{"title":"Kostant’s generating functions and Mckay–Slodowy correspondence","authors":"Naihuan Jing, Zhijun Li, Danxia Wang","doi":"10.1142/s0219498825501713","DOIUrl":null,"url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>N</mi><mo>⊴</mo><mi>G</mi></math></span><span></span> be a pair of finite subgroups of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">SL</mtext></mstyle></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℂ</mi><mo stretchy=\"false\">)</mo></math></span><span></span> and <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>V</mi></math></span><span></span> a finite-dimensional fundamental <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span>-module. We study Kostant’s generating functions for the decomposition of the <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">SL</mtext></mstyle></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℂ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>-module <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>V</mi><mo stretchy=\"false\">)</mo></math></span><span></span> restricted to <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>N</mi><mo>◃</mo><mi>G</mi></math></span><span></span> in connection with the McKay–Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219498825501713","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a pair of finite subgroups of and a finite-dimensional fundamental -module. We study Kostant’s generating functions for the decomposition of the -module restricted to in connection with the McKay–Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.