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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.10786 |
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| _version_ | 1866914871500603392 |
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| author | Kenyon, Richard Ovenhouse, Nicholas |
| author_facet | Kenyon, Richard Ovenhouse, Nicholas |
| contents | We study pairs of matrices $A,B\in GL_n({\mathbb C})$ such that the eigenvalues of $A$, of $B$ and of the product $AB$ are specified in advance. We show that the space of such pairs $(A,B)$ under simultaneous conjugation has dimension $(n-1)(n-2)$, and give an explicit parameterization.
More generally let $Σ$ be a surface of genus $g$ with $k$ punctures. We find a parameterization of the space $Ω_{g,k,n}$ of flat $GL_n({\mathbb C})$-structures on $Σ$ whose holonomies around the punctures have prescribed eigenvalues.
We show furthermore that, for $3\le k\le 2g+6$ (or $3\le k\le 9$ if $g=1$, or $3\le k$ if $g=0$), the space $Ω_{g,k,n}$ has an explicit symplectic structure and an associated Liouville integrable system, equivalent to a leaf of a Goncharov-Kenyon dimer integrable system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10786 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eigenvalues of matrix products Kenyon, Richard Ovenhouse, Nicholas Combinatorics 05E10 We study pairs of matrices $A,B\in GL_n({\mathbb C})$ such that the eigenvalues of $A$, of $B$ and of the product $AB$ are specified in advance. We show that the space of such pairs $(A,B)$ under simultaneous conjugation has dimension $(n-1)(n-2)$, and give an explicit parameterization. More generally let $Σ$ be a surface of genus $g$ with $k$ punctures. We find a parameterization of the space $Ω_{g,k,n}$ of flat $GL_n({\mathbb C})$-structures on $Σ$ whose holonomies around the punctures have prescribed eigenvalues. We show furthermore that, for $3\le k\le 2g+6$ (or $3\le k\le 9$ if $g=1$, or $3\le k$ if $g=0$), the space $Ω_{g,k,n}$ has an explicit symplectic structure and an associated Liouville integrable system, equivalent to a leaf of a Goncharov-Kenyon dimer integrable system. |
| title | Eigenvalues of matrix products |
| topic | Combinatorics 05E10 |
| url | https://arxiv.org/abs/2407.10786 |