi>s</mi> <mi>μ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. Meanwhile, the equivalence between our algebraic rule and the combinatorial one is also established. As an application, we provide a simple way to compute the generalized Waring formula.</p>

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Plethystic Murnaghan–Nakayama rule via vertex operators

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月7日

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  • Yue Cao,
  • Naihuan Jing,
  • Ning Liu

摘要

Based on the vertex operator realization of the Schur functions, a determinant-type plethystic Murnaghan–Nakayama rule is obtained and utilized to derive a general formula of the expansion coefficients of \(s_{\nu }\) s ν in the plethysm product \((p_{n}\circ h_{k})s_{\mu }\) ( p n ∘ h k ) s μ . Meanwhile, the equivalence between our algebraic rule and the combinatorial one is also established. As an application, we provide a simple way to compute the generalized Waring formula.