{"title":"论低级截断多项式环的模块不变式","authors":"Le Minh Ha, Nguyen Dang Ho Hai, Nguyen Van Nghia","doi":"arxiv-2408.16250","DOIUrl":null,"url":null,"abstract":"We verify the conjectures due to Lewis, Reiner and Stanton about the Hilbert\nseries of the invariant ring of the truncated polynomial ring for all parabolic\nsubgroups up to rank $3$. This is done by constructing an explicit set of\ngenerators for each invariant ring in question. We also propose a conjecture\nconcerning the action of the Steenrod algebra and the Dickson algebra on a\ncertain naturally occurring filtration of the invariant ring under the general\nlinear group.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"2012 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Modular Invariants of Truncated Polynomial Rings in low ranks\",\"authors\":\"Le Minh Ha, Nguyen Dang Ho Hai, Nguyen Van Nghia\",\"doi\":\"arxiv-2408.16250\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We verify the conjectures due to Lewis, Reiner and Stanton about the Hilbert\\nseries of the invariant ring of the truncated polynomial ring for all parabolic\\nsubgroups up to rank $3$. This is done by constructing an explicit set of\\ngenerators for each invariant ring in question. We also propose a conjecture\\nconcerning the action of the Steenrod algebra and the Dickson algebra on a\\ncertain naturally occurring filtration of the invariant ring under the general\\nlinear group.\",\"PeriodicalId\":501136,\"journal\":{\"name\":\"arXiv - MATH - Rings and Algebras\",\"volume\":\"2012 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Rings and Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.16250\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.16250","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On Modular Invariants of Truncated Polynomial Rings in low ranks
We verify the conjectures due to Lewis, Reiner and Stanton about the Hilbert
series of the invariant ring of the truncated polynomial ring for all parabolic
subgroups up to rank $3$. This is done by constructing an explicit set of
generators for each invariant ring in question. We also propose a conjecture
concerning the action of the Steenrod algebra and the Dickson algebra on a
certain naturally occurring filtration of the invariant ring under the general
linear group.