{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/towards-landau-ginzburg-models-for","title":"Towards Landau-Ginzburg models for cominuscule spaces via the exceptional cominuscule family","arxiv_id":"2204.03548","date":"2022-04-07","proceeding":null,"authors":["Peter Spacek","Charles Wang"],"abstract":"We present projective Landau-Ginzburg models for the exceptional cominuscule homogeneous spaces $\\mathbb{OP}^2 = E_6^\\mathrm{sc}/P_6$ and $E_7^\\mathrm{sc}/P_7$, known respectively as the Cayley plane and the Freudenthal variety. These models are defined on the complement $X^\\vee_\\mathrm{can}$ of an anti-canonical divisor of the \"Langlands dual homogeneous spaces\" $\\mathbb{X}^\\vee = P^\\vee\\backslash G^\\vee$ in terms of generalized Pl\\\"ucker coordinates, analogous to the canonical models defined for Grassmannians, quadrics and Lagrangian Grassmannians in arXiv:1307.1085, arXiv:1404.4844, arXiv:1304.4958. We prove that these models for the exceptional family are isomorphic to the Lie-theoretic mirror models defined in arXiv:math/0511124 using a restriction to an algebraic torus, also known as the Lusztig torus, as proven in arXiv:1912.09122. We also give a cluster structure on $\\mathbb{C}[\\mathbb{X}^\\vee]$, prove that the Pl\\\"ucker coordinates form a Khovanskii basis for a valuation defined using the Lusztig torus, and compute the Newton-Okounkov body associated to this valuation. Although we present our methods for the exceptional types, they generalize immediately to the members of other cominuscule families.","url_abs":"https://arxiv.org/abs/2204.03548v2","url_pdf":"https://arxiv.org/pdf/2204.03548v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"towards-landau-ginzburg-models-for","repo_url":"https://github.com/foxflo/exceptionalcominusculespaces","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}