{"title":"A generalisation of bar-core partitions","authors":"Dean Yates","doi":"10.5802/alco.231","DOIUrl":null,"url":null,"abstract":"When p and q are coprime odd integers no less than 3, Olsson proved that the q -bar-core of a p -bar-core is again a p -bar-core. We establish a generalisation of this theorem: that the p -bar-weight of the q -bar-core of a bar partition λ is at most the p -bar-weight of λ . We go on to study the set of bar partitions for which equality holds and show that it is a union of orbits for an action of a Coxeter group of type ˜ C ( p − 1) / 2 × ˜ C ( q − 1) / 2 . We also provide an algorithm for constructing a bar partition in this set with a given p -bar-core and q -bar-core.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebraic Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5802/alco.231","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 2
Abstract
When p and q are coprime odd integers no less than 3, Olsson proved that the q -bar-core of a p -bar-core is again a p -bar-core. We establish a generalisation of this theorem: that the p -bar-weight of the q -bar-core of a bar partition λ is at most the p -bar-weight of λ . We go on to study the set of bar partitions for which equality holds and show that it is a union of orbits for an action of a Coxeter group of type ˜ C ( p − 1) / 2 × ˜ C ( q − 1) / 2 . We also provide an algorithm for constructing a bar partition in this set with a given p -bar-core and q -bar-core.
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