Parabolic-equivariant modules over polynomial rings in infinitely many variables
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911942330810368 |
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| author | Yu, Teresa |
| author_facet | Yu, Teresa |
| contents | We study the category of $\mathbf{P}$-equivariant modules over the infinite variable polynomial ring, where $\mathbf{P}$ denotes the subgroup of the infinite general linear group $\mathbf{GL}(\mathbf{C}^\infty)$ consisting of elements fixing a flag in $\mathbf{C}^\infty$ 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 $\mathbf{FI}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02588 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parabolic-equivariant modules over polynomial rings in infinitely many variables Yu, Teresa Commutative Algebra Representation Theory 13A50, 13E05 We study the category of $\mathbf{P}$-equivariant modules over the infinite variable polynomial ring, where $\mathbf{P}$ denotes the subgroup of the infinite general linear group $\mathbf{GL}(\mathbf{C}^\infty)$ consisting of elements fixing a flag in $\mathbf{C}^\infty$ 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 $\mathbf{FI}$. |
| title | Parabolic-equivariant modules over polynomial rings in infinitely many variables |
| topic | Commutative Algebra Representation Theory 13A50, 13E05 |
| url | https://arxiv.org/abs/2407.02588 |