Eigenvalues of matrix products

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-04-10 DOI:10.1112/blms.70071

Richard Kenyon, Nicholas Ovenhouse

{"title":"Eigenvalues of matrix products","authors":"Richard Kenyon, Nicholas Ovenhouse","doi":"10.1112/blms.70071","DOIUrl":null,"url":null,"abstract":"We study pairs of matrices <math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <mo>,</mo>\n <mi>B</mi>\n <mo>∈</mo>\n <msub>\n <mi>GL</mi>\n <mi>n</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>C</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$A,B\\in \\mathrm{GL}_n({\\mathbb {C}})$</annotation>\n </semantics></math> such that the eigenvalues of <math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>, of <math>\n <semantics>\n <mi>B</mi>\n <annotation>$B$</annotation>\n </semantics></math> and of the product <math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <mi>B</mi>\n </mrow>\n <annotation>$AB$</annotation>\n </semantics></math> are specified in advance. We show that the space of such pairs <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>A</mi>\n <mo>,</mo>\n <mi>B</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(A,B)$</annotation>\n </semantics></math> under simultaneous conjugation has dimension <math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n <mo>)</mo>\n <mo>(</mo>\n <mi>n</mi>\n <mo>−</mo>\n <mn>2</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$(n-1)(n-2)$</annotation>\n </semantics></math>, and give an explicit parameterization. More generally, let <math>\n <semantics>\n <mi>Σ</mi>\n <annotation>$\\Sigma$</annotation>\n </semantics></math> be a surface of genus <math>\n <semantics>\n <mi>g</mi>\n <annotation>$g$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mi>k</mi>\n <annotation>$k$</annotation>\n </semantics></math> punctures. We find a parameterization of the space <math>\n <semantics>\n <msub>\n <mi>Ω</mi>\n <mrow>\n <mi>g</mi>\n <mo>,</mo>\n <mi>k</mi>\n <mo>,</mo>\n <mi>n</mi>\n </mrow>\n </msub>\n <annotation>$\\Omega _{g,k,n}$</annotation>\n </semantics></math> of flat <math>\n <semantics>\n <mrow>\n <msub>\n <mi>GL</mi>\n <mi>n</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>C</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\mathrm{GL}_n({\\mathbb {C}})$</annotation>\n </semantics></math>-structures on <math>\n <semantics>\n <mi>Σ</mi>\n <annotation>$\\Sigma$</annotation>\n </semantics></math> whose holonomies around the punctures have prescribed eigenvalues. We show furthermore that, for <math>\n <semantics>\n <mrow>\n <mn>3</mn>\n <mo>⩽</mo>\n <mi>k</mi>\n <mo>⩽</mo>\n <mn>2</mn>\n <mi>g</mi>\n <mo>+</mo>\n <mn>6</mn>\n </mrow>\n <annotation>$3\\leqslant k\\leqslant 2g+6$</annotation>\n </semantics></math> (or <math>\n <semantics>\n <mrow>\n <mn>3</mn>\n <mo>⩽</mo>\n <mi>k</mi>\n <mo>⩽</mo>\n <mn>9</mn>\n </mrow>\n <annotation>$3\\leqslant k\\leqslant 9$</annotation>\n </semantics></math> if <math>\n <semantics>\n <mrow>\n <mi>g</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$g=1$</annotation>\n </semantics></math>, or <math>\n <semantics>\n <mrow>\n <mn>3</mn>\n <mo>⩽</mo>\n <mi>k</mi>\n </mrow>\n <annotation>$3\\leqslant k$</annotation>\n </semantics></math> if <math>\n <semantics>\n <mrow>\n <mi>g</mi>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <annotation>$g=0$</annotation>\n </semantics></math>), the space <math>\n <semantics>\n <msub>\n <mi>Ω</mi>\n <mrow>\n <mi>g</mi>\n <mo>,</mo>\n <mi>k</mi>\n <mo>,</mo>\n <mi>n</mi>\n </mrow>\n </msub>\n <annotation>$\\Omega _{g,k,n}$</annotation>\n </semantics></math> has an explicit symplectic structure and an associated Liouville integrable system, equivalent to a leaf of a Goncharov–Kenyon dimer integrable system.","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 7","pages":"1938-1951"},"PeriodicalIF":0.9000,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.70071","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We study pairs of matrices $A, B \in {GL}_{n} (C)$ such that the eigenvalues of $A$ , of $B$ and of the product $A B$ 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 ${GL}_{n} (C)$ -structures on $Σ$ whose holonomies around the punctures have prescribed eigenvalues. We show furthermore that, for $3 ⩽ k ⩽ 2 g + 6$ (or $3 ⩽ k ⩽ 9$ if $g = 1$ , or $3 ⩽ k$ 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.

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矩阵乘积的特征值

我们研究矩阵A， B∈GL n (C) $A,B\in \mathrm{GL}_n({\mathbb {C}})$对，使得A的特征值$A$，提前确定产品B $B$和产品A B $AB$的规格。我们证明了这样的对(A，B) $(A,B)$在同时共轭下具有维数（n−1）（n−2）$(n-1)(n-2)$，并给出了一个显式参数化。更一般地说，设Σ $\Sigma$是一个有k个$k$点的g $g$属曲面。我们找到了空间的参数化Ω g， k，平面GL的n $\Omega _{g,k,n}$ n (C) $\mathrm{GL}_n({\mathbb {C}})$ -在Σ $\Sigma$上的结构，其全域围绕穿孔有规定的特征值。我们进一步证明，对于3≤k≤2g + 6 $3\leqslant k\leqslant 2g+6$(或3≤k≤9 $3\leqslant k\leqslant 9$，如果g =1 $g=1$，或3≥k $3\leqslant k$（如果g = 0 $g=0$），空间Ω g，k, n $\Omega _{g,k,n}$具有显式辛结构和相应的Liouville可积系统，相当于Goncharov-Kenyon二聚体可积系统的一个叶。

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