{"title":"Catalan numbers and noncommutative Hilbert schemes","authors":"Valery Lunts, Špela Špenko, Michel Van Den Bergh","doi":"10.4310/pamq.2024.v20.n3.a10","DOIUrl":null,"url":null,"abstract":"We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $\\mathbb{R}^n$, combinatorially equivalent to the permutahedron, and the set of m-parking functions of length n. This bijection restricts to a bijection between the regular $S_n$-orbits and $(m, n)$-Dyck paths, the number of which is given by the Fuss–Catalan number $A_n (m, 1)$. Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"48 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Pure and Applied Mathematics Quarterly","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/pamq.2024.v20.n3.a10","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $\mathbb{R}^n$, combinatorially equivalent to the permutahedron, and the set of m-parking functions of length n. This bijection restricts to a bijection between the regular $S_n$-orbits and $(m, n)$-Dyck paths, the number of which is given by the Fuss–Catalan number $A_n (m, 1)$. Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes.
期刊介绍:
Publishes high-quality, original papers on all fields of mathematics. To facilitate fruitful interchanges between mathematicians from different regions and specialties, and to effectively disseminate new breakthroughs in mathematics, the journal welcomes well-written submissions from all significant areas of mathematics. The editors are committed to promoting the highest quality of mathematical scholarship.