The equivariant cohomology ring of the representation variety $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C}))$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866909981595402240 |
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| author | Gritschacher, Simon |
| author_facet | Gritschacher, Simon |
| contents | We give a presentation of the $\mathrm{GL}_n(\mathbb{C})$-equivariant cohomology ring with $\mathbb{Z}$-coefficients of the variety $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C})) \subseteq \mathrm{GL}_n(\mathbb{C})^2$ for any $n$. It is torsion free and minimally generated as a $H^\ast B\mathrm{GL}_n(\mathbb{C})$-algebra by $3n$ elements. The ideal of relations is the saturation of an $n$-generator ideal by even powers of the Vandermonde polynomial. For coefficients in a field whose characteristic does not divide $n!$, we also give a presentation of the non-equivariant cohomology ring of $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C}))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00683 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The equivariant cohomology ring of the representation variety $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C}))$ Gritschacher, Simon Algebraic Topology 58D27, 55N91, 55N45 We give a presentation of the $\mathrm{GL}_n(\mathbb{C})$-equivariant cohomology ring with $\mathbb{Z}$-coefficients of the variety $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C})) \subseteq \mathrm{GL}_n(\mathbb{C})^2$ for any $n$. It is torsion free and minimally generated as a $H^\ast B\mathrm{GL}_n(\mathbb{C})$-algebra by $3n$ elements. The ideal of relations is the saturation of an $n$-generator ideal by even powers of the Vandermonde polynomial. For coefficients in a field whose characteristic does not divide $n!$, we also give a presentation of the non-equivariant cohomology ring of $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C}))$. |
| title | The equivariant cohomology ring of the representation variety $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C}))$ |
| topic | Algebraic Topology 58D27, 55N91, 55N45 |
| url | https://arxiv.org/abs/2601.00683 |