Minimal Faithful Representations of Split Extensions by Abelian Groups

[2508.00687] Minimal Faithful Representations of Split Extensions by Abelian Groups Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Representation Theory arXiv:2508.00687 (math) [Submitted on 1 Aug 2025 ( v1 ), last revised 14 Aug 2026 (this version, v2)] Title: Minimal Faithful Representations of Split Extensions by Abelian Groups Authors: Charles Daly , Justin Kingsnorth View a PDF of the paper titled Minimal Faithful Representations of Split Extensions by Abelian Groups, by Charles Daly and Justin Kingsnorth View PDF HTML (experimental) Abstract: Every finite group admits a representation which is 'most efficient' in the sense that the representation is faithful and the dimension is minimal. We call such representations minimal faithful representations. The image of this representation retains all the information about the abstract group and can come with some additional geometric structure. For example, a minimal faithful representation could preserve a symplectic form or act on a fixed basis by permutations. It is the purpose of this paper to address these ideas for finite semi-direct products of groups split by an abelian group whose splitting homomorphism is faithful. We calculate the minimal faithful dimensions of such representations over both the complex and real numbers, and we express them in terms of group invariants to argue that in some sense these groups are extensions of tori. In the real case, the dimension of a minimal faithful representation and the structure it preserves encode the possible types of orbifold singularities of the group in question. We apply our framework to some families of groups to calculate their minimal faithful dimensions and analyze some of their orbifold singularities. Comments: 24 pages, 3 figures. Significant changes, correction to Proposition 4.3 Subjects: Representation Theory (math.RT) MSC classes: 20C15, 20C30 Cite as: arXiv:2508.00687 [math.RT] (or arXiv:2508.00687v2 [math.RT] for this version) https://doi.org/10.48550/arXiv.2508.00687 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Justin Kingsnorth [ view email ] [v1] Fri, 1 Aug 2025 15:03:14 UTC (443 KB) [v2] Fri, 14 Aug 2026 16:20:49 UTC (170 KB) Full-text links: Access Paper: View a PDF of the paper titled Minimal Faithful Representations of Split Extensions by Abelian Groups, by Charles Daly and Justin Kingsnorth View PDF HTML (experimental) TeX Source view license Current browse context: math.RT < prev | next > new | recent | 2025-08 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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