{"title":"关于钩形的$$\\frac{n!}{2}$$猜想的证明","authors":"Sam Armon","doi":"10.1007/s00026-022-00613-3","DOIUrl":null,"url":null,"abstract":"<div><p>A well-known representation-theoretic model for the transformed Macdonald polynomial <span>\\({\\widetilde{H}}_\\mu (Z;t,q)\\)</span>, where <span>\\(\\mu \\)</span> is an integer partition, is given by the Garsia–Haiman module <span>\\({\\mathcal {H}}_\\mu \\)</span>. We study the <span>\\(\\frac{n!}{k}\\)</span> conjecture of Bergeron and Garsia, which concerns the behavior of certain <i>k</i>-tuples of Garsia–Haiman modules under intersection. In the special case that <span>\\(\\mu \\)</span> has hook shape, we use a basis for <span>\\({\\mathcal {H}}_\\mu \\)</span> due to Adin, Remmel, and Roichman to resolve the <span>\\(\\frac{n!}{2}\\)</span> conjecture by constructing an explicit basis for the intersection of two Garsia–Haiman modules.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-022-00613-3.pdf","citationCount":"1","resultStr":"{\"title\":\"A Proof of the \\\\(\\\\frac{n!}{2}\\\\) Conjecture for Hook Shapes\",\"authors\":\"Sam Armon\",\"doi\":\"10.1007/s00026-022-00613-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A well-known representation-theoretic model for the transformed Macdonald polynomial <span>\\\\({\\\\widetilde{H}}_\\\\mu (Z;t,q)\\\\)</span>, where <span>\\\\(\\\\mu \\\\)</span> is an integer partition, is given by the Garsia–Haiman module <span>\\\\({\\\\mathcal {H}}_\\\\mu \\\\)</span>. We study the <span>\\\\(\\\\frac{n!}{k}\\\\)</span> conjecture of Bergeron and Garsia, which concerns the behavior of certain <i>k</i>-tuples of Garsia–Haiman modules under intersection. In the special case that <span>\\\\(\\\\mu \\\\)</span> has hook shape, we use a basis for <span>\\\\({\\\\mathcal {H}}_\\\\mu \\\\)</span> due to Adin, Remmel, and Roichman to resolve the <span>\\\\(\\\\frac{n!}{2}\\\\)</span> conjecture by constructing an explicit basis for the intersection of two Garsia–Haiman modules.</p></div>\",\"PeriodicalId\":50769,\"journal\":{\"name\":\"Annals of Combinatorics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2022-10-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00026-022-00613-3.pdf\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Combinatorics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00026-022-00613-3\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-022-00613-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
A Proof of the \(\frac{n!}{2}\) Conjecture for Hook Shapes
A well-known representation-theoretic model for the transformed Macdonald polynomial \({\widetilde{H}}_\mu (Z;t,q)\), where \(\mu \) is an integer partition, is given by the Garsia–Haiman module \({\mathcal {H}}_\mu \). We study the \(\frac{n!}{k}\) conjecture of Bergeron and Garsia, which concerns the behavior of certain k-tuples of Garsia–Haiman modules under intersection. In the special case that \(\mu \) has hook shape, we use a basis for \({\mathcal {H}}_\mu \) due to Adin, Remmel, and Roichman to resolve the \(\frac{n!}{2}\) conjecture by constructing an explicit basis for the intersection of two Garsia–Haiman modules.
期刊介绍:
Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board.
The scope of Annals of Combinatorics is covered by the following three tracks:
Algebraic Combinatorics:
Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices
Analytic and Algorithmic Combinatorics:
Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms
Graphs and Matroids:
Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches