{"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/on-reductive-subgroups-of-reductive-groups","title":"On reductive subgroups of reductive groups having invariants in almost all representations","arxiv_id":"2110.11066","date":"2021-10-21","proceeding":null,"authors":["Valdemar Tsanov","Yana Staneva"],"abstract":"Let $G$ and $\\tilde G$ be connected complex reductive Lie groups, $G$ semisimple. Let $\\Lambda^+$ be the monoid of dominant weights for a positive root system $\\Delta^+$, and let $l(w)$ be the length of a Weyl group element $w$. Let $V_\\lambda$ denote an irreducible $G$-module of highest weight $\\lambda\\in\\Lambda^+$. For any closed embedding $\\iota:\\tilde G\\subset G$, we consider Property (A): $\\quad\\forall\\lambda\\in\\Lambda^+,\\exists q\\in\\mathbb{N}$ such that $V_{q\\lambda}^{\\tilde G}\\ne0$. A necessary condition for (A) is for $G$ to have no simple factors to which $G$ projects surjectively. We show that this condition is sufficient if $\\tilde G$ is of type ${\\bf A}_1$ or ${\\bf E}_8$. We define and study an integral invariant of a root system, $\\ell_G=\\min\\{\\ell^\\lambda:\\lambda\\in\\Lambda^+\\setminus\\{0\\}\\}$, where $\\ell^\\lambda=\\min\\{l(w):w\\lambda\\notin{\\rm Cone}(\\Delta^+)\\}$. We derive the following sufficient condition for (A), independent of $\\iota$: $$ \\ell_G - \\#\\tilde\\Delta^+ > 0 \\;\\Longrightarrow\\; (A). $$ We compute $\\ell_G$ and related data for all simple $G$, except ${\\bf E}_8$, where we obtain lower and upper bounds. We consider a stronger property (A-$k$) defined in terms of Geometric Invariant Theory, related to extreme values of codimensions of unstable loci, and derive a sufficient condition in the form $\\ell_G - \\#\\tilde\\Delta^+ > k$. The invariant $\\ell_G$ proves too week to handle $G=SL_n$ and we employ a companion $\\ell_G^{\\rm sd}$ to infer (A-$k$) for a larger class of subgroups. We derive corollaries on Mori-theoretic properties of GIT-quotients.","url_abs":"https://arxiv.org/abs/2110.11066v1","url_pdf":"https://arxiv.org/pdf/2110.11066v1.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":"on-reductive-subgroups-of-reductive-groups","repo_url":"https://github.com/yanastaneva8/codimensions-unstable-loci","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}