{"title":"Homotopy properties of the complex of frames of a unitary space","authors":"Kevin I. Piterman, Volkmar Welker","doi":"10.1112/jlms.12978","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mi>V</mi>\n <annotation>$V$</annotation>\n </semantics></math> be a finite-dimensional vector space equipped with a nondegenerate Hermitian form over a field <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>${\\mathbb {K}}$</annotation>\n </semantics></math>. Let <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n </mrow>\n <annotation>${\\mathcal {G}}(V)$</annotation>\n </semantics></math> be the graph with vertex set the one-dimensional nondegenerate subspaces of <span></span><math>\n <semantics>\n <mi>V</mi>\n <annotation>$V$</annotation>\n </semantics></math> and adjacency relation given by orthogonality. We give a complete description of when <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n </mrow>\n <annotation>${\\mathcal {G}}(V)$</annotation>\n </semantics></math> is connected in terms of the dimension of <span></span><math>\n <semantics>\n <mi>V</mi>\n <annotation>$V$</annotation>\n </semantics></math> and the size of the ground field <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>${\\mathbb {K}}$</annotation>\n </semantics></math>. Furthermore, we prove that if <span></span><math>\n <semantics>\n <mrow>\n <mo>dim</mo>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n <mo>></mo>\n <mn>4</mn>\n </mrow>\n <annotation>$\\dim (V) &gt; 4$</annotation>\n </semantics></math>, then the clique complex <span></span><math>\n <semantics>\n <mrow>\n <mi>F</mi>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n </mrow>\n <annotation>${\\mathcal {F}}(V)$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n </mrow>\n <annotation>${\\mathcal {G}}(V)$</annotation>\n </semantics></math> is simply connected. For finite fields <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>${\\mathbb {K}}$</annotation>\n </semantics></math>, we also compute the eigenvalues of the adjacency matrix of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n </mrow>\n <annotation>${\\mathcal {G}}(V)$</annotation>\n </semantics></math>. Then, by Garland's method, we conclude that <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mover>\n <mi>H</mi>\n <mo>∼</mo>\n </mover>\n <mi>m</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>F</mi>\n <mrow>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n </mrow>\n <mo>;</mo>\n <mi>k</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <annotation>$\\tilde{H}_m({\\mathcal {F}}(V);{\\mathbb {k}}) = 0$</annotation>\n </semantics></math> for all <span></span><math>\n <semantics>\n <mrow>\n <mn>0</mn>\n <mo>⩽</mo>\n <mi>m</mi>\n <mo>⩽</mo>\n <mo>dim</mo>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n <mo>−</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$0\\leqslant m\\leqslant \\dim (V)-3$</annotation>\n </semantics></math>, where <span></span><math>\n <semantics>\n <mi>k</mi>\n <annotation>${\\mathbb {k}}$</annotation>\n </semantics></math> is a field of characteristic 0, provided that <span></span><math>\n <semantics>\n <mrow>\n <mo>dim</mo>\n <msup>\n <mrow>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n </mrow>\n <mn>2</mn>\n </msup>\n <mo>⩽</mo>\n <mrow>\n <mo>|</mo>\n <mi>K</mi>\n <mo>|</mo>\n </mrow>\n </mrow>\n <annotation>$\\dim (V)^2 \\leqslant |{\\mathbb {K}}|$</annotation>\n </semantics></math>. Under these assumptions, we deduce that the barycentric subdivision of <span></span><math>\n <semantics>\n <mrow>\n <mi>F</mi>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n </mrow>\n <annotation>${\\mathcal {F}}(V)$</annotation>\n </semantics></math> deformation retracts to the order complex of the certain rank selection of <span></span><math>\n <semantics>\n <mrow>\n <mi>F</mi>\n <mo>(</mo>\n <mi>V</mi>\n <mo>)</mo>\n </mrow>\n <annotation>${\\mathcal {F}}(V)$</annotation>\n </semantics></math> that is Cohen–Macaulay over <span></span><math>\n <semantics>\n <mi>k</mi>\n <annotation>${\\mathbb {k}}$</annotation>\n </semantics></math>. Finally, we apply our results to the Quillen poset of elementary abelian <span></span><math>\n <semantics>\n <mi>p</mi>\n <annotation>$p$</annotation>\n </semantics></math>-subgroups of a finite group and to the study of geometric properties of the poset of nondegenerate subspaces of <span></span><math>\n <semantics>\n <mi>V</mi>\n <annotation>$V$</annotation>\n </semantics></math> and the poset of orthogonal decompositions of <span></span><math>\n <semantics>\n <mi>V</mi>\n <annotation>$V$</annotation>\n </semantics></math>.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12978","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a finite-dimensional vector space equipped with a nondegenerate Hermitian form over a field . Let be the graph with vertex set the one-dimensional nondegenerate subspaces of and adjacency relation given by orthogonality. We give a complete description of when is connected in terms of the dimension of and the size of the ground field . Furthermore, we prove that if , then the clique complex of is simply connected. For finite fields , we also compute the eigenvalues of the adjacency matrix of . Then, by Garland's method, we conclude that for all , where is a field of characteristic 0, provided that . Under these assumptions, we deduce that the barycentric subdivision of deformation retracts to the order complex of the certain rank selection of that is Cohen–Macaulay over . Finally, we apply our results to the Quillen poset of elementary abelian -subgroups of a finite group and to the study of geometric properties of the poset of nondegenerate subspaces of and the poset of orthogonal decompositions of .
设 V $V$ 是一个有限维向量空间,其上有一个域 K ${mathbb {K}}$ 的非enerate 赫米提形式。让 G ( V ) ${mathcal {G}}(V)$ 是顶点集为 V $V$ 的一维非enerate 子空间且邻接关系由正交性给出的图。我们用 V $V$ 的维数和基场 K ${mathbb {K}}$ 的大小给出了 G ( V ) ${mathcal {G}}(V)$ 连接时的完整描述。此外,我们证明如果 dim ( V ) > 4 $\dim (V) > 4$ ,那么 G ( V ) ${\mathcal {F}}(V)$ 的簇复数 F ( V ) ${\mathcal {G}}(V)$ 是简单相连的。对于有限域 K ${{mathbb {K}}$ ,我们也计算 G ( V ) ${\mathcal {G}}(V)$ 的邻接矩阵的特征值。然后,根据加兰方法,我们得出 H ∼ m ( F ( V ) ; k ) = 0 $\tilde{H}_m({\mathcal {F}}(V);{mathbb {k}}) = 0$ for all 0 ⩽ m ⩽ dim ( V ) - 3 $0\leqslant m\leqslant \dim (V)-3$, where k $\{mathbb {k}}$ is a field of characteristic 0、条件是 dim ( V ) 2 ⩽ | K | $\dim (V)^2 \leqslant |{\mathbb {K}}|$ 。在这些假设下,我们推导出 F ( V ) ${mathcal {F}}(V)$ 的重心细分形变回缩到 F ( V ) ${mathcal {F}}(V)$ 的一定秩选择的阶复数,该阶复数是在 k ${mathbb {k}} 上的 Cohen-Macaulay 。最后,我们将结果应用于有限群的基本无性 p $p$ 子群的奎伦正集,以及 V $V$ 的非enerate 子空间正集和 V $V$ 的正交分解正集的几何性质研究。
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.