[1801.09033] Affine Schubert calculus and double coinvariants Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:1801.09033 (math) [Submitted on 27 Jan 2018 ( v1 ), last revised 15 May 2026 (this version, v6)] Title: Affine Schubert calculus and double coinvariants Authors: Erik Carlsson , Alexei Oblomkov View a PDF of the paper titled Affine Schubert calculus and double coinvariants, by Erik Carlsson and 1 other authors View PDF HTML (experimental) Abstract: We define an action of the double coinvariant algebra $DR_n$ on the equivariant Borel-Moore homology of the affine flag variety $\widetilde{Fl}_n$ in type $A$, which has an explicit form in terms of the left and right action of the (extended) affine Weyl group and multiplication by Chern classes. Up to first order in the augmentation ideal, we show that it coincides with the action of the Cherednik algebra on the equivariant homology of the homogeneous affine Springer fiber $\widetilde{S}_{n,n+1} \subset \widetilde{Fl}_n$ due to Yun and the second author, and therefore preserves the non-equivariant Borel-Moore homology groups $H_*(\widetilde{S}_{n,n+1})\hookrightarrow H_*(\widetilde{Fl}_n)$. We then define a geometric filtration $F_{a} H_*(\widetilde{S}_{n,n+1})=H_*(\widetilde{S}(a))$ by closed subspaces $\widetilde{S}(a)\subset \widetilde{S}_{n,n+1}$, which we prove recovers the Garsia-Stanton descent order on $DR_n$. We use this to deduce an explicit monomial basis of $DR_n$, as well as an independent proof of the (non-compositional) Shuffle Theorem. Comments: 71 pages, correccted typos, improved exposition, explained and used previously-known results by Iarrobino and Goettsche on the geometry of the spaces that we use; some typos are corrected Subjects: Combinatorics (math.CO) ; Algebraic Geometry (math.AG) Cite as: arXiv:1801.09033 [math.CO] (or arXiv:1801.09033v6 [math.CO] for this version) https://doi.org/10.48550/arXiv.1801.09033 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Alexei Oblomkov [ view email ] [v1] Sat, 27 Jan 2018 04:44:01 UTC (32 KB) [v2] Tue, 2 Apr 2019 19:35:38 UTC (36 KB) [v3] Sat, 29 Jun 2019 15:30:17 UTC (36 KB) [v4] Sun, 16 Feb 2025 00:01:13 UTC (59 KB) [v5] Fri, 24 Apr 2026 02:59:55 UTC (80 KB) [v6] Fri, 15 May 2026 14:58:29 UTC (80 KB) Full-text links: Access Paper: View a PDF of the paper titled Affine Schubert calculus and double coinvariants, by Erik Carlsson and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2018-01 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