dorsal/arxiv
View SchemaCategories of split filtrations and graded quiver varieties
| Authors | Ricardo Canesin |
|---|---|
| Categories | |
| ArXiv ID | 2601.09509vv1 |
| URL | https://arxiv.org/abs/2601.09509 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
By the work of Hernandez-Leclerc, Leclerc-Plamondon, and Keller-Scherotzke, affine graded Nakajima quiver varieties associated with a Dynkin quiver $Q$ admit an algebraic description in terms of modules over the singular Nakajima category $\mathcal{S}$ and a stratification functor to the derived category of $Q$. In this paper, we extend this framework to Nakajima's $n$-fold affine graded tensor product varieties, which allow one to geometrically realize $n$-fold tensor products of standard modules over the quantum affine algebra. We introduce a category of filtrations with splitting of length $n$ of modules over a category and show that it is equivalent to the module category of a triangular matrix category. Applied to the singular Nakajima category, this yields a category $\mathcal{S}^{n\operatorname{-filt}}$ whose modules are parametrized by the points of the $n$-fold tensor product varieties. Generalizing the results of Keller-Scherotzke from $\mathcal{S}$ to $\mathcal{S}^{n\operatorname{-filt}}$, we prove that the stable category of finitely generated Gorenstein projective $\mathcal{S}^{n\operatorname{-filt}}$-modules is triangle equivalent to the derived category of the algebra of $n \times n$ upper triangular matrices over the path algebra of $Q$, and we obtain a corresponding stratification functor.
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"abstract": "By the work of Hernandez-Leclerc, Leclerc-Plamondon, and Keller-Scherotzke, affine graded Nakajima quiver varieties associated with a Dynkin quiver $Q$ admit an algebraic description in terms of modules over the singular Nakajima category $\\mathcal{S}$ and a stratification functor to the derived category of $Q$. In this paper, we extend this framework to Nakajima\u0027s $n$-fold affine graded tensor product varieties, which allow one to geometrically realize $n$-fold tensor products of standard modules over the quantum affine algebra. We introduce a category of filtrations with splitting of length $n$ of modules over a category and show that it is equivalent to the module category of a triangular matrix category. Applied to the singular Nakajima category, this yields a category $\\mathcal{S}^{n\\operatorname{-filt}}$ whose modules are parametrized by the points of the $n$-fold tensor product varieties. Generalizing the results of Keller-Scherotzke from $\\mathcal{S}$ to $\\mathcal{S}^{n\\operatorname{-filt}}$, we prove that the stable category of finitely generated Gorenstein projective $\\mathcal{S}^{n\\operatorname{-filt}}$-modules is triangle equivalent to the derived category of the algebra of $n \\times n$ upper triangular matrices over the path algebra of $Q$, and we obtain a corresponding stratification functor.",
"arxiv_id": "2601.09509",
"authors": [
"Ricardo Canesin"
],
"categories": [
"math.RT"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Categories of split filtrations and graded quiver varieties",
"url": "https://arxiv.org/abs/2601.09509",
"version": "v1"
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