On the normality of commuting scheme for general linear Lie algebra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908370434260992 |
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| author | Sheshmani, Artan Xia, Xiaopeng Yuan, Beihui |
| author_facet | Sheshmani, Artan Xia, Xiaopeng Yuan, Beihui |
| contents | The commuting scheme $\mathfrak{C}^{d}_{\mathfrak{g}}$ for reductive Lie algebra $\mathfrak{g}$ over an algebraically closed field $\mathbb{K}$ is the subscheme of $\mathfrak{g}^{d}$ defined by quadratic equations, whose $\mathbb{K}$-valued points are $d$-tuples of commuting elements in $\mathfrak{g}$ over $\mathbb{K}$. There is a long-standing conjecture that the commuting scheme $\mathfrak{C}^{d}_{\mathfrak{g}}$ is reduced. Moreover, a higher dimensional analog of Chevalley restriction conjecture was conjectured by Chen-Ngô. We show that the commuting scheme of $\mathfrak{C}^{2}_{\mathfrak{g}l_{n}}$ is Cohen-Macaulay and normal. As a corollary, we prove a 2-dimensional Chevalley restriction theorem for general linear group in positive characteristic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_13013 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the normality of commuting scheme for general linear Lie algebra Sheshmani, Artan Xia, Xiaopeng Yuan, Beihui Algebraic Geometry Commutative Algebra Representation Theory Primary 14L30, Secondary 13C14, 13H10, 14B25, 15A30, 20G05 The commuting scheme $\mathfrak{C}^{d}_{\mathfrak{g}}$ for reductive Lie algebra $\mathfrak{g}$ over an algebraically closed field $\mathbb{K}$ is the subscheme of $\mathfrak{g}^{d}$ defined by quadratic equations, whose $\mathbb{K}$-valued points are $d$-tuples of commuting elements in $\mathfrak{g}$ over $\mathbb{K}$. There is a long-standing conjecture that the commuting scheme $\mathfrak{C}^{d}_{\mathfrak{g}}$ is reduced. Moreover, a higher dimensional analog of Chevalley restriction conjecture was conjectured by Chen-Ngô. We show that the commuting scheme of $\mathfrak{C}^{2}_{\mathfrak{g}l_{n}}$ is Cohen-Macaulay and normal. As a corollary, we prove a 2-dimensional Chevalley restriction theorem for general linear group in positive characteristic. |
| title | On the normality of commuting scheme for general linear Lie algebra |
| topic | Algebraic Geometry Commutative Algebra Representation Theory Primary 14L30, Secondary 13C14, 13H10, 14B25, 15A30, 20G05 |
| url | https://arxiv.org/abs/2505.13013 |