On rational representations and rational group algebra of $\operatorname{GL}_2(q)$

[2606.02415] On rational representations and rational group algebra of $\operatorname{GL}_2(q)$ Skip to main content Search Submit Donate Log in Mathematics > Representation Theory arXiv:2606.02415 (math) [Submitted on 1 Jun 2026 ( v1 ), last revised 22 Aug 2026 (this version, v2)] Title: On rational representations and rational group algebra of $\operatorname{GL}_2(q)$ Authors: Ram Karan Choudhary , Sunil Kumar Prajapati View PDF HTML (experimental) Abstract: In this article, we study rational representations of $G=\operatorname{GL}_2(q)$, where $q$ is a prime power. Let $\rho$ be an irreducible representation of $G$ over $\mathbb{Q}$. Then $\rho$ affords the character \[ \Omega(\chi)=m_{\mathbb{Q}}(\chi)\sum_{\sigma\in\operatorname{Gal}(\mathbb{Q}(\chi)/\mathbb{Q})}\chi^{\sigma}, \] for some irreducible complex character $\chi$ of $G$, where $m_{\mathbb{Q}}(\chi)$ denotes the Schur index of $\chi$ over $\mathbb{Q}$, and conversely, every character of this form is afforded by an irreducible representation of $G$ over $\mathbb{Q}$. We obtain a combinatorial description for the counting of inequivalent irreducible $\mathbb{Q}$-representations of $G$ of each distinct degree. Furthermore, we briefly determine the rational character table of $G$ and present a method for constructing an irreducible rational matrix representation $\rho$ of $G$ affording the character $\Omega(\chi)$, where $\chi$ is an irreducible complex character of $G$ arising from parabolic induction. Finally, using the results on the rational representations of $G$, we derive an explicit combinatorial formula, depending only on $q$, for the Wedderburn decomposition of $\mathbb{Q}G$. Subjects: Representation Theory (math.RT) ; Combinatorics (math.CO); Group Theory (math.GR); Rings and Algebras (math.RA) MSC classes: [2020]{primary 20C15, secondary 20C33, 20C05} Cite as: arXiv:2606.02415 [math.RT] (or arXiv:2606.02415v2 [math.RT] for this version) https://doi.org/10.48550/arXiv.2606.02415 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ram Karan Choudhary [ view email ] [v1] Mon, 1 Jun 2026 15:57:38 UTC (27 KB) [v2] Sat, 22 Aug 2026 09:07:30 UTC (27 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.RT < prev | next > new | recent | 2026-06 Change to browse by: math math.CO math.GR math.RA References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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)

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