{"title":"用对称行列式描述经典李代数的带参数capelli元","authors":"Shotaro KAWATA","doi":"10.2206/kyushujm.77.43","DOIUrl":null,"url":null,"abstract":"We construct the Capelli elements Ck(u) = C(1k)(u) (k = 1, . . . , n) with a parameter u for the symplectic Lie algebras and orthogonal Lie algebras. They correspond to factorial Schur functions with parameter u attached to the column partitions (1k). We also give explicit formulas for Ck(u) arising from the expansion of Cn(u) = C(1n)(u) with respect to the parameter u. We use the Jacobi-Trudi formula for the factorial Schur functions Rλ(x; u) to construct the higher Capelli elements Cλ(u). They are expressed as determinants of matrices whose entries are Capelli elements Ck(u) attached to the column partitions.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"THE DESCRIPTION OF THE CAPELLI ELEMENTS WITH A PARAMETER FOR CLASSICAL LIE ALGEBRAS IN TERMS OF SYMMETRIZED DETERMINANT\",\"authors\":\"Shotaro KAWATA\",\"doi\":\"10.2206/kyushujm.77.43\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We construct the Capelli elements Ck(u) = C(1k)(u) (k = 1, . . . , n) with a parameter u for the symplectic Lie algebras and orthogonal Lie algebras. They correspond to factorial Schur functions with parameter u attached to the column partitions (1k). We also give explicit formulas for Ck(u) arising from the expansion of Cn(u) = C(1n)(u) with respect to the parameter u. We use the Jacobi-Trudi formula for the factorial Schur functions Rλ(x; u) to construct the higher Capelli elements Cλ(u). They are expressed as determinants of matrices whose entries are Capelli elements Ck(u) attached to the column partitions.\",\"PeriodicalId\":49929,\"journal\":{\"name\":\"Kyushu Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Kyushu Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2206/kyushujm.77.43\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Kyushu Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2206/kyushujm.77.43","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
THE DESCRIPTION OF THE CAPELLI ELEMENTS WITH A PARAMETER FOR CLASSICAL LIE ALGEBRAS IN TERMS OF SYMMETRIZED DETERMINANT
We construct the Capelli elements Ck(u) = C(1k)(u) (k = 1, . . . , n) with a parameter u for the symplectic Lie algebras and orthogonal Lie algebras. They correspond to factorial Schur functions with parameter u attached to the column partitions (1k). We also give explicit formulas for Ck(u) arising from the expansion of Cn(u) = C(1n)(u) with respect to the parameter u. We use the Jacobi-Trudi formula for the factorial Schur functions Rλ(x; u) to construct the higher Capelli elements Cλ(u). They are expressed as determinants of matrices whose entries are Capelli elements Ck(u) attached to the column partitions.
期刊介绍：
The Kyushu Journal of Mathematics is an academic journal in mathematics, published by the Faculty of Mathematics at Kyushu University since 1941. It publishes selected research papers in pure and applied mathematics. One volume, published each year, consists of two issues, approximately 20 articles and 400 pages in total.
More than 500 copies of the journal are distributed through exchange contracts between mathematical journals, and available at many universities, institutes and libraries around the world. The on-line version of the journal is published at "Jstage" (an aggregator for e-journals), where all the articles published by the journal since 1995 are accessible freely through the Internet.
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