{"title":"Yangian y2 $Y_2$的模表示","authors":"Hao Chang, Jinxin Hu, Lewis Topley","doi":"10.1112/jlms.70056","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <msub>\n <mi>Y</mi>\n <mn>2</mn>\n </msub>\n <annotation>$Y_2$</annotation>\n </semantics></math> be the Yangian associated to the general linear Lie algebra <span></span><math>\n <semantics>\n <msub>\n <mi>gl</mi>\n <mn>2</mn>\n </msub>\n <annotation>$\\mathfrak {gl}_2$</annotation>\n </semantics></math>, defined over an algebraically closed field <span></span><math>\n <semantics>\n <mi>k</mi>\n <annotation>$\\mathbb {k}$</annotation>\n </semantics></math> of characteristic <span></span><math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n <annotation>$p>0$</annotation>\n </semantics></math>. In this paper, we study the representation theory of the restricted Yangian <span></span><math>\n <semantics>\n <msubsup>\n <mi>Y</mi>\n <mn>2</mn>\n <mrow>\n <mo>[</mo>\n <mi>p</mi>\n <mo>]</mo>\n </mrow>\n </msubsup>\n <annotation>$Y^{[p]}_2$</annotation>\n </semantics></math>. This leads to a description of the representations of <span></span><math>\n <semantics>\n <msub>\n <mi>gl</mi>\n <mrow>\n <mn>2</mn>\n <mi>n</mi>\n </mrow>\n </msub>\n <annotation>$\\mathfrak {gl}_{2n}$</annotation>\n </semantics></math>, whose <span></span><math>\n <semantics>\n <mi>p</mi>\n <annotation>$p$</annotation>\n </semantics></math>-character is nilpotent with Jordan type given by a two-row partition <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>n</mi>\n <mo>,</mo>\n <mi>n</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$(n, n)$</annotation>\n </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70056","citationCount":"0","resultStr":"{\"title\":\"Modular representations of the Yangian \\n \\n \\n Y\\n 2\\n \\n $Y_2$\",\"authors\":\"Hao Chang, Jinxin Hu, Lewis Topley\",\"doi\":\"10.1112/jlms.70056\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <msub>\\n <mi>Y</mi>\\n <mn>2</mn>\\n </msub>\\n <annotation>$Y_2$</annotation>\\n </semantics></math> be the Yangian associated to the general linear Lie algebra <span></span><math>\\n <semantics>\\n <msub>\\n <mi>gl</mi>\\n <mn>2</mn>\\n </msub>\\n <annotation>$\\\\mathfrak {gl}_2$</annotation>\\n </semantics></math>, defined over an algebraically closed field <span></span><math>\\n <semantics>\\n <mi>k</mi>\\n <annotation>$\\\\mathbb {k}$</annotation>\\n </semantics></math> of characteristic <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$p>0$</annotation>\\n </semantics></math>. In this paper, we study the representation theory of the restricted Yangian <span></span><math>\\n <semantics>\\n <msubsup>\\n <mi>Y</mi>\\n <mn>2</mn>\\n <mrow>\\n <mo>[</mo>\\n <mi>p</mi>\\n <mo>]</mo>\\n </mrow>\\n </msubsup>\\n <annotation>$Y^{[p]}_2$</annotation>\\n </semantics></math>. This leads to a description of the representations of <span></span><math>\\n <semantics>\\n <msub>\\n <mi>gl</mi>\\n <mrow>\\n <mn>2</mn>\\n <mi>n</mi>\\n </mrow>\\n </msub>\\n <annotation>$\\\\mathfrak {gl}_{2n}$</annotation>\\n </semantics></math>, whose <span></span><math>\\n <semantics>\\n <mi>p</mi>\\n <annotation>$p$</annotation>\\n </semantics></math>-character is nilpotent with Jordan type given by a two-row partition <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>n</mi>\\n <mo>,</mo>\\n <mi>n</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(n, n)$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":\"111 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-12-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70056\",\"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.70056\",\"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.70056","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let be the Yangian associated to the general linear Lie algebra , defined over an algebraically closed field of characteristic . In this paper, we study the representation theory of the restricted Yangian . This leads to a description of the representations of , whose -character is nilpotent with Jordan type given by a two-row partition .
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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.
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