{"title":"The plethystic inverse of the odd Lie representations","authors":"S. Sundaram","doi":"10.1090/proc/15938","DOIUrl":null,"url":null,"abstract":"The Frobenius characteristic of $Lie_n,$ the representation of the symmetric group $S_n$ afforded by the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum $\\sum_{n\\geq 0} Lie_{2n+1}$ of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of $S_n$ in terms of irreducibles indexed by hooks, and the Lie representations.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"99 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/proc/15938","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
The Frobenius characteristic of $Lie_n,$ the representation of the symmetric group $S_n$ afforded by the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum $\sum_{n\geq 0} Lie_{2n+1}$ of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of $S_n$ in terms of irreducibles indexed by hooks, and the Lie representations.