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<p>We study the category of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{P}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">P</mi> </math></EquationSource> </InlineEquation>-equivariant modules over the infinite variable polynomial ring, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textbf{P}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">P</mi> </math></EquationSource> </InlineEquation> denotes the subgroup of the infinite general linear group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textbf{GL}(\textbf{C}^\infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">GL</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="bold">C</mi> <mi>∞</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> consisting of elements fixing a flag in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\textbf{C}^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">C</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> with each graded piece infinite-dimensional. We decompose the category into simpler pieces that can be described combinatorially, and prove a number of finiteness results, such as finite generation of local cohomology and rationality of Hilbert series. Furthermore, we show that this category is equivalent to the category of representations of a particular combinatorial category generalizing <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\textbf{FI}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">FI</mi> </math></EquationSource> </InlineEquation>.</p>

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Parabolic-equivariant modules over polynomial rings in infinitely many variables

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  • 发表日期：

    2025年12月6日

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  • Teresa Yu

摘要

We study the category of \(\textbf{P}\) P -equivariant modules over the infinite variable polynomial ring, where \(\textbf{P}\) P denotes the subgroup of the infinite general linear group \(\textbf{GL}(\textbf{C}^\infty )\) GL ( C ∞ ) consisting of elements fixing a flag in \(\textbf{C}^\infty \) C ∞ with each graded piece infinite-dimensional. We decompose the category into simpler pieces that can be described combinatorially, and prove a number of finiteness results, such as finite generation of local cohomology and rationality of Hilbert series. Furthermore, we show that this category is equivalent to the category of representations of a particular combinatorial category generalizing \(\textbf{FI}\) FI .