[2404.02846] On the Springer correspondence for wreath products Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Representation Theory arXiv:2404.02846 (math) [Submitted on 3 Apr 2024 ( v1 ), last revised 30 Apr 2026 (this version, v4)] Title: On the Springer correspondence for wreath products Authors: You-Hung Hsu , Chun-Ju Lai View a PDF of the paper titled On the Springer correspondence for wreath products, by You-Hung Hsu and Chun-Ju Lai View PDF HTML (experimental) Abstract: We establish a Bruhat decomposition indexed by the wreath product $\Sigma_m\wr \Sigma_d$ between two symmetric groups -- note that $\Sigma_m\wr \Sigma_d$ is not a Coxeter group in general. We show that such a decomposition affords a geometric variant in terms of the Bialynicki-Birula decomposition for varieties with $\mathbb{C}^*$-actions. Next, we construct a Steinberg variety whose top Borel-Moore homology realizes the group algebra $\mathbb{Q}[\Sigma_m\wr \Sigma_d]$ as a proper subalgebra. Such a geometric realization leads to a Springer-type correspondence which identifies the irreducible representations of $\Sigma_m\wr \Sigma_d$ with isotypic components of certain unconventional Springer fibers using type A geometry. In other words, we obtain a geometric counterpart of the (algebraic) Clifford theory, for the first time. Consequently, we obtain a new Springer correspondence of Weyl groups of type B/C/D using essentially type A geometry. Comments: 26 pages. v4: to appear in JPAA. Expositions improved. v3: Stronger results are proved on the (classification of) simple modules to address and correct a flaw identified in the previous version Subjects: Representation Theory (math.RT) ; Algebraic Geometry (math.AG) Cite as: arXiv:2404.02846 [math.RT] (or arXiv:2404.02846v4 [math.RT] for this version) https://doi.org/10.48550/arXiv.2404.02846 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Chun-Ju Lai [ view email ] [v1] Wed, 3 Apr 2024 16:26:01 UTC (49 KB) [v2] Thu, 18 Apr 2024 08:03:58 UTC (51 KB) [v3] Mon, 11 Nov 2024 09:43:42 UTC (54 KB) [v4] Thu, 30 Apr 2026 01:39:18 UTC (56 KB) Full-text links: Access Paper: View a PDF of the paper titled On the Springer correspondence for wreath products, by You-Hung Hsu and Chun-Ju Lai View PDF HTML (experimental) TeX Source view license Current browse context: math.RT < prev | next > new | recent | 2024-04 Change to browse by: math math.AG References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from