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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2507.17362 |
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| _version_ | 1866913956110532608 |
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| author | Marc-Zwecker, Arielle |
| author_facet | Marc-Zwecker, Arielle |
| contents | The multiplicative Horn problem is the following question: given three conjugacy classes $\mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3$ in a Lie group $G$, do there exist elements $(A,B,C)\in\mathcal{C}_1\times\mathcal{C}_2\times\mathcal{C}_3$ such that $ABC=\operatorname{Id}$? In this paper, we study the multiplicative Horn problem restricted to the elliptic classes of the group $G={\rm PU}(n,1)$ for $n\geq 1$, which is the isometry group of the $n$-dimensional complex hyperbolic space. We show that the solution set of Horn's problem in PU$(n,1)$ is a finite union of convex polytopes in the space of elliptic conjugacy classes. We give a complete description of these polytopes when $n=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17362 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Horn's problem in PU$(n,1)$ Marc-Zwecker, Arielle Geometric Topology Representation Theory The multiplicative Horn problem is the following question: given three conjugacy classes $\mathcal{C}_1, \mathcal{C}_2, \mathcal{C}_3$ in a Lie group $G$, do there exist elements $(A,B,C)\in\mathcal{C}_1\times\mathcal{C}_2\times\mathcal{C}_3$ such that $ABC=\operatorname{Id}$? In this paper, we study the multiplicative Horn problem restricted to the elliptic classes of the group $G={\rm PU}(n,1)$ for $n\geq 1$, which is the isometry group of the $n$-dimensional complex hyperbolic space. We show that the solution set of Horn's problem in PU$(n,1)$ is a finite union of convex polytopes in the space of elliptic conjugacy classes. We give a complete description of these polytopes when $n=2$. |
| title | Horn's problem in PU$(n,1)$ |
| topic | Geometric Topology Representation Theory |
| url | https://arxiv.org/abs/2507.17362 |