酉群中交换元空间的稳定分裂

IF 1 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2025-02-19 DOI:10.1112/jlms.70084

Alejandro Adem, José Manuel Gómez, Simon Gritschacher

{"title":"酉群中交换元空间的稳定分裂","authors":"Alejandro Adem, José Manuel Gómez, Simon Gritschacher","doi":"10.1112/jlms.70084","DOIUrl":null,"url":null,"abstract":"We prove an analogue of Miller's stable splitting of the unitary group <math>\n <semantics>\n <mrow>\n <mi>U</mi>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$U(m)$</annotation>\n </semantics></math> for spaces of commuting elements in <math>\n <semantics>\n <mrow>\n <mi>U</mi>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$U(m)$</annotation>\n </semantics></math>. After inverting <math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>!</mo>\n </mrow>\n <annotation>$m!$</annotation>\n </semantics></math>, the space <math>\n <semantics>\n <mrow>\n <mo>Hom</mo>\n <mo>(</mo>\n <msup>\n <mi>Z</mi>\n <mi>n</mi>\n </msup>\n <mo>,</mo>\n <mi>U</mi>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{Hom}(\\mathbb {Z}^n,U(m))$</annotation>\n </semantics></math> splits stably as a wedge of Thom-like spaces of bundles of commuting varieties over certain partial flag manifolds. Using Steenrod operations, we prove that our splitting does not hold integrally. Analogous decompositions for symplectic and orthogonal groups as well as homological results for the one-point compactification of the commuting variety in a Lie algebra are also provided.","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 2","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A stable splitting for spaces of commuting elements in unitary groups\",\"authors\":\"Alejandro Adem, José Manuel Gómez, Simon Gritschacher\",\"doi\":\"10.1112/jlms.70084\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove an analogue of Miller's stable splitting of the unitary group <math>\\n <semantics>\\n <mrow>\\n <mi>U</mi>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$U(m)$</annotation>\\n </semantics></math> for spaces of commuting elements in <math>\\n <semantics>\\n <mrow>\\n <mi>U</mi>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$U(m)$</annotation>\\n </semantics></math>. After inverting <math>\\n <semantics>\\n <mrow>\\n <mi>m</mi>\\n <mo>!</mo>\\n </mrow>\\n <annotation>$m!$</annotation>\\n </semantics></math>, the space <math>\\n <semantics>\\n <mrow>\\n <mo>Hom</mo>\\n <mo>(</mo>\\n <msup>\\n <mi>Z</mi>\\n <mi>n</mi>\\n </msup>\\n <mo>,</mo>\\n <mi>U</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{Hom}(\\\\mathbb {Z}^n,U(m))$</annotation>\\n </semantics></math> splits stably as a wedge of Thom-like spaces of bundles of commuting varieties over certain partial flag manifolds. Using Steenrod operations, we prove that our splitting does not hold integrally. Analogous decompositions for symplectic and orthogonal groups as well as homological results for the one-point compactification of the commuting variety in a Lie algebra are also provided.\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":\"111 2\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2025-02-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70084\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70084","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

对于U(m)$ U(m)$中交换元的空间，证明了Miller稳定分裂酉群U(m)$ U(m)$的一个类似。求m的倒数！m美元!$，空间hm (zn)，U(m))$ \operatorname{hm}(\mathbb {Z}^n，U(m))$稳定地分裂为若干部分标志流形上交换变体束的类thomas空间的楔子。利用Steenrod运算，我们证明了我们的分裂是不成立的。给出了李代数中交换簇的一点紧化的类似分解和正交群的同调结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

A stable splitting for spaces of commuting elements in unitary groups

We prove an analogue of Miller's stable splitting of the unitary group $U (m)$ for spaces of commuting elements in $U (m)$ . After inverting $m!$ , the space $Hom (Z^{n}, U (m))$ splits stably as a wedge of Thom-like spaces of bundles of commuting varieties over certain partial flag manifolds. Using Steenrod operations, we prove that our splitting does not hold integrally. Analogous decompositions for symplectic and orthogonal groups as well as homological results for the one-point compactification of the commuting variety in a Lie algebra are also provided.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.