Leaving the Hall: explicit formulas for Neguţ operators

[2608.14836] Leaving the Hall: explicit formulas for Negut operators Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2608.14836 (math) [Submitted on 14 Aug 2026 ( v1 ), last revised 16 Sep 2026 (this version, v2)] Title: Leaving the Hall: explicit formulas for Negut operators Authors: Michele D'Adderio , Giovanni Interdonato , Alessandro Iraci , Roberto Pagaria View a PDF of the paper titled Leaving the Hall: explicit formulas for Negut operators, by Michele D'Adderio and 3 other authors View PDF HTML (experimental) Abstract: Recent major breakthroughs in $q,t$-combinatorics include the introduction of the Dyck path algebra $\mathbb{A}_{q,t}$ by Carlsson and Mellit and of the Catalanimals by Blasiak et al., both of which led, among other things, to independent proofs of different extensions of the rational shuffle conjecture of Bergeron et al. The first main contribution of this paper is a simple, explicit formula inside the algebra $\mathbb{A}_{q,t}$ for the Negut operators, yielding a direct, elementary connection between the original operators of the rational shuffle conjecture and the corresponding Catalanimals. Our formula bypasses the elliptic Hall algebra, turning these operators into transparent, workable tools whose action we can compute exactly and efficiently on any symmetric function, not just constants. Our second main contribution consists of a series of explicit formulas relating the Negut operators to the Theta operators introduced by D'Adderio et al. To prove these formulas, we provide an extension of the aforementioned Theta operators to the entire algebra $\mathbb{A}_{q,t}$, allowing us to obtain a series of new combinatorial results. The algebraic computations underlying this extension have been formalized in Lean. To showcase the power of our results, we give a proof, also partially formalized in Lean, of the Theta conjecture of D'Adderio et al., first stated in 2019. Subjects: Combinatorics (math.CO) MSC classes: 05E05, 05A19, 05E10 Cite as: arXiv:2608.14836 [math.CO] (or arXiv:2608.14836v2 [math.CO] for this version) https://doi.org/10.48550/arXiv.2608.14836 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Alessandro Iraci [ view email ] [v1] Fri, 14 Aug 2026 19:20:42 UTC (42 KB) [v2] Wed, 16 Sep 2026 09:32:19 UTC (45 KB) Full-text links: Access Paper: View a PDF of the paper titled Leaving the Hall: explicit formulas for Negut operators, by Michele D'Adderio and 3 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2026-08 Change to browse by: math 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

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