Homological properties of determinantal arrangements

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• 作者：Arnold Yim¹ ; ^{ayim@purdue.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：13N15 ; 32S22
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：471
• 期：Complete
• 页码：220-239
• 全文大小：396 K

文摘

We explore a natural extension of braid arrangements in the context of determinantal arrangements. We show that these determinantal arrangements are free divisors. Additionally, we prove that free determinantal arrangements defined by the minors of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303350&_mathId=si1.gif&_user=111111111&_pii=S0021869316303350&_rdoc=1&_issn=00218693&md5=155be5dfaee06451c61ee65ab598c48d" title="Click to view the MathML source">2×nclass="mathContainer hidden">class="mathCode">$2\times n$ matrices satisfy nice combinatorial properties.

We also study the topology of the complements of these determinantal arrangements, and prove that their higher homotopy groups are isomorphic to those of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303350&_mathId=si2.gif&_user=111111111&_pii=S0021869316303350&_rdoc=1&_issn=00218693&md5=5b2a936bf0bdc29fba10f7dcdce603f0" title="Click to view the MathML source">S³class="mathContainer hidden">class="mathCode">msup>S3msup>. Furthermore, we find that the complements of arrangements satisfying those same combinatorial properties above have Poincaré polynomials that factor nicely.