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<p>The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudo-reflection groups over regular domains. More precisely, let <i>A</i> be a regular domain and let <i>K</i> be its field of fractions. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(G\subseteq GL_n(A)\)</EquationSource> </InlineEquation> be a finite group. Let <i>G</i> act linearly on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(A[X_1,X_2,\dots , X_n]\)</EquationSource> </InlineEquation> (fixing <i>A</i>). Assume that |<i>G</i>| is invertible in <i>A</i>. We prove that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(G\subseteq GL_n(K)\)</EquationSource> </InlineEquation> is generated by pseudo-reflections if and only if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((A[X_1,X_2,\dots , X_n])^G\)</EquationSource> </InlineEquation> is regular.</p>

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The Regular Property of Invariant Rings Over Regular Domains

  • 期刊论文

  • 发表日期：

    2026年3月19日

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  • Shubham Jaiswal,
  • Tony J. Puthenpurakal

摘要

The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudo-reflection groups over regular domains. More precisely, let A be a regular domain and let K be its field of fractions. Let \(G\subseteq GL_n(A)\) be a finite group. Let G act linearly on \(A[X_1,X_2,\dots , X_n]\) (fixing A). Assume that |G| is invertible in A. We prove that \(G\subseteq GL_n(K)\) is generated by pseudo-reflections if and only if \((A[X_1,X_2,\dots , X_n])^G\) is regular.