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<p>We consider a derivation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textsf{D}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">D</mi> </math></EquationSource> </InlineEquation> on the ring <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> of symmetric functions and investigate its combinatorial, algebraic and geometric properties. More precisely, we show that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textsf{D}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">D</mi> </math></EquationSource> </InlineEquation> restricts to a quasi-isometry, with respect to the Hall product, on the graded component of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> of each positive degree and provide a chain-rule formula with respect to the plethysm operation. Furthermore, we relate the geometry of the Schur functions supporting <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\textsf{D}(f)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">D</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(f\in \Lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi mathvariant="normal">Λ</mi> </mrow> </math></EquationSource> </InlineEquation> is a homogeneous element, to that of <i>f</i>.</p>

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A Plethystic Chain Rule

  • 期刊论文

  • 发表日期：

    2026年7月28日

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  • Alessandro D’Andrea,
  • Enrico Fatighenti,
  • Claudio Onorati

摘要

We consider a derivation \(\textsf{D}\) D on the ring \(\Lambda \) Λ of symmetric functions and investigate its combinatorial, algebraic and geometric properties. More precisely, we show that \(\textsf{D}\) D restricts to a quasi-isometry, with respect to the Hall product, on the graded component of \(\Lambda \) Λ of each positive degree and provide a chain-rule formula with respect to the plethysm operation. Furthermore, we relate the geometry of the Schur functions supporting \(\textsf{D}(f)\) D ( f ) , where \(f\in \Lambda \) f ∈ Λ is a homogeneous element, to that of f.